tag:blogger.com,1999:blog-25698926063152031172024-03-13T15:18:11.026-04:00Essays & Endnotesstephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comBlogger56125tag:blogger.com,1999:blog-2569892606315203117.post-55790104853004925922018-06-13T18:23:00.000-04:002018-06-13T18:23:23.923-04:00On the Value of Music Analysis<div style="text-align: center;">
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. . . . if you think about it, you'll miss it . . . . </div>
<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com0tag:blogger.com,1999:blog-2569892606315203117.post-41798179136959702942017-06-11T07:09:00.000-04:002017-06-17T10:09:23.926-04:00Motivations<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
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<i style="mso-bidi-font-style: normal;"><span style="font-size: 10.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 8.0pt;">I teach composition class at the
Conservatoire where, for the past forty years, I've spent my time decorticating
musical works, trying to figure out what happe</span></i><i style="mso-bidi-font-style: normal;"><span style="font-size: 10.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 12.0pt;">ns in them.</span></i><i style="mso-bidi-font-style: normal;"><span style="font-size: 10.0pt; mso-bidi-font-size: 8.0pt;"> <o:p></o:p></span></i></div>
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<span style="font-size: 10.0pt; mso-bidi-font-family: Times; mso-bidi-font-size: 16.5pt;">– </span><span style="font-size: 10.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 12.0pt;">Olivier Messiaen</span><span class="MsoEndnoteReference"><span style="font-size: 10.0pt; mso-bidi-font-family: Times; mso-bidi-font-size: 16.5pt;"> <span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[i]<!--[endif]--></span></span></span><span style="font-size: 10.0pt; mso-bidi-font-family: Times; mso-bidi-font-size: 16.5pt;"><o:p></o:p></span></div>
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<i style="mso-bidi-font-style: normal;"><span style="color: black; font-size: 10.0pt;">The idea of the series was engaging [Messiaen's]
maximum attention during these years, and it was probably the influence of this
fact that caused him to reflect on the possible strict, and strictly
calculated, relationships on which his music might depend; there are many
instances in these works of a clear conflict between spontaneity and
organization, the one unwilling to abdicate and the other determined to become
all powerful. This conflict, or antinomy, is reflected even in the titles of
the different pieces written between 1949 and 1951 – </span></i><span style="color: black; font-size: 10.0pt;">Les
Yeux dans les roues, Les Mains de l'abíme, Ile de feu<i style="mso-bidi-font-style: normal;">.<o:p></o:p></i></span></div>
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<span style="color: black; font-size: 10.0pt;">–
Pierre Boulez<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[ii]<!--[endif]--></span></span><o:p></o:p></span></div>
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<i style="mso-bidi-font-style: normal;"><span style="font-size: 10.0pt; mso-bidi-font-family: Times; mso-bidi-font-size: 16.0pt;">If nature were all lawfulness then every phenomenon would share the full
symmetry of the universal laws of nature .... The mere fact that this is not so
proves that contingency is an essential feature of the world.<o:p></o:p></span></i></div>
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<span style="font-size: 10.0pt; mso-bidi-font-family: Times; mso-bidi-font-size: 16.0pt;">–
Herman Weyl<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[iii]<!--[endif]--></span></span><o:p></o:p></span></div>
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<i style="mso-bidi-font-style: normal;"><span style="font-size: 10.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 16.0pt;">The main fallacy [of] the reductionist hypothesis [is that
it] does not by any means imply a “constructionist” one: The ability to reduce
everything to simple fundamental laws does not imply the ability to start from
those laws and reconstruct the universe. . . . The constructionist hypothesis
breaks down when confronted with the twin difficulties of scale and complexity.
. . . [A]t each level of complexity entirely new properties appear. . . . [T]he
whole becomes not only more than the sum of but very different from the sum of
the parts. . . . </span></i><i style="mso-bidi-font-style: normal;"><span style="font-size: 10.0pt; mso-bidi-font-family: Times; mso-bidi-font-size: 16.0pt;">[T]he
new symmetry – now called broken symmetry because the original symmetry is
no longer evident – may be an entirely unexpected kind and extremely
difficult to visualize. ... [T]he whole becomes not only more than but very
different from the sum of its parts. . . . <span style="mso-bidi-font-style: italic;">At some point we have to stop talking about decreasing symmetry and
start calling it increasing complication</span>. </span></i><i style="mso-bidi-font-style: normal;"><span style="font-size: 10.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 16.0pt;"><o:p></o:p></span></i></div>
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<span style="font-size: 10.0pt; mso-bidi-font-family: Times; mso-bidi-font-size: 16.0pt;">–
P.W. Anderson<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[iv]<!--[endif]--></span></span><o:p></o:p></span></div>
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<i style="mso-bidi-font-style: normal;"><span style="color: #141414; font-size: 10.0pt;">A symmetry can be exact,
approximate, or broken. Exact means unconditionally valid; approximate means
valid under certain conditions; broken can mean different things, depending on
the object considered and its context. . . . Generally, the breaking of a
certain symmetry does not imply that no symmetry is present, but rather that
the situation where this symmetry is broken is characterized by a lower
symmetry than the original one.<o:p></o:p></span></i></div>
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<span style="color: #141414; font-size: 10.0pt;">– Stanford Encyclopedia of Philosophy<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[v]<!--[endif]--></span></span><o:p></o:p></span></div>
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<i style="mso-bidi-font-style: normal;"><span style="color: #141414; font-size: 10.0pt;">Asymmetry is what creates a phenomenon.<o:p></o:p></span></i></div>
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<span style="color: #141414; font-size: 10.0pt;">– Pierre Curie<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[vi]<!--[endif]--></span></span><o:p></o:p></span></div>
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<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[i]<!--[endif]--></span></span> Messiaen, O.
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<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[ii]<!--[endif]--></span></span> <span style="color: black; mso-bidi-font-size: 13.5pt;">Pierre Boulez. 'Olivier Messiaen'
('Une classe et ses chimères', tribute to Messiaen on his fiftieth birthday
from the programme for the Domaine musical concert of 15 April 1959. Tr. by
Martin Cooper, 1986.) In</span><span style="color: black;"> </span><i style="mso-bidi-font-style: normal;"><span style="color: black; mso-bidi-font-size: 10.0pt;">Orientations: collected writings</span></i><span style="color: black; mso-bidi-font-size: 13.5pt;">. Faber & Faber, 1990. p.414.</span><span style="mso-bidi-font-size: 10.0pt;"><o:p></o:p></span></div>
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<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[iii]<!--[endif]--></span></span> Hermann
Weyl. <i style="mso-bidi-font-style: normal;">Symmetry</i>. Princeton UP, 1983.
p.26</div>
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<span class="MsoEndnoteReference"><span style="font-family: "cambria"; font-size: 12.0pt;"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[iv]<!--[endif]--></span></span></span><span style="font-family: "cambria"; font-size: 12.0pt;"> </span><span style="font-family: "cambria"; font-size: 12.0pt;">P.W. Anderson. 'More Is Different: Broken Symmetry and the Nature
of the Hierarchical Structure of Science'. <i>Science</i>, New Series,
Vol. 177, No. 4047. (Aug. 4, 1972), pp. 393-396. </span><a href="https://emergentpublications.com/ECO/ECO_other/Issue_16_3_7_CP.pdf?AspxAutoDetectCookieSupport=1"><span style="font-family: "cambria"; font-size: 12.0pt;">Republished in </span><i><span style="font-family: "cambria"; font-size: 12.0pt;">E:CO </span></i><span style="font-family: "cambria"; font-size: 12.0pt;">2014 16(3): 117-134</span></a><span style="font-family: "cambria"; font-size: 12.0pt;"> with an introduction</span><span style="font-family: "cambria"; font-size: 12.0pt;"> by J</span><span style="font-family: "cambria"; font-size: 12.0pt;">effrey A. Goldstein</span><span style="font-family: "cambria"; font-size: 12.0pt;">, '</span><span style="font-family: "cambria"; font-size: 12.0pt;">Reduction,
construction, and emergence in P. W. Anderson’s "More is different"'</span><span style="font-family: "cambria"; font-size: 12.0pt;"> available on-line at
https://emergentpublications.com/ECO/ECO_other/Issue_16_3_7_CP.pdf?AspxAutoDetectCookieSupport=1
(accessed 30.11.16)<o:p></o:p></span></div>
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<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[v]<!--[endif]--></span></span> 'Symmetry
and Symmetry Breaking'. <i style="mso-bidi-font-style: normal;">Stanford
Encyclopedia of Philosophy.</i>
http://plato.stanford.edu/entries/symmetry-breaking/#4 (accessed 30.11.16)</div>
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<span class="MsoEndnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]-->[vi]<!--[endif]--></span></span> Pierre
Curie. <span style="background: white; mso-bidi-font-size: 9.5pt;">'Sur la symétrie
dans les phénomènes physiques.' 1894.</span><span style="mso-bidi-font-size: 10.0pt;"><o:p></o:p></span></div>
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-55293385171118811742017-06-09T10:10:00.000-04:002017-06-09T10:26:39.207-04:00Return of Broken SymmetriesI am reposting below the series of entries on Messiaen's <i>Île de feu II</i> with no changes to the originals posted April 2015–May 2016. These were done as a kind of experiment in 'stream-of-consciousness analysis', following my nose to what caught my interest in a single score each time I returned to it, and publishing a few selectively. Much material never made it into the blog. What I ended up with resembles random pages from a laboratory notebook with observations only partially organized. Still, disorganized as they are, I am in the process of turning some of the material into a more finished form, and there may be some things here that may also trigger ideas from readers.<br />
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I am currently working up several short papers that connect <i>Île de feu I </i>and<i> </i><i>Île de feu II.</i> The first of these, an expansion of the blog post 'Broken Symmetries 3.1', is a comparative analysis of rhythmic structures based on a theory of duration string contours. It will be finished soon, and I will post a link to it in a separate post.<br />
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<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-42848371874272267132017-06-09T10:09:00.000-04:002018-10-12T10:38:00.280-04:00Broken Symmetries 1<div style="text-align: center;">
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[T]he new symmetry – now called broken symmetry because the original symmetry is no longer evident – may be an entirely unexpected kind and extremely difficult to visualize. ... [T]he whole becomes not only more than but very different from the sum of its parts. ... <i>At some point we have to stop talking about decreasing symmetry and start calling it increasing complication</i>. <span style="text-align: center;">–P.W. Anderson </span><span style="font-size: x-small; text-align: center;"><b>[1]</b></span> </blockquote>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEuIN3bI9_7HS2eBp3xBPrysxyuSeAfMc4581RTEne7ypbO_buRL6jXHAx5wjCyaN4SMIiCMEb6u-WjnDv0hf7xzIHXz1BJr-oXyxbJG9ZCUp6Wt59a6W2THRq7YCcKz14bHs_psAciNg/s1600/IDF+diagram+C1+C2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEuIN3bI9_7HS2eBp3xBPrysxyuSeAfMc4581RTEne7ypbO_buRL6jXHAx5wjCyaN4SMIiCMEb6u-WjnDv0hf7xzIHXz1BJr-oXyxbJG9ZCUp6Wt59a6W2THRq7YCcKz14bHs_psAciNg/s640/IDF+diagram+C1+C2.png" style="cursor: move;" width="640" /></a></td></tr>
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<tr><td class="tr-caption" style="font-size: 13px;">Diagram 1<br />
Compositional scheme of <i>Île de feu 2</i><br />
<i>(Timings refer to the Loriod recording)</i><br />
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Olivier Messiaen</div>
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<a href="http://www.worldcat.org/title/ile-de-feu-ii-no-4-de-quatre-etudes-de-rythme-pour-piano/oclc/56577632&referer=brief_results">Île de feu II : no 4 de Quatre études de rythme : pour piano</a></div>
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Performed by Yvonne Loriod</div>
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<span style="font-size: large;">Descriptive Analysis (C1)</span></div>
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Quite a bit of analytical ink has been spilled over the spiral (a.k.a. fan or wedge) "interversions" that Olivier Messiaen himself called attention to in the score of <i>Île de feu 2</i> (sections B1, B2 and A4+B3 in Diagram 1).<span style="font-size: x-small;"> </span>Less analytical effort has been spent on the unmarked interversions in a six-bar passage in the same piece, mm. 70-75 (C1 in Diagram 1) shown in Example 1 below, and less yet on the analytically problematic 40-bar passage mm. 92–131 (C2). C1 is relatively easy to "count notes" on, but hides a delicious compositional dilemma. C2, the problem passage, at first appears to have nothing to do with the interversion process, but it demonstrates so much mirror symmetry that it's difficult to dismiss it into the analyst's last resort, the through-composed bin. But before beginning a discussion of both of these still-open questions, I must make a brief comment about Messiaen's nomenclature.<br />
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Evidently Messiaen picked up the term "interversion" from Rudolph Reti who conceived it loosely as a (any?) reordering of pitches/pitch classes in some significant "cell" identified during the <i>analytical</i> process, the cell usually being a smaller motivic subset of the total chromatic/diatonic. However, Messiaen's <i>compositional</i> use focusses Reti's fuzzier analytical tool as a fairly well defined play with the basic mathematical idea of group action. For now I'll try to stick with Messiaen's use of the term "interversion" as the result of applying some permutation to some set. In the examples below, we'll take this set to be the 12-tone chromatic scale.<br />
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Applying a permutation function <i><b>p</b></i> once to the chromatic scale yields the first interversion, also referred to here as the <i>seed row</i>. Applying <i><b>p</b></i> again to the result (the seed row) yields the second interversion, etc., until inevitably the seed row is returned by repeated actions of <i><b>p</b></i>. We'll pick up other information as we go along, but this is enough to start with.</div>
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In Example 1 I've assigned the usual integers to each note in the right and left hands in m.70 (C=0, C#=1, ..., B=11).</div>
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<tr><td class="tr-caption" style="text-align: center;">Example 1.<br />
<b><span style="font-size: x-small;">Olivier Messiaen,</span></b><br />
<b><span style="font-size: x-small;"><i style="text-align: left;">Île de feu 2</i><span style="text-align: left;">,</span></span></b><br />
<b><span style="font-size: x-small; text-align: left;">mm.70–75</span></b><br />
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If we were to place the integers 0 through 11 above the row in the RH of m.70 we would get a "generation" of that row as a permutation of the chromatic scale which I'll label <b><i>f</i><span style="font-size: x-small;">1</span></b>. In simple 2-line notation <b><i>f</i><span style="font-size: x-small;">1</span></b>=</div>
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Applying this permutation again to the result, we get the LH of m.70: 8-0-1-9-10-3-4-11-7-6-5-2. Applied again, the result is the RH of m.71: 3-5-10-11-0-7-1-6-9-2-8-4. And so on. It's helpful to express this permutation in the alternative cycle notation also, which reads:<br />
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<b style="text-align: left;"><i>f</i><span style="font-size: x-small;">1</span></b> = ( 0 5 8 3 7 9 11 6 2 4 1 10 )<br />
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In either representation, pc0→pc5, 5→8, 8→3, ..., 1→10, and around the corner, pc10→pc0.</div>
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Since the length of the cycle is 12, after 12 actions of the permutation on the seed row [5<span style="text-align: center;">, </span>10<span style="text-align: center;">,</span><span style="text-align: center;"> </span>4<span style="text-align: center;">,</span><span style="text-align: center;"> </span>7<span style="text-align: center;">,</span><span style="text-align: center;"> </span>1<span style="text-align: center;">,</span><span style="text-align: center;"> </span>8<span style="text-align: center;">,</span><span style="text-align: center;"> </span>2<span style="text-align: center;">,</span><span style="text-align: center;"> </span>9<span style="text-align: center;">,</span><span style="text-align: center;"> </span>3<span style="text-align: center;">,</span><span style="text-align: center;"> </span>11<span style="text-align: center;">,</span><span style="text-align: center;"> </span>0<span style="text-align: center;">,</span><span style="text-align: center;"> </span>6] we know we will return to a restatement of the seed row; Messiaen ends the procedure after obtaining the ascending chromatic scale on the 11th repetition (Table 1) which leads to the octave C-natural in the next bar (not shown) which begins the next section (A4+B3).<br />
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<tr><td class="tr-caption" style="text-align: center;">Table 1.</td></tr>
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So far there is nothing different in what I have presented from what the reader can find in several other more sophisticated sources.<span style="font-size: x-small;">[2]</span> From a single measure (we also could have discovered <b><i>f</i><span style="font-size: x-small;">1</span></b> from RH →LH in m.70) we know that this is precisely how Messiaen composed-out this brief passage. So it is tempting to say that we have "solved" these six measures and simply stop here. But all we've done so far has amounted to no more than an exercise in counting notes.<br />
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So let's now ask: <i>Why</i> did Messiaen, lured so often by "the charm of impossibility," choose that particular seed row (permutation)? Is there anything special about this row? Did Messiaen pick the notes out of a hat? Or did his ear tell him it "just sounds right." Or did an angel dictate it to him? Are there any patterns here that might suggest this row was not merely accidental, but a conscious, pragmatic choice?<br />
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Published analyses of these six measures that I have encountered so far agree that the seed row, [5<span style="text-align: center;">, </span>10<span style="text-align: center;">,</span><span style="text-align: center;"> </span>4<span style="text-align: center;">,</span><span style="text-align: center;"> </span>7<span style="text-align: center;">,</span><span style="text-align: center;"> </span>1<span style="text-align: center;">,</span><span style="text-align: center;"> </span>8<span style="text-align: center;">,</span><span style="text-align: center;"> </span>2<span style="text-align: center;">,</span><span style="text-align: center;"> </span>9<span style="text-align: center;">,</span><span style="text-align: center;"> </span>3<span style="text-align: center;">,</span><span style="text-align: center;"> </span>11<span style="text-align: center;">,</span><span style="text-align: center;"> </span>0<span style="text-align: center;">,</span><span style="text-align: center;"> </span>6] is arbitrary and unpatterned; or, like Messiaen himself, they make no statement at all about its structure or derivation.<br />
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In fact this row <i><u>is</u></i> patterned, and it would be difficult to believe that Messiaen did not consciously design this pattern. The seed row is generated from a single trichord, and that seed row, disappearing from the surface after its appearance in m.70, will reappear (transposed) in the midst of the "recalcitrant" passage, mm. 92–131 (C2 in Diagram 1).<br />
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The trichord generator built into the permutation function <b><i>f</i><span style="font-size: x-small;">1</span></b> is SC-016 (Forte 3-5 if you still insist). The resulting row <i style="font-weight: bold;">S</i> = [5<span style="text-align: center;">, </span>10<span style="text-align: center;">,</span><span style="text-align: center;"> </span>4<span style="text-align: center;">,</span><span style="text-align: center;"> </span>7<span style="text-align: center;">,</span><span style="text-align: center;"> </span>1<span style="text-align: center;">,</span><span style="text-align: center;"> </span>8<span style="text-align: center;">,</span><span style="text-align: center;"> </span>2<span style="text-align: center;">,</span><span style="text-align: center;"> </span>9<span style="text-align: center;">,</span><span style="text-align: center;"> </span>3<span style="text-align: center;">,</span><span style="text-align: center;"> </span>11<span style="text-align: center;">,</span><span style="text-align: center;"> </span>0<span style="text-align: center;">,</span><span style="text-align: center;"> </span>6] can be partitioned and the constituent (internally <i>unordered</i>) subsets labelled <b>A</b>, <b>B</b>, <b>C</b>, <b>D</b>:</div>
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{4,5,10} <b>:</b> {7,8,1} <b>::</b> {9,2,3} <b>:</b> {6,11,0}</div>
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<b> A : B :: C : D</b></div>
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Note that the left and right hexachords and their constituent SC-016 trichords are mirror related, producing a nice set-class symmetry:</div>
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</b> <b>A</b>∪<b>B</b> ← I<span style="font-size: x-small;">7</span> → <b>C</b>∪<b>D</b></div>
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<b>A</b> ← I<span style="font-size: x-small;">4</span> → <b>D</b></div>
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<b>B</b> ← I<span style="font-size: x-small;">10</span> → <b>C</b><br />
<b>A</b>∪<b>B</b> = {4,5,7,8,10,1} = <b>O1</b>–{11,2}<br />
<b>C</b>∪<b>D</b> = {9,11,0,2,3,6} = <b>O2</b>–{5,8}<br />
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where <b style="text-align: center;">O1</b><span style="text-align: center;"> = {1,2,4,5,7,8,10,11} and </span> <b style="text-align: center;">O2</b><span style="text-align: center;"> = {2,3,5,6,8,9,11,0}, two of the three possible transpositions of Messiaen's second mode of limited transposition, commonly known as the octatonic scale. The dyad "remainders," {11,2}⊂(</span><b style="text-align: center;">C</b><span style="text-align: center;">∪</span><b style="text-align: center;">D</b><span style="text-align: center;">) and {5,8}⊂(</span><b style="text-align: center;">A</b><span style="text-align: center;">∪</span><b style="text-align: center;">B</b><span style="text-align: center;">), together making up a diminished-seventh chord and forming the complement of the third transposition of the octatonic, could possibly be heard as spanning-vector (IFUNC) connectors between the two hexachords that further emphasize the seed row's structural saturation with ics 1, 5, and 6.<span style="font-size: x-small;">[4]</span></span><span style="text-align: center;"> </span><br />
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<span style="text-align: center;">But all of this <i>set-class</i> symmetry<i> for constructing the seed row</i> <i style="font-weight: bold;">S</i> is broken as soon as the seed row is permuted, i.e., as soon as </span><b><i>f</i><span style="font-size: x-small;">1</span> </b><span style="text-align: center;">is applied to the seed row, generating the next row </span><i style="font-weight: bold; text-align: center;">S</i><span style="text-align: center;"><b style="font-style: italic;">'</b> = </span><i style="font-weight: bold;">f</i><b style="font-size: small;">1</b>(<i style="font-weight: bold;">S</i><span style="text-align: center;">) = [8, 0</span><span style="text-align: center;">, </span><span style="text-align: center;">1</span><span style="text-align: center;">, </span><span style="text-align: center;">9</span><span style="text-align: center;">, </span><span style="text-align: center;">10</span><span style="text-align: center;">, </span><span style="text-align: center;">3</span><span style="text-align: center;">, </span><span style="text-align: center;">4</span><span style="text-align: center;">, </span><span style="text-align: center;">11</span><span style="text-align: center;">, </span><span style="text-align: center;">7</span><span style="text-align: center;">, </span><span style="text-align: center;">6</span><span style="text-align: center;">, </span><span style="text-align: center;">5</span><span style="text-align: center;">, </span><span style="text-align: center;">2]. Parsing by consecutive trichords again, this time for <b><i>S', </i></b></span><span style="text-align: center;">we get [{8,0,1}, {9,10,3}, {4,7,11}, {2,5,6}] with the set-class string [015, 016, 037, 014]. In fact, each interversion yields a different string of set classes until the 12th returns to the seed row (which doesn't occur in the music).</span><br />
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<span style="font-size: large;">Fictive Analysis (C1)</span></div>
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<span style="text-align: center;"></span> <span style="text-align: center;">By choosing to use a generalized interversion technique, Messiaen certainly understood that, outside of the <i>T</i>, <i>I</i> and <i>R</i> mappings of "standard" 12-tone technique, set-class invariance would nearly always be lost. However, as we shall see, other relationships, whether we wish to think of them technically as symmetries or not, will be "counterpoint invariant" under interversion. A broken symmetry can produce or reveal new symmetry and increasing complexity.</span><br />
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</span><span style="text-align: center;"><span style="text-align: left;">Next note that, prior to any permutation, the <u>potential</u> </span><i style="text-align: left;">interval-string</i><span style="text-align: left;"> symmetry for Messiaen's set-class-symmetric seed row is unrealized, not due to the action of a function or transformation, but due either to a mistake (hardly likely) or to </span>Messiaen's conscious choice in ordering the seed row's pitch-class elements; that is to say, he avoided the symmetry on purpose in order to ...? When we go from considering the seed row as a symmetric string of set classes to an asymmetric string of pitch classes, the plot thickens.</span></div>
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In Example 2, the succession of the first three trichord interval strings <b><A></b>, <b><B></b>, <b><C></b> sets up an expectation for a <b><D></b> to "complete" the symmetry if <b><D></b> := <b><<i>S></i></b>, so the "ideal" run to symmetry – the row Messiaen did <i><u>not</u></i> choose – would be <b><A></b>, <b><B></b>, <b><C></b>, <b><</b><b style="font-style: italic;">S>.</b> (Angle brackets indicate interval strings: <b><A></b> = <+5,-6>, <b><B></b> = <-6,+7>, <b><C></b> = <+7,-6>, <b><</b><b style="font-style: italic;">S></b> ("symmetry completion") = <-6,+5>.)</div>
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The ordered interval strings within the row's trichords "<i>should</i>" be:<br />
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<+5,–6> <b>:</b> <–6,+7> <b>::</b> <+7,–6> <b>:</b> <–6,+5>.</div>
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Another way to spot all this is by comparing the position and direction of the arrows following each trichord's ascending minor second as shown in Example 2. Also note that I chose the octave placement of the pitches for <b><<i>S</i>></b> to keep Messiaen's dyadic relationships: ↑ ↓ ↑ ↓ .... But all of this potential pitch-order symmetry in our ideal seed row turns out to be the row not taken.<br />
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<b>. . . . . </b><span style="font-size: x-small;">[5]</span></div>
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Given that Messiaen chose to generate the seed row by partitioning the chromatic into consecutive SC-016 trichords, and given the internal pitch-class orderings he chose for the first three of those trichords, he certainly knew what the pc order of the <b><D></b> trichord <i>ought</i> to be <i>in order to</i> generate a row that retains the mirror symmetry of the corresponding harmonies. The question is: Why <i>didn't</i> he do it that way? Why did he rotate the final trichord in the seed row from the "obvious" [0,6,11] to [11,0,6], breaking the symmetry by setting <<b>D> </b>:= <span style="color: red;"><<i>M</i><i style="font-weight: bold;">></i></span>?<br />
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If (Condition 1:) Messiaen wanted to order the final trichord to make it "correct" (i.e., to attain a pc-symmetric seed row to reflect its harmonic symmetry), <i>and</i> (Condition 2:) he also wanted to head for the same final interversion of an ascending chromatic scale, then he would have been forced to use the permutation <b><i>f</i><span style="font-size: x-small;">2<i> </i></span></b>=<br />
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<b style="text-align: left;"><i>f<span style="font-size: x-small;">2</span></i></b> = ( 0 5 8 3 7 9 ) ( 1 10 6 2 4 ) <span style="color: red;"><b>( 11 )</b></span></div>
<br />
The asymmetrically-derived <i style="font-weight: bold;">f</i><b style="font-size: small;">1</b> actually chosen by Messiaen is a cyclic permutation of length 12 yielding 12 interversions counting the seed row. The symmetrically-derived <b><i>f</i><span style="font-size: x-small;">2</span><i> </i></b><span style="text-align: center;">has three disjoint cycles: a 6-cycle, a 5-cycle, and a fixed point. This would have led him to the following string of interversions which I've listed completely so the reader can immediately see the compositional situation he would have faced by starting from a pc-symmetric seed row (Table 2).</span><br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9xDHUU2FQpTmJ0kUZmNd9BcENiY49GlzWJ77Esgh45Mz_PtLuX8lTXSD4iFLPcIVpHIVU3XThwsEwC882PMIs8G-nNkMb2rtg56H6TqGalCtaKcIvx2bLabNjPLSkTqLk4v0zNktTsM4/s1600/ideal+perm+function.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9xDHUU2FQpTmJ0kUZmNd9BcENiY49GlzWJ77Esgh45Mz_PtLuX8lTXSD4iFLPcIVpHIVU3XThwsEwC882PMIs8G-nNkMb2rtg56H6TqGalCtaKcIvx2bLabNjPLSkTqLk4v0zNktTsM4/s640/ideal+perm+function.png" width="296" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Table 2</td></tr>
</tbody></table>
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<br />
So choosing <i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">2</span> for the sake of symmetry in the seed row would have committed him to dealing with 30 interversions<span style="font-size: x-small;">[6]</span>. But a tougher problem is that fixed point in <i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">2</span> – the 11 (B natural) which would have remained at the bottom of the 12-tone deck with every shuffle. If he were to use <i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">2</span> to make a seed row to use the same way he used the seed row he actually chose, (a) he either would have had to have a plan requiring or accommodating all 30 interversions or have planned to use only a portion of them <i>and</i> (b) he would have had a situation that required the same note to pop up at the end of every interversion. Instead of the six bars he wrote, he would have had 15 bars with a comical-bordering-on-annoying unison B sounding at the end of each bar. There certainly may be different situations where this would be musically possible, maybe even desirable, but it's hardly likely such a situation would be a passage of even triplets in two-voice "first species counterpoint."<br />
<br />
Ignoring the intentional fallacy (not a goal of fictive analysis but certainly a nice side benefit), we can take another tack, calling our next move "a likely scenario"; a.k.a., My Best Guess:<br />
<br />
Messiaen looked at the obvious symmetric ordering <i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">2</span>,<br />
immediately noticed the fixed point,<br />
scowled "This won't work for me here,"<br />
rotated the last trichord to turn <i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">2</span> into <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">1</span></span> (– aha! – a 12-cycle at that),<br />
and "Problem solved."<br />
He wrote it out,<br />
went over to the piano, and<br />
played mm. 70-75 right out.<br />
His ear was satisfied.<br />
<br />
Simpler explanations notwithstanding regarding the composer's actual cognitive process (and it's still only a guess, after all), what if we insist on investigating alternatives – whether Messiaen himself considered them or not. Messiaen now goes from composer to foil.<br />
<br />
Given conditions 1 & 2 as before, Messiaen still had four other options that would have kept the harmonic symmetry of the seed row. We can see this by looking at all six possible permutations of the <b>D</b> trichord. (See Table 3.)<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnROuZw1lgtyCth2PvtWyoflBqNtK40RhQ7OfZlWllExZ6rkS3-wg9zB7OWGONH833bgRShC87t9TUinZY-ZL1KB-QsTUHDvCqPNc3YfoPptBXkEpmQPJ0Dk_uwj-OQ6avK0_lJ5XD5M0/s1600/competition.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnROuZw1lgtyCth2PvtWyoflBqNtK40RhQ7OfZlWllExZ6rkS3-wg9zB7OWGONH833bgRShC87t9TUinZY-ZL1KB-QsTUHDvCqPNc3YfoPptBXkEpmQPJ0Dk_uwj-OQ6avK0_lJ5XD5M0/s640/competition.png" width="457" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Table 3.</td></tr>
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The last three permutations can be eliminated for presumed compositional reasons similar to those discussed above: the presence of a fixed point, too many, too few, or an odd number of interversions. But, then we come to the third from the top, (0 5 8 3 7 9 6 2 4 1 10 11). Like <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">1</span></span> it's a 12-cycle and so has no fixed points. Let's call it <i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">3</span><i style="font-weight: bold;">.</i><br />
<br />
<i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">3</span> can be derived in precisely the same way that we conjectured Messiaen's <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">1</span></span> may have been derived, <i>except</i> instead of rotating the final trichord of the <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">2</span></span> seed row one click "clockwise<i>"</i>, [0, 6, 11] → [11, 0, 6] yielding <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">1,</span></span> we rotate it one click "counterclockwise", [0, 6, 11] → [6, 11, 0] yielding <span style="font-weight: bold;"><i>f</i><span style="font-size: x-small;">3</span></span>. Example 3 shows the conjectural composed-out version of mm.70-75 had Messiaen discovered and chosen <i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">3</span>.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD1JxwV0vT1wcCYbZVcI4ScEG5BNaFvTaurMZLbmfZsrzLHkmQJWCB08PEp0crsDW4EXpSOoO2zb0ORKo1ysCscUGD_lTvONVfPrRbzM6qBbSJThKoEDVTmtRkJQODUsnp_UlCMVmaPNs/s1600/Messiaen+IdF+70-75+%2528f3%2529x.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="388" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD1JxwV0vT1wcCYbZVcI4ScEG5BNaFvTaurMZLbmfZsrzLHkmQJWCB08PEp0crsDW4EXpSOoO2zb0ORKo1ysCscUGD_lTvONVfPrRbzM6qBbSJThKoEDVTmtRkJQODUsnp_UlCMVmaPNs/s640/Messiaen+IdF+70-75+%2528f3%2529x.png" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 3</td></tr>
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Well, what's the difference? The harmonic symmetry is broken differently by successive interversions, but both choices end up with a measure in which the LH of m.75 is an ascending chromatic scale spanning C–B and with little distinction between the penultimate interversions in the RH of m.75. Looking at just <i>how</i> the harmonic symmetry is broken and the way that affects the counterpoint in each case might be interesting, but does it matter in the end? Let's see.<br />
<br />
I think Grant Sawatzky was the first to note about these bars that the choice of the seed row dictates that the <i>same</i> dyadic content will be repeated between the two voices in each bar, but they will come in a different order each time due to the interversions.<span style="font-size: x-small;">[2][7]</span> His analysis suggests, but he stops short of explicitly stating (probably not wanting to stray too far from a descriptive focus), that this not only works for this particular interversion set, but will be the case with virtually <i>any</i> permutation.<br />
<br />
With the right choice of permutation combined with a bit of manipulation, this can generate "first species interversion counterpoint" in three, four, or any number of voices. Loosen that up a bit and you have the interversion process as a source for generating matrix strings. Loosen it up a bit more and you get a way to generate strings of multisets. Put another way, you arrive at a large scale form generation based on interversionally derived matrices. For an intuitive start on this notion, look back at Figure 1. Take any four consecutive rows (interversions) and chose any column from those rows, say (reading down), 5-8-3-7 in the upper left whose neighbor to the right is 10-0-5-8. No matter which four consecutive rows you now move to, they will contain a column reading 5-8-3-7, sometimes with the same neighbor to the right and sometimes not, but always in a different position in different tetrachord rows. We now have a compositional technique/theory to explore. We'll leave it undeveloped here. Perhaps more on that at another time. Back to <i>Île de feu 2 ....</i><br />
<br />
Here <span style="font-size: x-small;">[2, p.89]</span> is Sawatzky's description of the dyadic counterpoint in this passage, beginning and ending with a claim relevant to our question about the difference compositionally between <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">1</span></span> and <i style="font-weight: bold;">f</i><span style="font-size: x-small;"><b>3.</b></span><br />
<blockquote class="tr_bq">
[There is] an <i>intervallic consistency</i> between all superpositions of adjacent interversions: within this S[ymmetric]P[ermutation] orbit, [it is] not possible to superpose two adjacent interversions and obtain a dyad of interval classes 0, 1 or 6. This is because, when superposing two interversions at a time, it is only possible to obtain the interval classes that are found between the pitch classes that are adjacent in the cycle notation (expressed in terms of pcs rather than order positions ...). <i>That three of the possible seven interval classes (0-6) do not occur makes the [six]-measure passage sound quite uniform</i>: one continuous episode, rather than six consecutive double statements of the aggregate. [My italics.]</blockquote>
We might resolve the <b> <i>f</i><span style="font-size: x-small;">1/</span><i>f</i><span style="font-size: x-small;">3</span></b> conundrum by dueling "sequence vectors." Referring to Table 3 for easy comparison, we see that every bar in the <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">1</span></span> version (Example 1) is an arrangement of the multiset of interval classes {5,3,2,4,3,3,4,2,5,2,2,5} which we'll collect into a multiplicity vector [[0,0,4,3,2,3,0]]. Every bar in the <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">3</span></span> version (Example 3) is an arrangement of the multiset of interval classes {5,3,2,4,3,3,4,2,5,3,1,1} which can be collected into the vector [[0,2,2,4,2,2,0]]. And now we're into the netherworld of similarity measures in music, but instead of abstract comparisons in "outside time" theory, we're knee-deep into "in-time" real music-in-context. The only meaningful outside-time distinction left is that two ic2's and one ic5 in <span style="font-weight: bold;"><i>f</i><span style="font-size: x-small;">1</span></span><b> </b>interversions are replaced by two ic1's and one ic3 in <span style="font-weight: bold;"><i>f</i><span style="font-size: x-small;">3</span></span><span style="font-size: x-small; font-weight: bold;"> </span>interversions, i.e., only interval classes 0 and 6 are "not possible." This might make a discernible difference in other contexts. But here once again: <i>does it matter?</i> There may have been other reasons for Messiaen's choice which we have yet to discover, but was his ear a significant deciding factor in this case?<br />
<br />
It's time for an ear test.<br />
<br />
Below are audio realizations of the two choices for measures 70–75. No score, just the music. Although with a little effort any musician should be able to discover quickly which is which, I have labelled these two realizations C1–A and C1–B without indicating which is real (<i style="font-weight: bold;">f</i><span style="font-size: x-small; font-weight: bold;">1</span>-generated<b>)</b> and which is the pretender (<span style="font-weight: bold;"><i>f</i><span style="font-size: x-small;">3</span></span>-generated<b>)</b>. Because the real (analytical) question is, after all, <b><i><u>IF</u></i></b> Messiaen <i>did</i> know that there were two nearly identical choices for this passage: why did he choose the one he did? And a not unrelated question for the listener: now that <i>you</i> have a choice, was Messiaen's the right one; would you prefer the other, or <i>does it matter</i> (which is to say, can you even <i>hear</i> the difference <u>at performance tempo</u>)? Here's the test:<br />
<br />
<iframe frameborder="no" height="20" scrolling="no" src="https://w.soundcloud.com/player/?url=https%3A//api.soundcloud.com/tracks/210270911&color=ff5500&inverse=false&auto_play=false&show_user=true" width="100%"></iframe><br />
<iframe frameborder="no" height="20" scrolling="no" src="https://w.soundcloud.com/player/?url=https%3A//api.soundcloud.com/tracks/210270863&color=ff5500&inverse=false&auto_play=false&show_user=true" width="100%"></iframe><br />
<br />
And again one last time, whether your ear can distinguish between them or not: Would it have mattered if Messiaen ultimately chose <span style="font-weight: bold;"><i>f</i><span style="font-size: x-small;">3</span></span> instead of <i style="font-weight: bold;">f</i><span style="font-weight: bold;"><span style="font-size: x-small;">1</span></span>?<br />
<br />
There is one last consideration I have purposely been putting off in order to concentrate on (to my ear) the nearly identical sound of both choices expressed as a blur of notes presaging the "magma dance" (C2) starting in bar 92. How does it fit into its surrounding architecture? What's the preparation for C1, and where is C1 heading as its immediate goal? I can detect no "technical" reason to prefer one over the other by analyzing the six measures in the score, though I could manufacture one or two reasons that are a real stretch not worth noting here. And there's the real possibility that I'm missing something – more later on that.<br />
<br />
This brings us to the only option left for deciding which choice is "right" for these six bars, given no other evidence from the work's structure. I'll let Messiaen speak for himself about this option:<br />
<blockquote class="tr_bq">
[A]side from all structures, it seems to me that each individual and every particular musician ... possesses what we call in philosophy "his accidents," his "tics," his personal habits. [Another composer], using the same structures, would certainly not obtain the same results. There is, then, a question of personal style. [Statement made as a member of the jury during Iannis Xenakis' thesis defense at the Sorbonne in 1976<span style="font-size: x-small;">[8]</span>]</blockquote>
Maybe. But now comes the really knotty problem when we meet Messiaen's choice again in C2 – where it returns at the center of that "magma dance."<br />
<br />
<br /></div>
<span style="font-family: "times" , "times new roman" , serif;"><b>____________________________</b></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br />
</span> <span style="font-family: "times" , "times new roman" , serif;">[1] P.W. Anderson. </span><a href="http://robotics.cs.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf" style="font-family: Times, 'Times New Roman', serif;">"More Is Different: Broken Symmetry and the Nature of the Hierarchical Structure of Science."</a><span style="font-family: "times" , "times new roman" , serif;"> </span><span style="font-family: "times" , "times new roman" , serif; font-style: italic;">Science</span><span style="font-family: "times" , "times new roman" , serif;">, New Series, Vol. 177, No. 4047. (Aug. 4, 1972), pp. 393-396.</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><b style="color: red;">[Added 11/24/2015:]</b> <b>See also </b></span>http://plato.stanford.edu/entries/symmetry-breaking/#4. <b style="font-family: times, 'times new roman', serif;">The concept of 'broken symmetry' used in this and subsequent posts was first <i><u>suggested</u></i> to me by Anderson's article, but the reader should not take its appearance here as an 'application' of this idea which would only result in adding to the collection of New Age pseudo-connections. There certainly are parallels in the way 'symmetry' is viewed and used in science and music, and these more substantial connections will be explored in a future post.</b><br />
<span style="font-family: "times" , "times new roman" , serif;"><br />
</span> <span style="font-family: "times" , "times new roman" , serif;">[2] The best technical analytical survey I have read on Messiaen's music from 1950–1992, and which I highly recommend, is <a href="https://circle.ubc.ca/handle/2429/43901">Grant Sawatzky, <i>Olivier Messiaen's <u>Permutations Symétriques</u> in Theory and Practice</i>, 2013</a>.</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br />
</span> <span style="font-family: "times" , "times new roman" , serif;">[4] V(11,A)=V(2,B)=V(8,C)=V(5,D)=V(A)=V(B)=V(C)=V(D)=[100011].</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br />
</span> <span style="font-family: "times" , "times new roman" , serif;">[5] Here is where we enter music theory's version of Boorstin's "Fertile Verge."</span><br />
<blockquote class="tr_bq">
<div style="letter-spacing: 0.25999999046325684px; line-height: 18px; margin-bottom: 10px; text-align: justify;">
<span style="font-family: "times" , "times new roman" , serif;">"American creativity…has flourished on what I call the Fertile Verge. A verge is a place of encounter between something and something else. America was a land of verges—all sorts of verges, between kinds of landscape or seascape, between stages of civilization, between ways of thought and ways of life…. The creativity, the hope, of the nation was in its verges, in its new mixtures and new confusions….</span></div>
<div style="letter-spacing: 0.25999999046325684px; line-height: 18px; margin-bottom: 10px; text-align: justify;">
<span style="font-family: "times" , "times new roman" , serif;">"On these verges—gifts of our geography, our history, our demography—we find three characteristic ways of thinking and feeling. First, there is our exaggerated <em>self-awareness</em>. On the verge we notice more poignantly who we are, how we are thinking, what we are doing. Second, there is a special <em>openness to novelty and change</em>. When we encounter something different, we become aware that things can be different, our appetite is whetted for novelty and its charms. Third, there is a strong <em>community-consciousness</em>. In the face of the different and the unfamiliar, we, the similars, lean on one another. We seek to reassure one another as we organize our new communities and new forms of community. These three tendencies are all both opportunities and temptations. They are sometimes complementary, sometimes contradictory. Creativity in our United States has been a harvest of these hypertrophied American attitudes stiffed on the Fertile Verge."</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br />
</span><br />
<div style="text-align: right;">
<span style="font-family: "times" , "times new roman" , serif;">– <a href="http://www.worldcat.org/title/fertile-verge-creativity-in-the-united-states-an-address-given-at-the-carnegie-symposium-on-creativity-the-inaugural-meeting-of-the-library-of-congress-council-of-scholars-november-19-20-1980/oclc/7170148">Daniel J. Boorstin, <i>The Fertile Verge: Creativity in the United States</i></a></span></div>
</div>
</blockquote>
<span style="font-family: "times" , "times new roman" , serif;"><i>Theoria</i> <span style="font-size: x-small;">[see 5.1 below]</span> goes off-road into that liminal region between or overlapping the beaten paths of analysis and the wilds of composition. The Fertile Verge is generally ignored by – if not forbidden to – the strict analytico-pedagogical "theorist". The ground here is constantly shifting: an unpredictably variable blend of the composer's intentions in creating the work and the complete set of choices available in attempting to fulfill that intention. The questions here are not of the impossible sort, such as "What did Messiaen <i>mean</i> (whatever that means) by X", let alone what "inspired" Messiaen or where did X come from, although even these questions are not disallowed in a fictive analysis. The final quote in this post notwithstanding, <u>we leave the Angel out of this entirely in this case</u> – "style" and automatic writing are not the same. Mostly the questions in the Fertile Verge are <i>openings</i> and revolve around a different sort of unanswerable. (Elsewhere I have called these <a href="https://www.academia.edu/2512174/Riemannian_Variations_on_a_Theme_by_Milton_Babbitt">"Babbittian questions."</a>) The goal here is not to find a static truth-as-fact – to answer any question definitively or to follow Messiaen up to the point where X is no longer applicable in Messiaen's work, publish it and call it a day. The goal in the Fertile Verge – where fictive analysis is most fruitful – is, having discovered X in Messiaen, to follow X <i>wherever</i> X leads (to the extent our imagination can take us) – <u><i>with or without</i> Messiaen or any other example of X that might be found in other musical works or even in things ostensibly unrelated to music</u>. It beckons to take a chance, to consider absurd ideas, to make connections, to create <i>new</i> work.</span><br />
<blockquote class="tr_bq">
<div style="text-align: justify;">
<span style="font-family: "times" , "times new roman" , serif;">[5.1] The theory/<i>theoria</i> distinction I am using is from David L. Hall, <i>Eros and Irony</i> (SUNY Press, c1982):</span></div>
</blockquote>
<blockquote class="tr_bq">
<blockquote class="tr_bq">
<div style="text-align: justify;">
<span style="font-family: "times" , "times new roman" , serif;">"[<i>T</i>]<i>heoria</i> is, above all, obedient to that sense of eros which lures toward completeness of understanding." (p.43)</span></div>
</blockquote>
<blockquote class="tr_bq">
<div style="text-align: justify;">
<span style="font-family: "times" , "times new roman" , serif;">"Strictly systematic theory [vs. <i>theoria</i>] is more often than not an ideological epiphenomenon functioning apologetically with respect to current modes of practice. Thus theory [vs. <i>theoria</i>] is practical by definition if one means no more by theoretical endeavor than that systematic, principled form of thinking shaped by the desire for application." (p.45)</span></div>
</blockquote>
</blockquote>
<span style="font-family: "times" , "times new roman" , serif;">[6] Figuring out the number of interversions before the seed is repeated is the same task as calculating the relationship between rhythmic n-tuples. In this case, the permutation function has three disjoint cycles of lengths 6, 5, and 1, so our answer will be the least common multiple of those three numbers: lcm(6,5,1)=30 repetitions, the same calculation we would make to determine a 5-against-6 rhythm.</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br />
</span> <span style="font-family: "times" , "times new roman" , serif;">[7] The reader might be confused about what appear to be discrepancies between Sawatzky's notation of permutations and mine, he is mapping <i>positions</i> of pcs which is probably more relatable to accepted mathematical usage; I am mapping pcs <i>directly</i>, which I hope will be more relatable to the way music theory is currently presented. I acknowledge my reason for this choice could be wrong, but it is also meant to prepare for my later treatment of permutations of strings of (music) events.</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br />
</span> <span style="font-family: "times" , "times new roman" , serif;">[8] In <a href="http://www.worldcat.org/title/arts-sciences-alloys-the-thesis-defense-of-iannis-xenakis-before-olivier-messiaen-michel-ragon-olivier-revault-dallonnes-michel-serres-and-bernard-teyssedre/oclc/8553270&referer=brief_results">Iannis Xenakis, </a><i><a href="http://www.worldcat.org/title/arts-sciences-alloys-the-thesis-defense-of-iannis-xenakis-before-olivier-messiaen-michel-ragon-olivier-revault-dallonnes-michel-serres-and-bernard-teyssedre/oclc/8553270&referer=brief_results">Arts/Sciences: Alloys</a>,</i> p.39-40.</span><br />
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-20947681805319500532017-06-09T10:08:00.000-04:002017-06-09T10:26:39.191-04:00Broken Symmetries 2<blockquote class="tr_bq" style="text-align: justify;">
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The idea of the series was engaging [Messiaen's] maximum attention during these years, and it was probably the influence of this fact that caused him to reflect on the possible strict, and strictly calculated, relationships on which his music might depend; there are many instances in these works of a clear conflict between spontaneity and organization, the one unwilling to abdicate and the other determined to become all powerful. This conflict, or antinomy, is reflected even in the titles of the different pieces written between 1949 and 1951 – <i>Les Yeux dans les roues</i>, <i>Les Mains de l'abíme</i>, <i>Ile de feu</i>.</blockquote>
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– Pierre Boulez *</div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/A45e4BRjk0g/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/A45e4BRjk0g?feature=player_embedded" width="320"></iframe></div>
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Olivier Messiaen</div>
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<a href="http://www.worldcat.org/title/ile-de-feu-ii-no-4-de-quatre-etudes-de-rythme-pour-piano/oclc/56577632&referer=brief_results">Île de feu II : no 4 de Quatre études de rythme : pour piano</a></div>
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Performed by Yvonne Loriod</div>
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<i><b>( A score may be located through <a href="http://www.worldcat.org/title/ile-de-feu-ii-no-4-de-quatre-etudes-de-rythme-pour-piano/oclc/56577632/editions?referer=di&editionsView=true">OCLC WorldCat</a> )</b></i></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLSkQNTe5Tk1_sO7eXAFczKFZWWR0KVL8zzs0L4BkHI0037X7z-HSRY0yVAOP3utCzfjeU_xxS6p2uq571Fr9ylEdOeLa9O3XZnLPIy4-YmwZRiaeiYJSXq08pfeZxoWJTgtC1NQSVKmo/s1600/IDF+diagram+C1+C2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLSkQNTe5Tk1_sO7eXAFczKFZWWR0KVL8zzs0L4BkHI0037X7z-HSRY0yVAOP3utCzfjeU_xxS6p2uq571Fr9ylEdOeLa9O3XZnLPIy4-YmwZRiaeiYJSXq08pfeZxoWJTgtC1NQSVKmo/s640/IDF+diagram+C1+C2.png" width="640" /></a></td></tr>
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<tr><td class="tr-caption" style="font-size: 13px;">Diagram 1<br />
Compositional scheme of <i>Île de feu 2</i><br />
<i>(Timings refer to the Loriod recording)</i><br />
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<span style="font-size: large;">Descriptive Analysis (C2)</span></div>
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The previous post, <a href="http://essaysandendnotes.blogspot.com/2015/06/broken-symmetries-1.html">Broken Symmetries 1</a>, concentrated on the analytical dilemmas posed by the brief passage labelled <b>C1</b> (mm. 70–75) in Diagram 1. <b>C1</b> appears, as if out of nowhere, almost exactly half way through the piece as a stream of even <i>pp</i> legato 16th-note triplets. Architectonically, <b>C1</b> only makes sense to me as a prefiguration of <b>C2</b>, the lengthy section I'm calling the 'Magma Dance' (this begins around 2:55 in the Loriod recording). Like <b>C1</b>, <b>C2</b> is a double stream of 16th notes, one stream running in the right hand, the other in the left hand.<br />
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When I referred to <b>C1</b> in the previous post as a passage of even triplets in two-voice 'first species counterpoint,' I wasn't entirely joking. Both <b>C1</b> and <b>C2</b> <i>are</i> instances of note-against-note counterpoint, but this is 'counterpoint-<i>by-other-means</i>.' I won't yet pursue, let alone formalize, the idea that is slowly emerging here and from other posts in this blog, but a better term for these passages – and many more in the contemporary literature – might be <i>counterset</i>. In our new, still tentative musical world, 'point' may refer to pitch, duration, dynamic, density, chord, key, silence, sample, noise, event, action, choice, instruction, function, feature, preference, intention – in short, any stuff that can make up a musical game of things-and-arrows from a mostly intuitive category theory.<br />
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But now to the story of Messiaen's notes and his (and my) 'conflict between spontaneity and organization.'<br />
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A <b>C1</b> characteristic that I saved to discuss for comparison to <b>C2</b> is the disjunct, narrow ranges occupied by each voice in <b>C1</b>. There the left hand spans the range <i>C</i><span style="font-size: x-small; font-style: italic;">3</span><i>–B</i><span style="font-size: x-small; font-style: italic;">3</span> while the right hand stays within the range <i>C<span style="font-size: x-small;">4</span></i><i>–B<span style="font-size: x-small;">4</span></i>, disjunct from the left hand (Example 1a).<span style="font-size: x-small;">[1]</span><br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixznAKr9RDxR4ZhdKyuZ1I5FOx50lN3lDaXT8nBzOCofuidXssQJAehrHeujP10SVv-TBTLRsuOX5rnToEoZQJudv-geraAXCvPSJCMau2pTHrTVJkliKRlRdeQUlHYZxFz0ornxEAKxA/s1600/idf+92-131+range+c1+c2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="189" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixznAKr9RDxR4ZhdKyuZ1I5FOx50lN3lDaXT8nBzOCofuidXssQJAehrHeujP10SVv-TBTLRsuOX5rnToEoZQJudv-geraAXCvPSJCMau2pTHrTVJkliKRlRdeQUlHYZxFz0ornxEAKxA/s320/idf+92-131+range+c1+c2.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 1a Example 1b</td></tr>
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The <b>C2</b> passage on the other hand was described by Messiaen as 'a <i>cross-handed</i> perpetual motion in the depths of the keyboard.'<span style="font-size: x-small;">[2]</span> Example 1b illustrates that the left hand's range in <b>C2</b> is again bounded within a single octave. Down a fifth from the left hand in <b>C1</b>, it spans <i>F<span style="font-size: x-small;">2</span></i><i>–E<span style="font-size: x-small;">3</span></i>. The <i>ceiling</i> of the right hand in <b>C2</b> also drops a fifth from what it was in <b>C1</b>, taking the r.h. ceiling down from <i>B<span style="font-size: x-small;">4</span></i> to <i>E<span style="font-size: x-small;">4</span></i>; but the r.h. <i>floor</i> drops from <i>C<span style="font-size: x-small;">4</span></i> down more than two octaves to <i>Bb<span style="font-size: x-small;">1</span></i>, a fifth below the l.h. floor.<i> </i>As we will soon see, all this 'down-a-fifth' into the depths of the keyboard activity associated with right- and left-hand ranges for the Magma Dance and its prefiguration will be accompanied by a related row transposition at the center of the big puzzle embedded in <i style="text-align: center;">Île de feu 2</i>.<br />
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A final contrasting element is that in <b>C1</b> both hands are marked <i>pp legato</i> – but in <b>C2</b> the right hand is marked <i>f staccato</i> against a 'ghosting' left hand that retains the <b>C1</b> character as <i>p legato</i>. We'll now focus on the ghost in the left hand.<br />
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<b><span style="font-size: large;">. . . .</span></b></div>
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Diagram 2 is a numeric pitch-class transliteration of mm 92–131 so that the lines of the crossing hands can be clearly distinguished for analysis. Each strip represents two measures; the red lines indicate bar lines, the black vertical lines indicate Messiaen's 16th-note groupings. The top row of each strip transliterates the right hand & the bottom transliterates the left hand.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ_EQ2vjE_G9tA0zaC9WwwqiOFcJfGSRHLA4245E-KxqdWKtC6jncPCgyT75TVJX-g6drurbijQiLPl_inKQmzlTPgK-wrfN2Sr0kLdLUVLRRet9p8AfP04c4Rz4kcaM7B2RkRxf9X2VY/s1600/IDF+92-131+1.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ_EQ2vjE_G9tA0zaC9WwwqiOFcJfGSRHLA4245E-KxqdWKtC6jncPCgyT75TVJX-g6drurbijQiLPl_inKQmzlTPgK-wrfN2Sr0kLdLUVLRRet9p8AfP04c4Rz4kcaM7B2RkRxf9X2VY/s1600/IDF+92-131+1.png" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Diagram 2</td></tr>
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Let's consider the left hand (bottom half of the strips) which as noted previously spans only one octave, <i>F<span style="font-size: x-small;">2</span></i><i>–E<span style="font-size: x-small;">3</span></i>. One is immediately struck by the fact that the l.h. in each pair of measures is a 23-note palindrome. So it is tempting to jump immediately to the conclusion that any given l.h. strip consists of a 'twelve-tone row' and its retrograde connected by a common tone as in, for example, the first strip (l.h. of mm. 92–93):<br />
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9–5–4–10–6–3–11–7–2–0–8–\ <b>1 </b> /–8–0–2–7–11–3–6–10–4–5–9</div>
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This symmetry has been pointed out elsewhere<span style="font-size: x-small;">[3]</span>, but I have yet to find a source that notes a second symmetry hiding in the left hand, let alone a source that asks where these 'rows' came from and whether/how they are related to one another and to the right hand. One thing is certain. It is quite misleading to suggest that these pitch-class strings will turn out to be well behaved 'twelve-tone rows' because, as a first quick scan reveals, there isn't a single canonic serial transformation in sight beyond the distinct palindromes we just identified. If <b>C2</b> were 'serial' in the usual sense that word is taught to be taken, it would mean <b>C2</b> is a simple concatenation of <u>ten</u> distinct 'tone rows' (which would be somewhat of a record for just one minute of music, and a triumph of compositional randomization.<br />
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So we've arrived at <i style="text-align: center;">Île de feu 2</i>'s big puzzle. Now let's see if we can solve it, or at least determine if it is solvable or not. We'll start by isolating the distinct pc-strings.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPdRycBGXOwLLNeAx1ppWu7n_8ADMBgOIl3hUD_yJXY_tWpQ5OhNDAKF2_ZmhdrfppUFCE2HcNeXukxDuWKR4GouyhC5EwMpJBSKt5SHf1m21YVZBsZUoQreV1tkP88OX6AU6rKdE7DvI/s1600/92-131+symmetries+chart.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="438" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPdRycBGXOwLLNeAx1ppWu7n_8ADMBgOIl3hUD_yJXY_tWpQ5OhNDAKF2_ZmhdrfppUFCE2HcNeXukxDuWKR4GouyhC5EwMpJBSKt5SHf1m21YVZBsZUoQreV1tkP88OX6AU6rKdE7DvI/s640/92-131+symmetries+chart.png" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Diagram 3</td></tr>
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Diagram 3 shows that there is a vertical reflective symmetry accompanying the horizontal reflective symmetry, turning <b>C2</b>'s left hand part into a house of mirrors. Folding Diagram 3 right to left and then bottom to top yields a 10×12 matrix (Diagram 4).<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzPnXmBEKPoKEHvBoGiLGN_XLCZWSNzMDFMqRlrkaCu0AA_gO56cW4sB4j413d53LY-6fO0G50p4BYHn4r1sKIhMeISN7kcHVyZG-gZBhTO1b1uPaQMr4WKykGWsZGR_96JpJijRn8pt8/s1600/idf+92-131+matrix.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="260" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzPnXmBEKPoKEHvBoGiLGN_XLCZWSNzMDFMqRlrkaCu0AA_gO56cW4sB4j413d53LY-6fO0G50p4BYHn4r1sKIhMeISN7kcHVyZG-gZBhTO1b1uPaQMr4WKykGWsZGR_96JpJijRn8pt8/s400/idf+92-131+matrix.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Diagram 4</td></tr>
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This simplifies the analytical focus since it means that, to discover any further relationships within <b>C2 </b>(beyond this two-fold mirror symmetry), we only need to look at any one of the quadrants, since any relationship between 'rows' (or columns, for that matter) in one quadrant will be duplicated or mirrored in the other three. Diagram 4 shows the upper left quadrant.<span style="font-size: x-small;">[4]</span> Any relationship we can find among these ten 12-tone strings will be duplicated in the other quadrants.</div>
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</span> <span style="font-size: large;">Antinomous Analysis (C2)</span></div>
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The ten interversions in sections <b>B1</b>, <b>B2</b> and <b>B3</b> (compositional scheme, Diagram 1) have been the focus of most commentary about this work. The passage <b style="text-align: center;">C2</b> (mm. 92-131, distilled into the matrix shown in Diagram 4), although it is one forth of the entire piece, has been ignored. But the <b>C2</b> passage is <i>Île de feu</i>'s maddeningly elusive big puzzle. Where do these ten strings come from and how are they related to one another (and/or possibly to material outside <b>C2</b>)? The story here is not, as in the fictive analysis of <b>C1</b> in the previous post, centered on a single anomaly for which we can make up a likely scenario. The analysis below will be seen (at least for now) to describe either an unsolved solvable puzzle or one that has no solution. Such a description is what I mean by 'antinomous analysis.'<span style="font-size: x-small;">[5]</span><br />
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All that our strictly descriptive analysis of <b>C2</b> has given us so far are compelling clues scattered about, any one of which might be ignored as anomalous, but which taken together strongly suggest an as yet undiscovered over-arching pattern which we are compelled to pursue, not knowing whether or not we are running down a blind alley.<br />
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<span style="text-align: center;">So let's begin by trying to find </span><span style="text-align: center;">in Diagram 4 </span><span style="text-align: center;">all the indisputably <i>non</i>-random features as well as noting the features that suggest randomness.<span style="font-size: x-small;">[6]</span> Following is a list of seven compositional features in </span><b>C2</b>.<span style="text-align: center;"> Each feature is followed by an analytical judgement:</span><br />
<b> [</b>✓<b>]</b> = functionally determined,<br />
<b> [?]</b> = '<i>non liquet</i>' ('it is not clear'), i.e., suggestive of a functional relationship that remains undiscovered, or<br />
<b> [</b>✕<b>]</b> = tentatively random (no obvious evidence for either <b>[</b>✓<b>]</b> or <b>[?]</b>).</div>
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<i><b>(1.)</b></i> To reiterate from the descriptive analysis above, the <i>entire</i> passage in the left hand has a two-fold mirror symmetry. And if you connect identical edges top to bottom and left to right in Diagram 3 you have a torus. <b>[</b>✓<b>]</b><br />
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<i><b>(2.)</b></i> String 1 in Diagram 4 can be derived as a 'double retrograde' followed by a <a href="http://www.emis.de/journals/DMTCS/pdfpapers/dm050111.pdf">(<i>k,n</i>)-perfect shuffle</a>.<br />
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Let's say we are given a 'deck' (or set) of 12 pitch classes 0 through 11, and we divide it into <i>k</i>=3 cuts (subsets), each containing <i>n</i>=4 pcs:<br />
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[1,2,3,4], [5,6,7,8], [9,10,11,0].</div>
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A straightforward 3-way perfect shuffle of these three subsets would take the 'top' remaining element from each ordered subset consecutively and result in the permutation<br />
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[1,5,9,2,6,10,3,7,11,4,8,0].</div>
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Harmonically, this shuffle would result in four consecutive, equally-spaced augmented triads. But a 'perfect shuffle' doesn't depend on the internal order of each subset, only the order of choosing from each subset. So instead let's begin with a set of pcs cut into three stacks this way:<br />
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[9,10,11,0], [5,6,7,8], [4,3,2,1].</div>
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This reverses the order of the initial three tetrachords, and then reverses the order of internal elements in the last of the three retrograded chords. We then proceed to interleave this result left to right as a 3-way perfect shuffle. The result is string 1:</div>
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[9,5,4,10,6,3,11,7,2,0,8,1].</div>
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Note that reversing the final ordered sub-set to read [4,3,2,1] instead of [1,2,3,4], breaks the expected symmetry of equally spaced augmented triads, but the broken symmetry is replaced with a new trichord symmetry:</div>
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<span style="text-align: center;">A : B :: B' : A'</span></div>
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<span style="text-align: center;">where the As are SC-015 and the Bs are SC-037. A={4,5,9}, A'={8,0,1}=I</span><span style="font-size: x-small; text-align: center;">5</span><span style="text-align: center;">(A), B={3,6,10}, B'={7,11,2}=</span><span style="text-align: center;">I</span><span style="font-size: x-small; text-align: center;">5</span><span style="text-align: center;">(B) (and, of course, the two hexachords are also related by </span><span style="text-align: center;">I</span><span style="font-size: x-small; text-align: center;">5</span><span style="text-align: center;">).</span><span style="text-align: center;"> Checking back to Diagram 2 (l.h.) we see that in mm. 92–93 Messiaen grouped the 16th notes in 3's.</span> His grouping appears to be intentionally drawing attention to the string's origin in a composite function drawing from these three sets since, by inspection, it can't possibly be functionally related to the right-hand line. <b>[</b>✓<b>]</b><br />
<i><<Added 11.10.15: String 1 is a variation of a pattern found in 'Mode de valeurs et d'intensités' identified as '7. Bars 86-96 (Group II)' (also '6. Bars 81-86 (Group I)' in retrograde) in Robert Sherlaw-Johnson (</i>Messiaen<i>, 1975, p.108, Table II)</i>>></div>
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<i><b>(3.)</b></i> String 2 results from successive spiral permutations. Start with the same initial chromatic clusters as we did in String 1, and label them A=[1,2,3,4], B=[5,6,7,8], C=[9,10,11,0]. The first spiral action (identical to a rotation) takes these tetrachords and rearranges them this way: A,B,C → B,C,A. The second spiraling takes the content of each tetrachord and rearranges it this way: [a,b,c,d] → [a,d,b,c]. So B,C,A= [5,6,7,8],[9,10,11,0],[1,2,3,4] becomes<br />
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[5,8,6,7],[9,0,10,11],[1,4,2,3],</div>
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which is string 2. The initial chromatic tetras are rearranged internally but kept intact harmonically (SC-0123) and related by T<span style="font-size: x-small;">4</span>. C<span style="text-align: center;">hecking back to Diagram 2 (l.h.) we see that in mm. 94–95 Messiaen grouped the 16th notes in 4's.</span> Again, it looks like he was grouping by quadruplets according to this string's generating function operating on sets of four elements. <b>[</b>✓<b>]</b><br />
<i><<Added 11.10.15: String 2 is a variation of a pattern found in 'Mode de valeurs et d'intensités' identified as '4. Bars 53-57 (Group I)' in Robert Sherlaw-Johnson (</i>Messiaen<i>, 1975, p.108, Table II)</i>>></div>
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<i><b><span style="text-align: center;"></span>(4.)</b></i> String 5 is generated by simple transposition, T<span style="font-size: x-small;">5</span>, from the seed string in passage <b>C1</b> discussed in <a href="http://essaysandendnotes.blogspot.com/2015/06/broken-symmetries-1.html">Broken Symmetries 1</a>.<br />
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[10,3,9,0,6,1,7,2,8,4,5,11] = <span style="text-align: start;">T</span><span style="font-size: xx-small; text-align: start;">5</span>([5,10,4,7,1,8,2,9,3,11,0,6]).</div>
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(... or was <b>C1</b>'s seed string derived from String 5 by T<span style="font-size: x-small;">7</span>? At any rate ....)<br />
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String 5 functionally connects <b>C2</b> to <b>C1</b>.<br />
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As shown in Diagram 2, in mm. 100-101, the 16ths are grouped in 3's, as they were generated in the entire <b>C1</b> passage, mm. 70–75. <b>[</b>✓<b>]</b><br />
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(It's as though Messiaen is leaving a trail for us by the way he groups sixteenths, relating that to the way the pitches were generated. But this idea is difficult to sustain beyond strings 1, 2, and 5.)<br />
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<span style="text-align: center;"><b><i>(5.)</i></b><i> </i>Strings 1 and 10 in Diagram 4 are related by a simple one-click rotation: the first element in string 1 (pc9) is moved to the end to generate string 10.</span><br />
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[9,5,4,10,6,3,11,7,2,0,8,1] → [5,4,10,6,3,11,7,2,0,8,1,9].<span style="text-align: start;"> <b>[</b></span><span style="text-align: start;">✓<b>]</b></span></div>
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<i style="font-weight: bold;">(6.)</i> (Hypothetical feature) So now we are left with strings 3, 4, 6, 7, 8, and 9 entirely unaccounted for. Is it possible there is a permutation that could relate all 10 strings, e.g., by interversion as in <b>C1</b>? Trivially, <i>any</i> pair of 12-strings could be treated as a permutation. But it would take a single covering permutation or group of related permutations to knit the matrix together such that we could claim a full solution to the question. For example, we can derive string 3 from string 2 by the permutation (5,4,9,3,11,0,6,7,2,10,8)(1), but this permutation doesn't relate any other pair of strings in the matrix. And the same for any other pair (just pick one element in any string and follow it through its position vis a vis all possible pairs – cf. Table 1 in <a href="http://essaysandendnotes.blogspot.com/2015/06/broken-symmetries-1.html">Broken Symmetries 1</a>). Also, none of these remaining strings can (thus far to me) be related by any permutation from an obvious/relevant external seed string as we found in strings 1, 2 and 5. The idea of knitting the whole matrix together by interversion (iterated permutation) or some obviously related group of permutations must be labeled <b>[</b>✕<b>]</b> as tentatively random. The reason the judgement remains tentative is due to the next feature observation. Examining the matrix <u>columns</u>, we find something else very suggestive that would seem to increase the odds against randomness for the matrix as a whole.<br />
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<i style="font-weight: bold;">(7.)</i> <i><u>Column</u></i> 5 reads <i>down</i> 6–9–3–5–6–9–3–5–6–3. Omitting the final element 3 which comes from the rotation of string 1 and might be redundant to the pattern, we get two identical <i>vertical</i> 4-string groupings with an extra element 6 tacked on the end: 6–9–3–5 / 6–9–3–5 / 6. Similarly, reading <u>up</u> we get the groupings 6–5–3–9 / 6–5–3–9 / 6. Another way to look at this is by isolating the 6's: 6 / 9–3–5 / 6 / 9–3–5 / 6 down or 6 / 5–3–9 / 6 / 5–3–9 / 6 up. The middle 6 is element number 5 in string 5. This suggests that string 5, the string related by transposition from the seed string in section <b>C1</b> is row-wise 'central' to the matrix and might somehow generate the four strings above and below it. Another (better?) guess is that any string pair (<i>n</i>, <i>n</i>+4), where <i>n</i>=1,2,3,4, is related by some yet to be discovered function/transformation/generating principle. I have no idea how to calculate the odds against this symmetry appearing coincidentally in a random draw of ten 12-strings, but I would imagine it's huge. There's enough demonstrable consistency and planning evident to encourage me to continue searching for something I'm overlooking. So I mark this feature highly suggestive (that there is an undiscovered pattern) but inconclusive (and certainly not 'solved'). <b>[?]</b><br />
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There are other features that suggest internal secondary patterns, but the ones above are the obvious ones. Pursuing them is quite a stretch, and the ones I know of are difficult to describe, far-fetched, often end in blind alleys, and (so far) wouldn't change or add anything to my present 'conclusion' about this passage. Obviously it is not possible at this point, given only the information above, to demonstrate that <b style="text-align: center;">C2</b> was consciously organized as <b style="text-align: center;">C1</b> clearly was – to wrap it up in a neat functional package where everything relates to everything else. <u>Neither</u> is it possible to declare that the unexplained portions are merely random.<br />
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<b><span style="font-size: large;">. . . .</span></b></div>
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My conclusion concerning the entire passage <b style="text-align: center;">C2</b> is: no conclusion at all – at least in the sense of 'mission accomplished.' The evidence strongly suggests to me that all ten of the ghost strings in the Magma Dance are derived and, less certainly, interrelated. I can't believe that Messiaen – <i>especially</i> Messiaen – would have functionally generated four 12-strings and then pulled in six random ones out of thin air. It is at least clear to me that in <b style="text-align: center;">C2</b> he is conducting his earliest experiments in functional derivation/manipulation of material that go beyond the canonic T, R and I.<br />
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I invite others to take up the <b style="text-align: center;">C2</b> challenge – and I will take it up again if I can think of an analytical attack I haven't tried – but as of now, I must follow in the steps of those befuddled jurists of old who wrote at the bottom of their undecidable cases:<br />
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<b>NL</b></div>
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_____________________<br />
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* Pierre Boulez. 'Olivier Messiaen' ('Une classe et ses chimères', tribute to Messiaen on his fiftieth birthday from the programme for the Domaine musical concert of 15 April 1959. Tr. by Martin Cooper, 1986.) In <i>Orientations: collected writings</i>. Edition quoted: Faber & Faber, 1990. p.414.<br />
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[1] Ranges are given in scientific notation: <i>C</i><i style="font-size: small;">4</i> = middle C. Apologies for the confusion. I labeled the compositional scheme (Figure 1) without thinking I was planning on also talking about voice ranges. I'll use bold for the compositional scheme & italics for pitch notation here.<br />
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[2] Olivier Messiaen. 1994. Programme note in booklet accompanying Koch International Classics 3-7267-2 H1 quoted in <a href="https://en.wikipedia.org/wiki/Quatre_%C3%A9tudes_de_rythme#CITEREFMessiaen1994">Wikipedia article 'Quatre études de rythme</a>'<br />
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[3] E.g., John M. Lee. 'Harmony in the Solo Works of Olivier Messiaen: The First Twenty Years.' In <i>College Music Symposium</i>. Vol. 23.<br />
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[4] In traditional notation, below is Diagram 4 with pitches as they appear in the score within the delimiting span <i>F<span style="font-size: x-small;">2</span></i><i>–E<span style="font-size: x-small;">3</span></i>:<br />
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[5]<br />
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It was tempting at first to call this type of analysis 'inconsistency-tolerant' or even <a href="http://plato.stanford.edu/entries/logic-paraconsistent/">paraconsistent</a>. Then, while reading the 1959 essay by Pierre Boulez from which the above quote is taken, I realized that the kind of analysis I was almost forced into by this section of <i style="text-align: center;">Île de feu 2</i> was more precisely seen as a <i>reflection</i> of the 'conflict, or antinomy' Messiaen himself must have faced in creating this passage. In other words, the analytical process itself turns out to be antinomous.<br />
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Boulez identified such a feature of a work as a <i>compositional</i> conflict between spontaneity and organization. (One can easily believe he was speaking sympathetically here! The tug between the two has become endemic to music composition – as well as analysis – for more than a century.) So I have identified it similarly, viewing it from the other side, as an <i>analytic-decision </i>conflict between randomness and functionally created pattern. Fitting the present theme, it is also a study in persistent vs. broken symmetries which necessitates discriminating between invariant/covariant and unrelateable features.<br />
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[6] Four meditations on 'random.'<br />
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(1) My use of 'random' in this context is not meant to imply its pejorative use accusing the composer of 'pick a note, any note, it doesn't matter', which, when reflected in analysis, is the oft used but seldom recognized academic's gloss, a wave of the hand signifying 'I have no idea, so let's move on.' I mean 'random' to be taken here in the sense of a placeholder for 'the analyst is stumped but [unlike the gloss]<i> can't leave it alone.</i>' This condition reminds one of the prince searching for Cinderella's foot which he assumes will eventually lead to Cinderella, not knowing in the light of day whether or not there is such a foot anywhere in the kingdom, or where the slipper came from, and increasingly facing the possibility that someone will finally dare tell him he spent the night dancing with a figment.<br />
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(2) The word 'random' associated with the arts is often used loosely as a pejorative, but it has objective, non-pejorative meanings in the sciences. My usage here is meant to dismiss the former and respect, if not live up to, the latter. Any feature (say, in the descriptively generated matrix in Diagram 4) that can be shown to have been functionally derived from another feature (inside or outside the matrix) is a <i>formally determined</i> (non-random) feature, whether or not the composer is fully aware of such a determination in this formal sense. The probability that a formally determined feature of a 12-string (not to be confused with the guitar of the same name) could <i>also</i> have been arrived at by blindly drawing 12 notes out of a hat – tripped over, so to speak – is close to zero. Conversely, any feature that <i>cannot</i> be shown to be formally determined in this sense becomes a <i>candidate</i> for being judged analytically as a <i>random</i> feature, and such a feature remains forever a candidate. I add "candidate" as a hedge because it is close to impossible to be certain a feature is random in the sense I am using. How would one provide evidence for creative indeterminacy – call it inspiration or the angel, if you will? Even the composer can't be believed except in the case when s/he is divulging a determined feature (which Messiaen often did). The analyst simply has to learn to live in this suspension.<br />
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(3) As I use the word 'random' in a music-analytic setting, any <i>strictly aural</i> ('it's just what I hear & I can't explain it') preference or other unalloyed preference (by the composer) for selecting one note or pitch-class row or interval string or chord or rhythm or other feature over another is a random selection that is potentially form-inducing. Again: my notion of randomness ignores the pejorative sense of 'pick any note, <i>it doesn't matter</i>'; but it preserves an essential place within <i>techne</i> for inspiration, serendipity, accident, and mistake.<br />
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(4) This sets the stage for analysis of inconsistent features found embedded in a work. Such a feature potentially arises when a preponderance of evidence strongly suggests that a given feature surely <span style="font-style: italic;">must</span> be determined, leading to the conclusion that either the formal determination (function, transformation, whatever) exists but cannot be found, or to the analyst's dreaded conclusion that the feature has no possible formal determination in a sense that relates to the context. It may persist in analytic limbo (undecidable) indefinitely, and its status as determined or random may never be decided definitively. Looked at one way a feature may be judged random, looked at another way it may be judged an un(re)solved determination.<br />
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-29174502459208441772017-06-09T10:07:00.000-04:002017-06-09T10:26:39.196-04:00Broken Symmetries 3.1<div class="separator" style="clear: both; text-align: center;">
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<span style="font-size: large;"><b><i>A Tale of Two Tunes</i></b></span></div>
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<span style="font-size: large;">1. Rhythmic structure</span></div>
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<span style="font-size: large;">in the themes of Messiaen's <i>Île de feu I</i> & <i>II</i></span></div>
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The following rhythmic analysis comparing the two themes in <i>Île de feu I</i> (herein <i>Idf1</i>) and <i>Île de feu II</i> (<i>Idf2</i>) is a rather unexceptional description based on the notion of fuzzy rhythmic contours. But while it doesn't present any surprises taken by itself, this rhythmic analysis lays the groundwork for the deeper connections that will be presented in the following two entries in this thread. The two themes are shown in Examples 1 and 2.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMWFGLwJD29XLLNHzYD_APxxe_WIuSyRLpOX7Hxhk-Po7nW8kaCzyRS4Z6w5nHjQwmnCqicohIvaI7cirlOBaGje22kNcgiV9T-fjYlAz6IXMAUAfqdy8JxwcTz65PaWhCQX0C1q-J5To/s1600/idf1+theme.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="142" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMWFGLwJD29XLLNHzYD_APxxe_WIuSyRLpOX7Hxhk-Po7nW8kaCzyRS4Z6w5nHjQwmnCqicohIvaI7cirlOBaGje22kNcgiV9T-fjYlAz6IXMAUAfqdy8JxwcTz65PaWhCQX0C1q-J5To/s640/idf1+theme.png" width="570" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 1<br />
Theme, Idf1 (mm.1-2)</td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPOnOumcM5emGho9CQWC4gMm1A_5NMs_dI13dMUbpzpI2vtNhXO2aArHMQM2FOQQ4WAMBm7m3MqWV33mxZnLq5UDzfUXCyGE8-6bx4g4QDtB2GfO12_GxqeNq9fFdHPDIGStK-JulEQmg/s1600/IdF2+theme.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="219" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPOnOumcM5emGho9CQWC4gMm1A_5NMs_dI13dMUbpzpI2vtNhXO2aArHMQM2FOQQ4WAMBm7m3MqWV33mxZnLq5UDzfUXCyGE8-6bx4g4QDtB2GfO12_GxqeNq9fFdHPDIGStK-JulEQmg/s640/IdF2+theme.png" width="570" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 2<br />
Theme, Idf2 (mm.1-7)</td></tr>
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The basic rhythmic cell in both themes is a pair of notes which we'll designate [>~], where '>' indicates a short stressed note and '~' indicates a long unstressed note.<span style="font-size: x-small;">[1]</span> And usually the accented note is higher in pitch than the unaccented one. These pairs can be heard in the comparatively brief theme of <i>Idf1</i>, but they are especially obvious in <i>Idf2</i>'s lengthier theme. They take three forms (Example 3). Both themes are built rhythmically from these pairs, the added-note features in both themes notwithstanding.<span style="font-size: x-small;">[2]</span><br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp7YtN5sMo18V5g8lW2V_ftTGc80DBSKWNOoxnyI_cLkaqkWOMnyLDhzUWLp7ZntZOt2_IPSnOHtz6ii25U8JfsNgqcEvxaizn7cFU6FMLTgd37_42set_tJ4EUDkBes_jzJbcBYEBFQ4/s1600/hiccups.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="95" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp7YtN5sMo18V5g8lW2V_ftTGc80DBSKWNOoxnyI_cLkaqkWOMnyLDhzUWLp7ZntZOt2_IPSnOHtz6ii25U8JfsNgqcEvxaizn7cFU6FMLTgd37_42set_tJ4EUDkBes_jzJbcBYEBFQ4/s400/hiccups.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 3.</td></tr>
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Added-note rhythmic segments as quasi-palindromes – especially obvious in the <i>Idf2</i> theme – are probably more in keeping within Messiaen's compositional conception.<span style="font-size: x-small;"> </span>This would make the basic building block [~>~], which is also the characteristic incipit in both themes. But trying to fit every foot into the shoe of Messiaen's theory of non-retrogradable rhythms here will leave out other significant structural features. E.g., the <i>Idf2</i> theme can be parsed further as [~>~~], a basic four-note rhythmic contour with the hammer stroke falling on the second note as marked in Example 4. There are two forms: those circled in blue all have similar basic pitch contours, [^vv], while the pitch contours circled in red are inversionally related [^v^] / [v^v]. This also approximates Messiaen's barring.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcyIaEJEXDb688vwRQSRr2ukQIxP0ukbtm2gsqxrQCQNRD_k2s_a6KQlYvRiIy8ynycVkxmBg-6e0qljPVgOW5yseYjGttYeTyZpgsY2_M2luE-3J0wCAKHUyaXGJDR0mpD7EBQ5gluBs/s1600/IDF2+theme+LSLL+contour+analysis.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="219" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcyIaEJEXDb688vwRQSRr2ukQIxP0ukbtm2gsqxrQCQNRD_k2s_a6KQlYvRiIy8ynycVkxmBg-6e0qljPVgOW5yseYjGttYeTyZpgsY2_M2luE-3J0wCAKHUyaXGJDR0mpD7EBQ5gluBs/s640/IDF2+theme+LSLL+contour+analysis.png" width="570" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 4.<br />
(<i>Idf2</i> theme)</td></tr>
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A comparable rhythmic parsing of the <i>Idf1</i> theme (Example 5) produces less straightforward and ultimately less convincing results.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4ivHsOb-uh1tXtv43tGu8KvSWY9V0H1qdcP6hr9tKgLJwDjZUMccYlLkbAd8MPO9yFOWMv4WqiZZYxwzAGG-bFc7vDsN3kSjDSvPAAwEf3UBz2l2BOI0LuJmmnlWPR2AoI4JMbFmzGIE/s1600/rhythmic+parse+idf1+final.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="142" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4ivHsOb-uh1tXtv43tGu8KvSWY9V0H1qdcP6hr9tKgLJwDjZUMccYlLkbAd8MPO9yFOWMv4WqiZZYxwzAGG-bFc7vDsN3kSjDSvPAAwEf3UBz2l2BOI0LuJmmnlWPR2AoI4JMbFmzGIE/s640/rhythmic+parse+idf1+final.png" width="570" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 5.<br />
(<i>Idf1</i> theme)</td></tr>
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The pairs circled in orange are [>~], as in <i>Idf2</i>'s theme, and that almost works. However, going beyond those rhythmic pairs in <i>Idf1</i> strains the basic analytical idea we're testing. Circled in purple at either end are [~>~] structures. Beyond that, things get implausible. We might try to force the theme to fit into this analytical mold by identifying [~>~]'s inverse, [>~>], enclosed by the purple dashed circle around E-F-C#. But then (assuming our ear hasn't stopped us by now) we have to account for the two consecutive sixteenths, F-F#. This rhythmic anomaly could be 'fixed' analytically by eliding the sixteenths into a single 'background' eighth note, reasserting the [~>~] analysis (gray dashed segment with the question mark). This would clearly be overanalyzing only for the sake of asserting a theory, and we can hear Messiaen laughing.</div>
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But there is no need to find in <i>Idf1</i>'s theme all the rhythmic symmetries found in <i>Idf2</i>'s theme in order to establish a strong rhythmic similarity. It is enough to assert conservatively that the two themes are built from [>~] pairs, and beyond that each takes on its own personality.</div>
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<span style="font-family: inherit; text-align: start;"><b><a href="http://essaysandendnotes.blogspot.com/2016/05/broken-symmetries-32.html" target="_blank">Broken Symmetries 3.2: Pitch content</a></b></span></div>
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[1] I was tempted to use terminology from scansion in poetry, but I quickly abandoned this idea. What I was after would be, in poetry, a confusion between the <i>iamb</i> and the <i>trochee </i>due to the accent falling on the short duration and lack of accent on the longer duration. As far as I know, adequate terminology is for the most part missing from analogous musical rhythmic analysis, at least since the <i>ars antiqua</i>. So I opted for using a binary n-tuple notation consisting of the symbol '>' for a variably short, accented hammer note and '~' for an unaccented (or relatively less accented) note of duration longer than or equal to its associated '>'. Beyond this fuzzy 'notation rule', specific measures of accent and duration (e.g., 'accented staccatissimo sixteenth note') are left undefined. This was a <i>pro tem</i> solution, but one might consider it a provisional basis for an analytical theory for rhythmic contour analogous to that for pitch contour.</div>
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[2] <span style="background-color: white; font-family: inherit;">One can also hear these characteristic pairs emerging in the rhythms generated by the pairing of interversions in </span><i style="background-color: white; font-family: inherit;">Idf2</i><span style="background-color: white; font-family: inherit;">'s </span><b style="background-color: white; font-family: inherit;">B</b><span style="background-color: white; font-family: inherit;"> material, but it is difficult to say whether the theme statement, after several hearings, has trained the ear to select for these pairs during the interversion sections, or whether they were pre-compositionally built into, or discovered in, the interversions by Messiaen.</span>stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-54722031564821195422017-06-09T10:06:00.000-04:002017-06-09T10:26:39.203-04:00Broken Symmetries 3.2<br />
<div style="text-align: center;">
<span style="font-size: large;"><b><i>A Tale of Two Tunes</i></b></span></div>
<div style="text-align: center;">
<span style="font-size: large;"><b><i><br />
</i></b></span></div>
<div style="text-align: center;">
<span style="font-size: large;">2. Pitch content</span></div>
<div style="text-align: center;">
<span style="font-size: large;">in the themes of Messiaen's <i>Île de feu I</i> & <i>II</i></span></div>
<br />
In the previous post the comparison of the two themes was based on similarity of rhythmic contours. We now consider a pitch set-class comparison which will provide a transformational relationship amongst the two themes and their initial accompaniments. Once again, for reference, here are the two themes:<br />
<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMWFGLwJD29XLLNHzYD_APxxe_WIuSyRLpOX7Hxhk-Po7nW8kaCzyRS4Z6w5nHjQwmnCqicohIvaI7cirlOBaGje22kNcgiV9T-fjYlAz6IXMAUAfqdy8JxwcTz65PaWhCQX0C1q-J5To/s1600/idf1+theme.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="142" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMWFGLwJD29XLLNHzYD_APxxe_WIuSyRLpOX7Hxhk-Po7nW8kaCzyRS4Z6w5nHjQwmnCqicohIvaI7cirlOBaGje22kNcgiV9T-fjYlAz6IXMAUAfqdy8JxwcTz65PaWhCQX0C1q-J5To/s640/idf1+theme.png" width="570" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 13px;">Example 1<br />
Theme, Idf1 (mm.1-2)</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPOnOumcM5emGho9CQWC4gMm1A_5NMs_dI13dMUbpzpI2vtNhXO2aArHMQM2FOQQ4WAMBm7m3MqWV33mxZnLq5UDzfUXCyGE8-6bx4g4QDtB2GfO12_GxqeNq9fFdHPDIGStK-JulEQmg/s1600/IdF2+theme.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="219" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPOnOumcM5emGho9CQWC4gMm1A_5NMs_dI13dMUbpzpI2vtNhXO2aArHMQM2FOQQ4WAMBm7m3MqWV33mxZnLq5UDzfUXCyGE8-6bx4g4QDtB2GfO12_GxqeNq9fFdHPDIGStK-JulEQmg/s640/IdF2+theme.png" width="570" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 13px;">Example 2<br />
Theme, Idf2 (mm.1-7)</td></tr>
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</div>
<br />
We first need to visualize the pitch-class material as nested sets. In Example 3 a black box indicates total pitch-class material in the indicated segment, a grey box indicates the characteristic set for that segment, and a red box indicates a linking tetrachord. (Technically, Example 3b should have a grey box labelled, say, <b>X</b> surrounding the red box and a black box labelled <b>X*</b> around that grey one, but considering that this tetrachord shares all three of those descriptions (<b>J'</b> = <b>X</b> = <b>X*</b>), just the red box is sufficient to show the transformation link we're headed for.)<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0xc3SmpM_snMpxRLRSQPaeXHbCG3tp7tOmquXUB1CTo4FFgYyaLITnB3E-9oWK_ARBwgVfLdQWllXabXNjTNE7-MLuTmCSek6RnsovnK0I8_Q7WJYQUk9VrSkWL3D1EIS_EVYvqnJHIw/s1600/Nests+Idf1%25262.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="582" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0xc3SmpM_snMpxRLRSQPaeXHbCG3tp7tOmquXUB1CTo4FFgYyaLITnB3E-9oWK_ARBwgVfLdQWllXabXNjTNE7-MLuTmCSek6RnsovnK0I8_Q7WJYQUk9VrSkWL3D1EIS_EVYvqnJHIw/s640/Nests+Idf1%25262.png" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example 3.</td></tr>
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The object of this game is to draw arrows that connect Examples 3a, 3b, 3c, and 3d. To begin, we note that the total pitch-class content of <i>Idf1</i>'s theme, enclosed by the black box in Example 3a, is the set <b>K*</b> = {C#, E, F, F#, G, G#}. (Ignore the parenthetic A# for now. We'll get to it shortly.) Note that the five-note subset of <b>K*</b> marked by the grey box is <b>K</b> = {E, F, F#, G, G#}, SC[01234], a segment of the chromatic collection. To emphasize this structural characteristic we can write<br />
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<div style="text-align: right;">
<div style="text-align: center;">
<b> K*</b> = {<b>K</b>, C#} <i>(1)</i></div>
</div>
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Next, the pitch-class content of <i>Idf2</i>'s theme (Example 3c) is <b>P*</b> = {E, G, A, A#, C, C#, D}. The characteristic subset of <b>P*</b> is the pentatonic collection <b>P</b> = {E, G, A, C, D}, SC[02479], so to emphasize that structural characteristic we can write a partition of <b>P*</b> as<br />
<br />
<div style="text-align: right;">
<div style="text-align: center;">
<b> P*</b> = {<b>P</b>, A#, C#} <i>(2)</i></div>
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<i><br />
</i></div>
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We know that the two set classes [01234] and [02479] are homothetically related by <i>M</i><span style="font-size: x-small; font-style: italic;">5</span><i>/M</i><span style="font-size: x-small;"><i>7</i></span> (mod12); so there must be some pitch-class transposition such that<span style="font-family: inherit; text-align: center;"> <i>T<span style="font-size: x-small;">n</span>M<span style="font-size: x-small;">5/7</span></i>(<b>K</b></span><span style="font-family: inherit; text-align: center;">) =</span><span style="font-family: inherit; text-align: center;"> </span><b style="font-family: inherit;">P</b><span style="font-family: inherit; text-align: center;">, and indeed:</span><br />
<span style="font-family: inherit; text-align: center;"><br />
</span> <span style="font-family: inherit; text-align: center;"></span><br />
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<span style="font-family: inherit;"><i> T<span style="font-size: x-small;">8</span>M<span style="font-size: x-small;">5</span></i>(<b>K</b>) = <b>P</b><span style="font-size: normal;">. <i>(3)</i></span></span></div>
<div>
<span style="font-family: inherit;"><span style="font-size: normal;"><i><br />
</i></span></span></div>
Now what about the C# and A#? The parenthetic A# doesn't appear in the <i>Idf1</i> theme at all. But when looking initially at the symmetric transformation relationship between the two themes, it is revealing to include it provisionally as an imaginary element or 'ghost presence' in the <i>Idf1</i> theme's pc content. So let's posit a set <b>K*+</b> that includes the A#,<br />
<br />
<div style="text-align: center;">
<span style="font-family: inherit;"><span style="text-align: center;"><b> K*+</b> = {<b>K*</b>, A#} = </span><span style="text-align: center;">{<b>K</b></span><span style="text-align: center;"><span style="font-size: normal;">, </span></span><span style="text-align: center;">A#, C#</span><span style="text-align: center;">} <i>(4)</i></span></span><br />
<div>
<span style="font-family: inherit;"><span style="text-align: center;"><i><br />
</i></span></span></div>
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and pretend, for now, that Messiaen didn't find any need to use that A# until <i>Idf2</i>'s theme; or, conversely, after its inclusion in the construction of the <i>Idf2</i> them, he dropped the A# in making <i>Idf1</i>'s theme. Now we can summarize the entire transformation relationship between the pitch content of the two themes (Examples 3a and 3c) as either<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: inherit;"><i> T<span style="font-size: x-small;">8</span>M<span style="font-size: x-small;">5</span></i>(<b>K*+</b>) = <b>P*</b><span style="font-size: normal;">, <i>(5)</i></span></span></div>
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<span style="font-family: inherit; font-size: normal;"><br />
</span><span style="font-family: inherit; font-size: normal;">or, going the other direction, its inverse</span></div>
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<span style="font-family: inherit; font-size: normal;"><i><br />
</i></span></div>
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<span style="font-family: inherit;"><span style="font-size: normal;"><i> M<span style="font-size: x-small;">5</span></i><i>T<span style="font-size: x-small;">4</span></i>(<b>P*</b></span>) = <b>K*+</b><span style="font-size: normal;">. <i>(6)</i></span></span></div>
<span style="font-family: inherit; font-size: normal;"></span><br />
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<span style="font-family: inherit; font-size: normal;"><br />
</span></div>
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This suggests at least one reason why (assuming Messiaen started (pre)compositionally from <i>Idf1</i>), after he applied the <i>M</i><span style="font-size: x-small; font-style: italic;">5</span> transformation to <b>K*</b> (or <b>K*+</b>), he then transposed it by <i>T</i><span style="font-size: x-small; font-style: italic;">8</span> to get the content for <i>Idf2</i>, because that's the only transposition that regains the C# and reifies the ghost element A# after <i>M</i><span style="font-size: x-small; font-style: italic;">5</span> multiplication.<br />
<br />
<span style="font-family: inherit;">Also note that <b>K*+</b> and<span style="text-align: center;"> <b>P*</b></span><span style="text-align: center;"> share a diminished seventh chord,</span></span><br />
<br />
<div style="text-align: center;">
<b> K*+</b> <span style="font-size: medium;"><span style="font-size: x-large;">∩</span><span style="font-size: large;"> </span></span><b>P*</b> = {A#, C# E, G}. <i>(7)</i></div>
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In the final post in this thread we will see that the A# marks a 'cut' of the theme that is tempting to call a half cadence. It's at the A# that Messiaen cuts the <i>Idf2</i> theme into two halves and reverses their order to use as part of connecting material leading into the final Magma Dance. The diminished triad {C#, E, G} is used to create quasi-cadential patterns in both themes.<br />
<br />
<span style="font-family: inherit;"><br />
</span> <span style="font-family: inherit;">Next let's consider the initial accompaniment for </span><i style="font-family: inherit;">Idf2</i><span style="font-family: inherit;">'s theme in the right hand in mm.1-7 (Example 4).</span><br />
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<span style="font-family: inherit;"><br />
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<tr><td><img border="0" height="628" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1aUwsd10_GzsUCRWSgeWd4XVp7fEp9Z97VZszi3OaK1pVS39eMprZbcQ9pD-BP96ECXy04Hr4XPSX5qypLl2kUPaFQNjQ1k40NtamnBDhzg7F5k7qAlxaosnPSV0hk7PY-Gm4vZ9mXus/s640/mm1-7.png" style="margin-left: auto; margin-right: auto;" width="640" /></td></tr>
<tr><td class="tr-caption" style="font-size: 13px;">Example 4.</td></tr>
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With the exception of the notes circled in red, the entire pitch-class content of the homorhythmic accompaniment in the right hand in mm.1-7 is the set <b>H</b> = {D#, E#, F#, G#, A#, B, C#}. This belongs to SC[013568t], the 'usual diatonic' set class, which is the set-theoretic complement of <i>Idf2</i> theme's characteristic set, the pentatonic SC[02479]:<br />
<br />
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<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: inherit;"><span style="font-size: normal;"><i> </i><b>H</b> = <i>COMPL</i>(<b>P)</b></span><span style="font-size: normal;"> <i>(8)</i></span></span></div>
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<span style="font-family: inherit; font-size: normal;"></span></div>
As noted, what ruins the analytical case for the entire accompaniment <b>H*</b> being a strictly diatonic set is the presence of those four spoilers circled in red: <b>Q'</b> = {G, A, C, D}, SC[0257]. So for the complete accompaniment content <b>H*</b> we have<br />
<br />
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<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: inherit;"><span style="font-size: normal;"><i> </i><b>H*</b> = {<b>H</b>, <b>Q'</b>}</span><span style="font-size: normal;"> <i>(9)</i></span></span></div>
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<span style="font-family: inherit; font-size: normal;"></span></div>
<b>H*</b> can also be expressed as a string of eleven equally tempered perfect fifths or minor seconds (short the E-natural it would take to make a complete circle), however, focussing on <b>Q'</b> in the accompaniment, which is also found as <b>Q</b> in the theme's characteristic set <b>P</b>, provides the key to unlocking relationships binding the two themes and their initial accompaniments.<br />
<br />
<span style="font-family: inherit;">Looking at mm1-7 (Example 4) only as a sequence of homorhythmic verticals will miss a salient feature of the accompaniment. First fix the elements of the unordered set </span><b style="font-family: inherit;">Q</b><span style="font-family: inherit;"> to create the descending string [D-C-A-G], then transpose up a minor second to form the string q = [D#-C#-A#-G#]. Tracing q and its retrograde and transpositions (Example 7) reveals that it knits together nearly all the pitch-class pairs making up the r.h. accompaniment for <i>Idf2</i>'s theme. The pairs untouched by q and its transformations, with the exception of m.7, can be reached by permuting the string [D-C-A-G] into q' = [C-G-A-D] (reading the C as B#). This finds all the notes circled in red in Example 4. Measure 7 is quasi-cadential in the r.h. The voice leading by interval-class 3, F→G# & F#→D#, is broken by Messiaen to create a final decisive hammer stroke – a gesture that lends greater force to the final two notes in the l.h.; keep in mind this motion by ic3, especially as a cadential move.</span><br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVvuGZ7RAuHNSybsP9FWabLkNqBwLWAkSNd2mYAtk4to2Y_ZPWHO4a6c8dUv9QM1bNZZ0ktwYy4F59afj1AuuLKOHtxyvYElQEDgGr4LZHizQUo2r1uz39zSksYWo0KLi0AuFQvKjcC9E/s1600/SC0257+in+idf2+acc.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="509" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVvuGZ7RAuHNSybsP9FWabLkNqBwLWAkSNd2mYAtk4to2Y_ZPWHO4a6c8dUv9QM1bNZZ0ktwYy4F59afj1AuuLKOHtxyvYElQEDgGr4LZHizQUo2r1uz39zSksYWo0KLi0AuFQvKjcC9E/s640/SC0257+in+idf2+acc.png" width="570" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 13px;"><span style="font-size: small;">Example 5<br />
q-strings in <i>Idf2</i>, mm1-7 (rh)<br />
(brought down 2 octaves for easier reading)</span><br />
<div>
<span style="font-size: small;"><br />
</span></div>
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<span style="font-family: inherit;">We are now nearly finished with a tour around an analytical schema that binds the significant features of the themes and accompaniments for <i>Idf1</i> and <i>Idf2.</i> Example 6 shows where we've been in this transformation analysis so far, as well as where we are now heading.</span><span style="font-family: inherit;"><br />
</span><br />
<span style="font-family: inherit;"></span><br />
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<span style="font-family: inherit;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfRBjqwtzukBi-lWVakWV2SFB0yeL3gt32_SKZ0ZWLJCoMTj-4afJMK2Jyst81iwZWvwmr7dfBdcgrbexJQki6KbrJ9UNg04jk9avgPC8ifC4BBjagqcRVr63llt584s5nDUxUm36bHEI/s1600/idf+1%25262+circle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: inherit;"></span></a></span></div>
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<img border="0" height="432" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfRBjqwtzukBi-lWVakWV2SFB0yeL3gt32_SKZ0ZWLJCoMTj-4afJMK2Jyst81iwZWvwmr7dfBdcgrbexJQki6KbrJ9UNg04jk9avgPC8ifC4BBjagqcRVr63llt584s5nDUxUm36bHEI/s640/idf+1%25262+circle.png" style="cursor: move; margin-left: auto; margin-right: auto;" width="640" /></div>
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<tr><td class="tr-caption" style="padding-top: 4px; text-align: center;"><div style="margin: 0px;">
<span style="font-size: small;">Example 6.<br />
Circle of transformations relating <i>Idf1</i> and <i>Idf2</i>.</span></div>
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<span style="font-family: inherit;">To close the circle, we 'flatten' <b>Q'</b> back to a four-note segment of the chromatic:</span><br />
<span style="font-family: inherit;"><br />
</span> <br />
<div style="text-align: center;">
<span style="font-family: inherit;"><i>M</i><span style="font-size: x-small; font-style: italic;">5</span><b>Q'</b> = {A, Bb, B, C} = <b>J'</b> <i>(10)</i></span></div>
<div style="text-align: left;">
<span style="font-family: inherit;"><i><br />
</i></span></div>
<div style="text-align: left;">
<span style="font-family: inherit;">Then returning to <i>Idf1</i>'s tetrachord is a simple transposition:</span></div>
<div style="text-align: left;">
<span style="font-family: inherit;"><br />
</span></div>
<div style="text-align: center;">
<span style="font-family: inherit;"> T<span style="font-size: x-small;">8</span><b>J'</b> = <b>J</b> <i>(11)</i></span></div>
<div style="text-align: left;">
<br /></div>
While <b>Q</b> is consistently expressed as a melodic cell in <i>Idf2</i>'s accompaniment, <b>J'</b> (without the C-natural that appears only in the 32nds that echo the 16ths up in the theme) is expressed as a tone cluster and treated as percussion in <i>Idf1</i>'s drumming accompaniment. (See Example 7 [C-flat was substituted for the published edition's B-natural for clarity].)<br />
<span style="font-family: inherit;"></span><br />
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<span style="font-size: small;">Example 7.<br />
<i>Idf1</i>, mm.1-2</span></div>
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<span style="background-color: #4c1130; color: white; font-family: inherit;"> The final installment in this thread, </span></div>
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<span style="background-color: #4c1130; color: white; font-family: inherit;"> 'Broken Symmetries 3.3: Tonality, a.k.a. Strange attractors', </span></div>
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<span style="background-color: #4c1130; color: white; font-family: inherit;"> is under construction as of 5 May 2016 </span></div>
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</span> <b>_____________________</b><br />
<b><br />
</b> [1] The letter-symbols I'm using here may be confusing but they are not entirely random. Since I am using only the QWERTY keyboard in order to avoid characters that will be unreadable for some browsers/devices, it's necessary to reserve the traditional A-through-G letter names for pitches as well as the now standard transformation symbols, T, I, R and M. In this section (if it will help remember and keep the sets straight) I've chosen 'suggestive' symbols: <b>K</b> is for 'chromatic', <b>P</b> is for 'pentatonic', <b>H</b> is for 'heptatonic', and <b>Q</b> is for the French 'quatre'. I have no idea where I got the <b>J</b> from (maybe from <b>J</b><<b>K</b> alphabetically).<br />
<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-87781413638416038872016-11-18T11:41:00.000-05:002016-11-18T11:41:11.313-05:00Removed postsFive posts analyzing Olivier Messiaen's <i style="font-family: inherit;">Île de feu II</i><span style="font-family: inherit;"> (thread title:</span> 'Broken Symmetries') have been temporarily removed from this blog while I rewrite and consolidate them with additional material for an article to be submitted for publication.stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-74711101600045101712016-10-24T17:35:00.000-04:002020-11-21T14:52:33.479-05:00Babbitt crypto<br />
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<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-18386289148082702016-09-20T22:55:00.001-04:002017-05-31T07:24:36.674-04:00Ode on a Cretan MazeThis one is just for fun.........<br />
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<blockquote class="tr_bq">
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<blockquote>
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<i>It may be that universal history is the history of the different intonations given a handful of metaphors.</i></blockquote>
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– Borges, 'The Fearful Sphere of Pascal'</div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6o5agfnFXKXEwY5XyDKBYKU4Hh9a-37QUCtEf2rsHXi7qNSZm-Ewg4JKHqtmrQ8oUdZq5nnEo_itPg34HF-kN9f_f6oLYz06P3XRshR8lUXiwfBBmUEtvJIPzxCySbPuDjbgKodxT87I/s1600/MazePylos.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6o5agfnFXKXEwY5XyDKBYKU4Hh9a-37QUCtEf2rsHXi7qNSZm-Ewg4JKHqtmrQ8oUdZq5nnEo_itPg34HF-kN9f_f6oLYz06P3XRshR8lUXiwfBBmUEtvJIPzxCySbPuDjbgKodxT87I/s320/MazePylos.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The Pylos Maze (tablet pre-1200 BCE)*</span></td></tr>
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<tr><td class="tr-caption" style="text-align: center;">(a) (b)<br />
Illustration 1.<br />
Sketch for analysis of the Pylos Maze,<br />
rotated clockwise 90 degrees from photo above<br />
(Prof. Tony Phillips)</td></tr>
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Illustration 1 is taken from the article '<a href="http://www.ams.org/samplings/feature-column/fc-2015-10">Hidden Symmetries of Labyrinths from Antiquity and the Middle Ages</a>' by <a href="http://www.math.stonybrook.edu/~tony/">Tony Phillips</a> appearing in the American Mathematical Society's October 2015 <a href="http://www.ams.org/samplings/feature-column/fc-editors">Feature Column: Monthly essays on mathematical topics</a>. As Phillips describes it:<br />
<blockquote class="tr_bq">
The oldest securely dated labyrinth design appears on the back of an accounting tablet found in the ruins of King Nestor's palace in Pylos, on the western shore of Greece. The tablet (7 X 5.7 cm) was baked when the palace burned in 1200 BC. The design on this tablet is often called the Cretan Maze. It appears on coins minted on Knossos on Crete during the 5th century BC and later, and represented the legendary labyrinth where King Minos kept the Minotaur.</blockquote>
The aim of the article as a whole is 'to point out that the great majority of labyrinth designs share a <i>topological</i> symmetry which, <i>while not obvious, cannot be accidental</i>.' Phillips works through examples of several labyrinths (the reader is encouraged to go to the entire article), but I'd like to concentrate on the Cretan Maze as another illustration of hidden or 'deep' form – one of the main themes in <i>Essays & Endnotes</i>.<br />
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If you stare at the labyrinth on the left as it appears on the tablet (Illustration 1a), you realize that, considered as a 2-dimensional rigid figure, it's <i>asymmetric</i> – you can't rotate or flip it to get an identical figure. So first, Phillips labels the <i>levels</i> of the labyrinth in red, starting on the outside with 0 and ending at the goal with 8 (Illustration 1b). Now enter the labyrinth at 0 (at the opening on the right) and follow the path. Each time you cross the line of red numbers, record the number you cross. The 'level sequence' you come up with will be <span style="text-align: center;">032147658. Phillips reveals the hidden symmetry by first noting that 'if each number <i>n</i> is eight's complement 8–<i>n</i> [in music theory terminology this is "inversion" I<span style="font-size: x-small;">8 </span>(mod 9)] the sequence becomes 856741230, which is the original sequence read backwards [retrograde]' – the way to back out of the labyrinth. He then "lifts" the labyrinth into three dimensions by wrapping it around four faces of a cube (see his illustrations in the article) which ultimately reveals its hidden rotational symmetry. </span><br />
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Using the pictorial representation of permutations that we've used previously (beginning with <a href="http://essaysandendnotes.blogspot.com/2013/04/the-form.html">spiral representations</a>), after some unscrambling of the original labyrinth using Phillips' labelling of the levels, we can see that Illustration 1b is topologically identical to Illustration 2.</div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj008mjVouCowG5z6K_49XmlaVUgX_9tYwyrFdjTymjAJXEezF3CafdXkTDca6hdD1oKAHzwwBU6XZxi22bnciCV_F-E5Uh41OxrH7akdbpn8K7pMeGhdAqfhB0AgqtJR488pCEK3XGGJY/s1600/Cretan+Maze+2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj008mjVouCowG5z6K_49XmlaVUgX_9tYwyrFdjTymjAJXEezF3CafdXkTDca6hdD1oKAHzwwBU6XZxi22bnciCV_F-E5Uh41OxrH7akdbpn8K7pMeGhdAqfhB0AgqtJR488pCEK3XGGJY/s320/Cretan+Maze+2.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Illustration 2.</td></tr>
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So the labyrinth path can be analyzed as a concatenation of two identical paths,<br />
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(<i>n</i>, <i>n</i>+3, <i>n</i>+2, <i>n</i>+1, <i>n</i>+4) <i>(1)</i></div>
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with <i>n</i>=0,4. Tracing these paths in <b>K</b><span style="font-size: x-small; font-weight: bold;">9</span> (mod 9 circle space) with red for <i>n</i>=0 and blue for <i>n</i>=4 (Illustration 3) gives us another view of labyrinth permutation.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcyTTdtwDwkroSSSX-23Ok5xsAuFDdX70nESr3h20a66BVn21IxrSQxmqeQz9hFbQaN2SNBH_audUqQYCBfQtabeont10ehqs6yTFIwpWj9ezlzZ7uruBH_drRisg5zMzAvtOML75fV8k/s1600/Cretan+circle.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcyTTdtwDwkroSSSX-23Ok5xsAuFDdX70nESr3h20a66BVn21IxrSQxmqeQz9hFbQaN2SNBH_audUqQYCBfQtabeont10ehqs6yTFIwpWj9ezlzZ7uruBH_drRisg5zMzAvtOML75fV8k/s320/Cretan+circle.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Illustration 3.</td></tr>
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Now if we let <i>n</i> take the values 0,4,8 we get the obvious extension of the Pylos labyrinth permutation (Illustration 4).<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEelLEQEdRl-k5MZJMtTZuSFzr46w5Ku6MOXoKEO11of9-hBxXAj_HUTxrgBVrkumJpYq0Pu865w0BXCaDsyHCGKFT0n3y-CyMn7-QXmv7FQWYpwb2E9du2jGa8mrfVGgQVThYpnOx9sI/s1600/Maze+mod+13.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="101" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEelLEQEdRl-k5MZJMtTZuSFzr46w5Ku6MOXoKEO11of9-hBxXAj_HUTxrgBVrkumJpYq0Pu865w0BXCaDsyHCGKFT0n3y-CyMn7-QXmv7FQWYpwb2E9du2jGa8mrfVGgQVThYpnOx9sI/s400/Maze+mod+13.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Illustration 4.</td></tr>
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And then, working backward, we can translate this into a maze of three 'levels' by surrounding the original two-level maze with another level:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnjsve6hMJK4Ln6djg5YwMw-J_JBnWkwwQeXVvH05kJXJM-OwkEX3CXZ8QykpjC21Jvm6O4Fb8SzVMAopdRJWPYruudp2lK-JkU2OHzr86dMGHlNrB0wfXD3PbpCnh4jFqZcVnBbHtT2k/s1600/maze+A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="274" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnjsve6hMJK4Ln6djg5YwMw-J_JBnWkwwQeXVvH05kJXJM-OwkEX3CXZ8QykpjC21Jvm6O4Fb8SzVMAopdRJWPYruudp2lK-JkU2OHzr86dMGHlNrB0wfXD3PbpCnh4jFqZcVnBbHtT2k/s320/maze+A.png" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZler57Db2nUeryPX9NSdPL0ejG18QGIYeqRI6F5NqTb0IUyTH4RSY5lrtKWykeWsyiwa07-wDvFSCk_9VlQCciYgL704lH7HJZHJCIboge_aoVa7vi8oMqoktzTe9LORr8lgPBiYDpkE/s1600/maze+B.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="263" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZler57Db2nUeryPX9NSdPL0ejG18QGIYeqRI6F5NqTb0IUyTH4RSY5lrtKWykeWsyiwa07-wDvFSCk_9VlQCciYgL704lH7HJZHJCIboge_aoVa7vi8oMqoktzTe9LORr8lgPBiYDpkE/s320/maze+B.png" width="320" /></a></div>
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<span style="font-size: x-large;">=</span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiP1Wlz0gvHsna1FrGmi2NkKZcG3vwA9kl3OWlNoTKq18HmS1mlt48PVOWUWfpdbflvB80MyNP0BIMW8OOybel0sqS1Yz9cJ-FnNM3NdrQ_8du-Uz2hheP4Lf1ezzW3G4tavOlrSPYAi5A/s1600/maze+C.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="263" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiP1Wlz0gvHsna1FrGmi2NkKZcG3vwA9kl3OWlNoTKq18HmS1mlt48PVOWUWfpdbflvB80MyNP0BIMW8OOybel0sqS1Yz9cJ-FnNM3NdrQ_8du-Uz2hheP4Lf1ezzW3G4tavOlrSPYAi5A/s320/maze+C.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Illustration 5.</td></tr>
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We can indefinitely extend expression <i>(1)</i> that generates the permutation (and the labyrinth) by letting <i>n</i>=4<i>k</i> with <i>k</i>=0,1,2,3,.... Do we then have a labyrinth that has a center but no outside? If we begin at the center and some demon keeps adding to <i>k</i> it will take us an infinite amount of time to get out; that much is clear. But, with a nod to Zeno and <i>pace</i> Cantor, exactly where could one enter such a labyrinth?<br />
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* Photograph from <a href="http://mathworld.wolfram.com/about/author.html" style="color: #006699; font-family: Arial, Helvetica, sans-serif; text-decoration: none;">Weisstein, Eric W.</a><span style="background-color: white; font-family: "arial" , "helvetica" , sans-serif;"> "Maze." From </span><a href="http://mathworld.wolfram.com/" style="color: #006699; font-family: Arial, Helvetica, sans-serif; text-decoration: none;"><i>MathWorld</i></a><span style="background-color: white; font-family: "arial" , "helvetica" , sans-serif;">--A Wolfram Web Resource. </span><a href="http://mathworld.wolfram.com/Maze.html" style="color: #006699; font-family: Arial, Helvetica, sans-serif; text-decoration: none;">http://mathworld.wolfram.com/Maze.html</a></div>
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<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-66350886509360191592015-04-16T11:16:00.000-04:002015-06-14T16:40:59.280-04:00The Spiral Form – Developing VariationsTo pick up a previous thread, we first need a quick review:<br />
<br />
Two years ago in Essays & Endnotes I began investigating a web of unlikely connections within and between seemingly unrelated areas. It began with "pure mathematics" inspired by a metaphor from nature –<br />
<div style="text-align: left;">
<blockquote class="tr_bq">
<a href="http://essaysandendnotes.blogspot.com/2013/04/the-form.html">The Form</a></blockquote>
This abstract pattern led to a connection to a poetic form invented by a 12th-century troubadour –<br />
<blockquote class="tr_bq">
<a href="http://essaysandendnotes.blogspot.com/2013/04/a-proper-sestina.html">The Sestina</a></blockquote>
The same pattern was also noted in card shuffling techniques that probably track back to the first known playing cards in the 9th century –<br />
<blockquote class="tr_bq">
<a href="http://essaysandendnotes.blogspot.com/2013/04/jeu-de-cartes.html">Jeu de Cartes</a></blockquote>
</div>
<div style="text-align: left;">
Then the form crossed paths with computer science in the 20th century –<br />
<blockquote class="tr_bq">
<a href="http://essaysandendnotes.blogspot.com/2013/07/dont-panic-parallel-processing-for.html">Parallel Processing</a></blockquote>
The pattern was also uncovered by a 19th-century musicologist as one basis for diatonic theory in music –<br />
<blockquote class="tr_bq">
<a href="http://essaysandendnotes.blogspot.com/2013/07/the-hauptmann-shuffle.html">Hauptmann Shuffle</a></blockquote>
Four further real-world appearances of the pattern –– (1) 3-card Monte, (2) tritina poetry, and (3) a string of triads in music generated by (4) a parallel processing routine from computer science –– all went into a single post to summarize and further illustrate the usefulness of the form in disparate contexts –<br />
<blockquote class="tr_bq">
<a href="http://essaysandendnotes.blogspot.com/2013/08/the-grifter-poet-composer.html">The Grifter, the Poet & the Composer</a></blockquote>
<br />
Earlier I spoke of two fairly recent applications of the spiral form in music. It's time now to take up this investigation once again to explore those applications and then close with potential compositional variations and strategies suggested by the spiral form. The summary question is: How far can we push the spiral form in compositional music theory?<br />
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Here is where the next post will eventually lead:</div>
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<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com0tag:blogger.com,1999:blog-2569892606315203117.post-65433302840819549162015-03-29T13:05:00.000-04:002015-04-06T13:47:34.014-04:00Notes from the Pluriverse {14–16} (A mythology for music theory today)<div style="text-align: center;">
<span style="font-weight: bold;"><br /></span>
<span style="font-size: large; font-weight: bold;"><i>A MYTHOLOGY FOR MUSIC THEORY TODAY</i></span><br />
<span style="font-weight: bold;"><br /></span></div>
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<b>{14}</b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidip5UMhT5S8k-zHHLEzDYnnz9D7ostcERUcOuP61DqQX9uXEF_KRxd1wVjrKEnuciA_n3pCj4PgK3BDwlXOEMybR7SvByq_IklYtA5clQ8UYFEd5ierQozxc4f1ZD-QgHfkVbbebVtbU/s1600/330px-Janus.xcf.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidip5UMhT5S8k-zHHLEzDYnnz9D7ostcERUcOuP61DqQX9uXEF_KRxd1wVjrKEnuciA_n3pCj4PgK3BDwlXOEMybR7SvByq_IklYtA5clQ8UYFEd5ierQozxc4f1ZD-QgHfkVbbebVtbU/s1600/330px-Janus.xcf.png" height="320" width="296" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik6SulUd8y70C5GpCknsxUXPnjadRhjWMii33dXAw9EIfJ4YkRrrBvTcGJtUqVnxOC-J7T-I-1243XvC05cCZjRv3pe3e1RZXGfSomM6Hs8zM_3SMQGP3oZjrCxGyf5JNxi5s47IRD7Qc/s1600/gradus_t_p.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik6SulUd8y70C5GpCknsxUXPnjadRhjWMii33dXAw9EIfJ4YkRrrBvTcGJtUqVnxOC-J7T-I-1243XvC05cCZjRv3pe3e1RZXGfSomM6Hs8zM_3SMQGP3oZjrCxGyf5JNxi5s47IRD7Qc/s1600/gradus_t_p.png" height="320" width="194" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6SrX36APUPRM-fzjRXvKO4q_i6MjhvVyp9DRZE2HlpxilWAPB3NTam-y52JA8REbkQ21IgszdqOAnrYLuF4PS4tXFfWg1AOHNGJKNiukYxCGncLBkKLwsbzfc5ZS3z8kSJsUh0Ed525U/s1600/Rameau_Traite_de_l%E2%80%99harmonie.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6SrX36APUPRM-fzjRXvKO4q_i6MjhvVyp9DRZE2HlpxilWAPB3NTam-y52JA8REbkQ21IgszdqOAnrYLuF4PS4tXFfWg1AOHNGJKNiukYxCGncLBkKLwsbzfc5ZS3z8kSJsUh0Ed525U/s1600/Rameau_Traite_de_l%E2%80%99harmonie.jpg" /></a></div>
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In just seven years we will celebrate the tercentenary of Jean-Philippe Rameau's <i>Traité de l'harmonie réduit à ses principes naturels </i>(1722).</div>
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Three years later, we will celebrate the tercentenary of Johann-Joseph Fux' <i>Gradus ad Parnassum </i>(1725).<br />
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Juxtaposition of Fux and Rameau offers a mid-stream snapshot of a fundamental bifurcation plaguing/driving (take your pick) the Common Practice Period. The Roman Janus keeps reappearing in countless guises in unexpected places. Janus was slain by Clio, the muse of history. (This is <i>my</i> mythology; I can write it however I want.) Today the Greek Hydra has been reborn from Janus' honored remains. She now lurks among us, both plaguing and driving contemporary composition and theory.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFas6NUhUSnYS3Z0DPVZx9CY2wyOdOwpn5Y1dc-dHcX9lTYZRCRWZKD7XT9lvl5lYepegE3-UxI2yRjLZzBXqj3AAfCrwg2K4vcpGfw9J0dGAd3b9y8Cnr3MGr1Rkd0eVnegjxi35TRm0/s1600/Hercules_slaying_the_Hydra.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFas6NUhUSnYS3Z0DPVZx9CY2wyOdOwpn5Y1dc-dHcX9lTYZRCRWZKD7XT9lvl5lYepegE3-UxI2yRjLZzBXqj3AAfCrwg2K4vcpGfw9J0dGAd3b9y8Cnr3MGr1Rkd0eVnegjxi35TRm0/s1600/Hercules_slaying_the_Hydra.jpg" height="220" width="320" /></a></div>
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<span style="font-weight: bold;">{15}</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinynU24-oa7OBKqmfR7l4ccToEj-yuMvGyp18KAXHxP_JwhBxgi0_-Yqpr-y9P1hZxR3R7LuQV-cZC4r7QdukpBQfMef2rZLh5-udX8eETvzg4DEDuKa1aep0DoT1AbU7q93ENjXVYELo/s1600/gradus+frontispiece.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinynU24-oa7OBKqmfR7l4ccToEj-yuMvGyp18KAXHxP_JwhBxgi0_-Yqpr-y9P1hZxR3R7LuQV-cZC4r7QdukpBQfMef2rZLh5-udX8eETvzg4DEDuKa1aep0DoT1AbU7q93ENjXVYELo/s1600/gradus+frontispiece.png" height="640" width="387" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 13px;">Frontispiece from the 1725 edition of<i> </i>Fux' <i>Gradus ad Parnasum</i><br />
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<u>Scene:</u></div>
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Josephus has completed his climb up the steps to the top of Mount Parnassus.</div>
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Surrounded by the Nine Muses, he receives the laurel wreath from Apollo.</div>
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In the background Pegasus charges over Mount Helicon</div>
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where his hoof strikes a rock, creating the Hippocrene spring,</div>
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fount of poetic inspiration.</div>
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<u>Question:</u></div>
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As he bows to receive the wreath,</div>
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is Josephus holding his first musical work</div>
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or his completed counterpoint exercises?<br />
– or –<br />
Can you reach Helicon from Parnassus?</div>
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<span style="font-weight: bold;">{16}</span><br />
<span style="font-weight: bold;">[added Apr 5 2015]</span><br />
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<blockquote style="-webkit-text-stroke-width: 0px; color: black; font-family: Times; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: center; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
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<a href="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Klee,_Angelus_novus.png/200px-Klee,_Angelus_novus.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Klee,_Angelus_novus.png/200px-Klee,_Angelus_novus.png" height="320" width="244" /></a> "A Klee painting named <i>Angelus Novus</i> shows an angel looking as though he is about to move away from something he is fixedly contemplating. His eyes are staring, his mouth is open, his wings are spread. This is how one pictures the angel of history. His face is turned toward the past. Where we perceive a chain of events, he sees one single catastrophe which keeps piling wreckage upon wreckage and hurls it in front of his feet. The angel would like to stay, awaken the dead, and make whole what has been smashed. But a storm is blowing from Paradise; it has got caught in his wings with such violence that the angel can no longer close them. The storm irresistibly propels him into the future to which his back is turned, while the pile of debris before him grows skyward. This storm is what we call progress." (Walter Benjamin)</div>
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com0tag:blogger.com,1999:blog-2569892606315203117.post-20970926563205808862015-03-11T17:46:00.000-04:002015-07-09T08:56:49.879-04:00Notes from the Pluriverse {7–13 }<div style="text-align: center;">
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{7}<br />
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<i>Seeing the nose on my face</i></div>
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"... to show the fly the way out of the fly-bottle." (Wittgenstein)</div>
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From within a/the world, I cannot see that world. A model is not a/the world, but a house of mirrors in which (imperfectly) to <i>view</i> a/the world. The model, flaunting the impossible, allows me to see into a/the world <i>from outside</i> – allows me to see (approximately) how that world<i> relates</i> itself to itself and, ideally, to other worlds (if they exist).<br />
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... but the fly is still in its bottle.<br />
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{8}</div>
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<i>Dueling ontologies</i></div>
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pluriverse:</div>
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I don't live in <i>the</i> world, but in <i>the</i> model of <i>a</i> world.</div>
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vs</div>
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universe:</div>
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I don't live in <i>a</i> world, but in <i>a</i> model of <i>the</i> world.</div>
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{9}</div>
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<i>Ethical dilemma</i></div>
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In the [music] pluriverse, models can be misinterpreted, misunderstood, misapplied.</div>
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But there are no <i>wrong</i> models.</div>
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This is not the case if [music] is a universe.</div>
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There the search is for the one <i>right</i> model.</div>
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{10}<br />
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<i>Janus again.</i></div>
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Relationships inside/outside time (warped from Xenakis):</div>
<div style="text-align: left;">
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<div style="text-align: left;">
we theorize ourselves <i>into</i> (–do (–experience (–hypostatize)))<br />
a possible music-world <i>inside</i> time</div>
<div style="text-align: left;">
as composer, performer, audience;</div>
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we theorize ourselves <i>from</i> (–undo (–observe (–hypothesize)))</div>
<div style="text-align: left;">
a possible music-world <i>outside</i> time<br />
as analyst, critic, politician.<br />
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Question: Is there a <i>corpus callosum </i>that allows communication between the two?</div>
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<div style="text-align: center;">
{11}</div>
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<div style="text-align: center;">
<i>Etymological shards from the Early Jargonic Period</i><br />
<br /></div>
<div style="text-align: center;">
(This started as a bit of dorky self-absorbed etymological play<br />
– just killing time – with no intention of posting it.<br />
Then I started to think about it.<br />
And from deep in the recesses of<br />
my aging brain came the whisper,<br />
"Bosch's egg.")</div>
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<i><br /></i>
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θέα – view</div>
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θεωρώ – consider, speculate</div>
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θεωρός – <b>envoy sent to consult an oracle</b></div>
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θεωρία – contemplation, speculation </div>
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– a looking at, viewing</div>
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– a sight, show, spectacle, things looked at.</div>
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theory<i> (</i>1590s) – conception, mental scheme.</div>
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<span style="font-size: large;"><b>____________________</b></span></div>
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<span style="font-size: large;"><b><br /></b></span></div>
<div style="text-align: right;">
up, throughout + a loosening – ανά + λύσης</div>
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analysis – ανάλυσης</div>
<div style="text-align: right;">
a breaking up, a loosening, releasing – <span style="color: white;">.</span></div>
<div style="text-align: right;">
unloose, release, set free – αναλύειν</div>
<div style="text-align: right;">
<b>to loose a ship from its moorings</b> – <span style="color: white;">.</span></div>
<div style="text-align: right;">
resolution of anything complex into simple elements – analysis (1580s) </div>
<div style="text-align: center;">
<span style="font-size: large;"><b>____________________</b></span></div>
<div style="text-align: right;">
<span style="color: white; font-size: large;">.</span></div>
<div style="text-align: left;">
composition (late 14c.) – action of combining</div>
<div style="text-align: left;">
– manner in which a thing is composed</div>
<i>compositus</i> – placed together<br />
<i>componere – to put together, to collect a whole from several parts</i><br />
<i>< com + ponere .......</i><br />
F. <i>pondre</i> <b>= lay an egg</b><br />
<blockquote class="tr_bq" style="text-align: justify;">
<span style="background-color: white;"><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">One of the most remarkable facts in F[rench] etymology is the extraordinary substitution whereby the Low Lat. </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">pausare</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;"> came to mean 'to make to rest, to set,' and so usurped the place of the Lat. </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">ponere</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, to place, set, with which it has no etymological connection. And this it did so effectually as to restrict the F. </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">pondre</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, the true equivalent of Lat. </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">ponere</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, to the sense of 'laying eggs;' whilst in all compounds it completely thrust it aside, so that </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">compausare</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;"> (i.e. F.</span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">composer</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">) took the place of Lat. </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">componere</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, and so on throughout. Hence the extraordinary result, that whilst the E. verbs </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">compose</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">depose</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">impose</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">propose</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, &c. exactly represent </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">in sense</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;"> the Lat. </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">componere</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">deponere</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">imponere</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">proponere</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;">, &c., we cannot </span><span class="foreign" style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px; font-style: italic;">derive</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;"> the E. verbs from the Lat. ones since they have (as was said) no real etymological connection. [W.W. Skeat, "Etymological Dictionary of the English Language," 1898]</span><span style="font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;"> </span></span><span style="background-color: white; font-family: Georgia, Garamond, 'Times New Roman', Times, serif; font-size: 16px;"> – From the Online Etymology Dictionary (retrieved Mar. 11, 2015)</span></blockquote>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://upload.wikimedia.org/wikipedia/commons/thumb/9/91/Concert_in_the_Egg.jpg/1215px-Concert_in_the_Egg.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/91/Concert_in_the_Egg.jpg/1215px-Concert_in_the_Egg.jpg" height="268" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="http://en.wikipedia.org/wiki/Concert_in_the_Egg">Concert in the Egg</a>, (follower of) Hieronymus Bosch (ca.1561)</td></tr>
</tbody></table>
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{12}</div>
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<div style="text-align: left;">
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<br />
<i>Analytical heresy</i><br />
<br />
<div style="text-align: left;">
If you want to become a clockmaker, a good place to start is to take clocks apart to find out how they work. But if you simply want to know what time it is, all you have to know is how to "read" the face of the clock. If you want to make your appointment on time, knowledge of the clocks's mechanism hiding behind the face does you no good except insofar as it moves the hands accurately over the face.<br />
<br />
The clock's mechanism becomes important (and vitally so) only when the clock no longer tells the right time.</div>
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<br />
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<br />
{13}</div>
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<br />
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<i>Back to the future: the "relevance" issue 25 years ago.</i></div>
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From David Lewin Collection, Administrative Papers, Library of Congress:</div>
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<blockquote>
<blockquote class="tr_bq">
Lewin: The [Harvard] Graduate Program in Theory, iv/21/90 </blockquote>
<blockquote class="tr_bq">
What do ‘music theorists’ study?<span style="mso-spacerun: yes;"> </span>Theorists’ opinions vary widely.<span style="mso-spacerun: yes;"> </span>Mine: <i>they study the vocabulary, concepts, and intellectual structures in general, through which people talk and have talked about the organization and coherence of music. </i></blockquote>
<blockquote class="tr_bq">
How do you go about studying this?<span style="mso-spacerun: yes;"> </span>Among other things,<br />
<b><span style="mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">1.<span style="font-family: 'Times New Roman';"> </span></span></span>You explore the systematic bases for contemporary compositional methods.</b><br />
<span style="mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">2.<span style="font-family: 'Times New Roman'; font-size: 7pt;"> </span></span></span>You study the history of music theory in our cultural tradition, and so far as possible in others.<span style="mso-spacerun: yes;"> </span>(There is a question to what extent the whole notion of ‘music theory’ is meaningful when applied beyond our own cultural tradition.)<br />
<span style="mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">3.<span style="font-family: 'Times New Roman'; font-size: 7pt;"> </span></span></span>You explore the systematic assumptions underlying received analytic methods.</blockquote>
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com0tag:blogger.com,1999:blog-2569892606315203117.post-37013789381700830952015-02-26T12:17:00.000-05:002017-05-31T12:20:23.119-04:00Desperately Seeking Relevance: Arts Administration TodayLike all arts, music has thrived on (or despite) its patrons, whether church or nobility or bourgeoisie or folk, and there is no disputing that there has always been a waxing and waning relationship between artist and patron affecting content. But artists, in particular musicians I think, have never before had to face anything like PBA-meets-NRH: professional business administrator meets nouveau riche hobbyist. Yes, of course there has always been a need for administration of the business side of music, from the Hurokian impresario to the entrepreneurial (a.k.a. "starving") musician left to his or her own devices. People have to live, whatever their true interest and calling & however their contributions are valued.<br />
<br />
The difference today, at least in art music's economics, is that the distance between patron and administrator has all but disappeared behind boardroom doors; audience has become converted (out of ignorance or cynicism) into a conveniently nebulous customer-who-is-always-right; and, before and during the design of next year's product, musicians are required to confer with the marketing department about the results from the latest focus group in order to make any necessary improvements before the next product release.<br />
<br />
This situation, which now seems to be accepted status quo, created the perceived need for a New Business Model. So the Master of Music (and other arts degrees) hopped into bed with the Master of Business Administration and started to reproduce: among other degrees, the MBA in Arts Administration. While I can't produce a smoking gun to connect MBAs with classical music's recent problems (one needn't have an MBA to screw up an orchestra or cause an art gallery to go belly up), it is at least interesting (correlation does not imply causation but only a fool would ignore it) that the list of problems in orchestras and other music and arts programs have increased EVEN AS Arts Admin degree programs have exploded – almost entirely within the U.S. It beggars the imagination. Seriously – go to these two links and contemplate the maps:<br />
<div style="margin: 6px 0px;">
<span style="font-family: inherit;">Graduate programs:<br />
<a href="http://www.artsadministration.org/find-a-program/graduate-other-programs/" rel="nofollow" style="color: #3b5998; cursor: pointer; text-decoration: none;" target="_blank">http://www.artsadministration.org/…/graduate-other-programs/</a><br />
Undergraduate programs:<br />
<a href="http://www.artsadministration.org/find-a-program/undergraduate-programs/" rel="nofollow" style="color: #3b5998; cursor: pointer; text-decoration: none;" target="_blank">http://www.artsadministration.org/f…/undergraduate-programs/</a></span></div>
<div style="margin: 6px 0px;">
<span style="font-family: inherit;">Out of 76(!) graduate programs in arts admin, 65 are in the U.S.; and of 42(!) undergraduate programs, 37 are in the U.S. Within the U.S. these programs are concentrated almost entirely in the East and Midwest, which just makes it all the weirder.</span></div>
-------<br />
<span style="color: red;">UPDATE (5/31/2017)</span><br />
<span style="color: red;">The URLs above no longer work. The new URL is:</span><br />
<span style="color: red;">http://www.artsadministration.org/programs/?programs=all</span><br />
<span style="color: red;">Also, they seem to be expanding:</span><br />
<span style="color: red;">'<span style="background-color: white; font-family: arial, sans-serif; font-size: 16px;">The Association of Arts Administration Educators (AAAE) consists of more than 150 member programs</span>'.</span><br />
------<br />
<br />
And what happens when the freshly or not so freshly minted MBA goes into the real world where he or she works with (often: is pitted against) local or national business leaders who got where they are by imposing their own interests and agendas on others? Then what good will that course, "Choosing and Managing Your Board of Directors," do for you – even in the unlikely circumstance that your professor has had demonstrable real-world experience. Will you take a principled stand on behalf of the arts, or will you concede that yes, it would be an excellent idea to have Madonna and Paul Simon sing Lied von der Erde?<br />
<br />
I'm not arguing that administration of the business side of the arts is unnecessary. It certainly IS. I'm simply asking: Is pumping a continuous supply of MBAs into the arts' infrastructures really the way to go about addressing the undeniable problems involved in arts support today?<br />
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<br /></div>
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<span style="font-size: x-large;">☛</span> Daniel Wolf has also written on this in his web site, <i>Renewable Music</i>. Go to: <a href="http://renewablemusic.blogspot.com.au/2015/02/slow-death-by-administration.html">"Slow Death by Administration."</a><br />
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<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.comtag:blogger.com,1999:blog-2569892606315203117.post-79527495396207960212015-01-12T10:42:00.000-05:002016-05-17T17:00:06.520-04:00Notes from the Pluriverse {1}<div style="text-align: left;">
<div style="text-align: center;">
<b>{1}</b></div>
<b><br /></b>
<b>THREE HATS</b></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
It was around 6:30 in the morning on July 18, 1997. This is one of those dates that I can pinpoint, not because I remember the exact date (I have a lousy memory for facts), but because it was the first day of the second Buffalo Music Theory Symposium – the dates are easily found on the web. I was there to present a paper on an unlikely topic, "The Z-Relation in Neo-Riemannian Transformations."<br />
<br />
I didn't really know why I was there. In the first place, I had (and have) no qualifications that would put me in the company of the small and highly distinguished group of scholars invited to attend, and I had no expectation that what I had to offer would be of any interest to anyone there. In the second place, I have a phobia involving euphemistically named "conferences" where you suddenly realize you've been trapped inside someone else's fable.<br />
<br />
I feel I can now admit that more than once I have fled a conference presentation on a topic of interest to me and rushed back to the sanctuary of my hotel room with a Snicker Bar and a Coke to watch The Price Is Right or Jerry Springer.<br />
<br />
Milton Babbitt may have had a touch of this phobia as well. I was once told, by the organizer of a smallish invitation-only conference, that when the first scheduled meeting was ready to begin, Milton was nowhere to be seen. They waited for a while, then the organizer called his room. Rather annoyed, Milton said to go ahead and start without him – he would be there as soon as the game he was watching was over. Well, maybe this wasn't my phobia, just a matter of Milton's priorities. In either case, the organizer who told me the story didn't seem to appreciate the humor and was obviously inviting me to share in his indignation. But I digress.<br />
<br />
The Buffalo conference was to turn out to be one of those rare meetings out of the admittedly few I have attended that lives up to the name "conference" (thanks to the synectic mix of participants & John Clough's sensitive planning ear). My mounting anxiety was to prove unfounded. Still, when I walked in to the hotel restaurant for breakfast the first morning, I was relieved to find no one else there yet. I just wanted to sit alone, eat my breakfast, and gather my thoughts while pretending to read my free copy of USA Today. I had just taken my first sip of coffee when a voice said, "May I join you?" I looked up to see David Lewin.<br />
<br />
Although we had corresponded, I had never really had a private conversation with David before that – only small talk at a conference dinner once. I can't say exactly that he grilled me, but he was curious and managed to get me to tell him about some of my adventures as a closet theorist (defined as a non-academic theorist who knows enough to keep his mouth shut when visiting the academy). Then came a question no one had asked me before.<br />
<br />
"Steve, do you compose?"<br />
Big G.P. while I chewed on a bite of toast.<br />
"Well, no, I don't ... I mean, not much any more. ... I used to. I used to try. ... There was a. ... It's not so easy with a 9-to-5 job. ... I just can't find the time. ... It's different than ...."<br />
<br />
He interrupted, quietly, almost conspiratorially:<br />
<br />
"You should make the time."</div>
<span style="font-family: inherit;"><br /></span>
<br />
<span style="font-family: inherit;">No one had ever before gotten to my well-guarded core.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Others began to straggle in and join us, and then we were all shuttled off to Buffalo (U) for the day.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">I had breakfast alone with David the next morning as well. Evidently we were the only two early risers in the lot. Over the few years left we never talked about "a composing life" again. So I never got the chance to ask the same question back at him – to get at the core that I now realized we shared – more importantly, to get at how he got over the wall of that amazingly beautiful cloister he had built and into the more dangerous exoteric world of personal expression. </span><span style="font-family: inherit;">It was much later, after his death, that I got an answer of sorts.</span><br />
<div>
<span style="font-family: inherit;"><br /></span></div>
<div>
<span style="font-family: inherit;">As I looked through his relatively sparse collection of compositions and noted the large gaps between their dates I realized that David's advice to me was advice he must have repeated again and again to himself. He wanted it all, but even he just couldn't find the time.</span></div>
<div>
<span style="font-family: inherit;"><br /></span></div>
<div>
<span style="font-family: inherit;">There is a Moses and Aaron tragedy that's played out by all those who seriously struggle through their art. The field for that struggle is what I've tried to describe quasi-metaphorically in the tri-partite model. I now confess my inspiration for that entire fantasy came from David Lewin. The following is from a letter David wrote to Oliver Neighbour that is now part of the David Lewin Collection at the Library of Congress.</span><br />
<blockquote class="tr_bq">
<blockquote class="tr_bq">
<div style="text-align: justify;">
<span style="font-family: "georgia" , "times new roman" , serif;"><i>Your overriding
interest is in the man </i>[Schoenberg]<i> and his music.
Mine is too, when I have my analysis hat on. That is when I make Dr. Jekyll type statements, from your
point of view. But I have at least
two other hats which I wear on occasion, which is when I say those narsty
things. One I would call my Theory
hat. When you get around to
Lewin/Cone [“Behind the Beyond: A Response to Edward T. Cone,” PNM 7:2
(Spring-Summer, 1969), pp.59-69], you’ll see what I mean by distinguishing this
from my Analysis one. You probably
will not agree with me that it is possible (much less desirable) to distinguish
the hats conceptually. On that
issue, you would be on Ed’s side and not mine. Incidentally, I have a great deal of respect for EC also;
among other things, I took several courses from him with great profit at P’ton
(or, as we used to call it, the Six and Twelve Store). Then I have still another bonnet which,
however, I don’t wear in print: my Composer hat. With that hat on, my interest in either AS or serialism is
as completely self-serving as my interest in Mozart or tonality … more so as
regards tonality in any case.
Baldly, what interests me then is “what’s in it for me to use.” From that point of view, my tendency is
also to try to separate “the system,” to the extent I can, from AS’s personal
musical profile; I am interested in using “the system” as a matter of public
domain, so to speak, but of course not interested in writing watered-down
pastiches of Schoenberg’s personal
discourse. And of course, in
between “the system” and AS’s personal manner lies a large area which one could
classify as the “usual” sorts of technical things a composer can learn by
studying the work of a great composer of another generation. This area contains such things as
control of rate-of-change that you cite (here one can learn much from Mozart
also, and beyond that, from concurrent study of both composers). And this area merges fuzzily. For me,
into “the system” at one extreme and personal manner at the other. Now one of these fuzzy boundaries
exists for any composer: the one between craft and personal manner. It seems to me that what we are
arguing, in this context, is whether or not there is also a fuzzy boundary at
the other end, between craft and “method” (to vary the terminology) in
Schoenberg’s case. I am claiming
that there is such, and you are claiming there isn’t (more or less, when all
the endless qualifications are made).
A lot of the reason I am prepared to maintain and defend that position,
personally, has to do with my intuition as a composer. That is, I feel that I can use “the
method” as a vehicle for my own expression, to a considerable extent without
feeling bound not only by Schbg’s personal manner, but more significantly by
his general “style,” the latter involving predilections for certain kinds of
musical situations, and certain ways of treating and working out their musical
implications. I don’t pretend to
Olympian stature as a composer, but I’m very sure that every composer who has
ever written twelve-tone music has experienced a similar feeling, if he is worth
his salt as a self-respecting artist, of whatever rank. (At least until recently, when it has
become possible and even fashionable to write serial music without having heard
any of Schbg’s music … or any music at all, for that matter.) I’m sure Webern felt this, and I’m sure
Berg did too, though he probably would never have dared admit it to
himself. It’s more than obvious
that Stravinsky felt it. Were/are
we all just kidding ourselves?
Very possibly, it may be that all “the method” amounts to is a certain
means by which obscure electrical circuits in the brains, or endocrine
secretions in the blood, of many composers at a certain period in history have
been stimulated, in such a way as to inspire creative results when the
composers play the appropriate mental games. I’m not being completely sarcastic about this, I think there
is probably at least a grain of truth in it, and possibly a good deal
more. I would however, argue that
even to the extent composers have been and are fooling themselves, in
considering that they can use “the method” without being bound by Schoenberg’s
“style” (as above), the illusion was/is artistically necessary, in order to
accomplish anything; and it has turned out to be quite productive. And then, to what extent can one
distinguish a tenet which is necessary and productive for artists, from one
which is artistically “true”?</i></span></div>
<div style="text-align: right;">
<span style="font-family: "georgia" , "times new roman" , serif;">February
26, 1974</span></div>
</blockquote>
<span style="font-family: "courier new" , "courier" , monospace;"> </span></blockquote>
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com1tag:blogger.com,1999:blog-2569892606315203117.post-22428912484118583932015-01-04T12:27:00.001-05:002015-01-04T14:29:27.341-05:00Desperately Seeking Relevance: Music Theory Today [6.3]<div style="background-color: white; font-family: Merriweather; font-size: 15px; line-height: 21px; text-align: center;">
<i><span style="font-size: medium;"><b>Rubbish Theory</b></span></i><br />
<i><span style="font-size: medium;"><b>and Music Theory Today</b></span></i></div>
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<i>How can the all-embracing logic which mirrors the world use such special catches and manipulations? Only because all these are connected into an infinitely fine network, to the great mirror. . . .</i></div>
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<i>[Laws] treat of the network and not what the network describes.</i></div>
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–<a href="http://people.umass.edu/phil335-klement-2/tlp/tlp.html">Wittgenstein. </a><i><a href="http://people.umass.edu/phil335-klement-2/tlp/tlp.html">Tractatus Logico-Philosophicus</a></i>[5.511 & 6.35] (Ogden Translation)<br />
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<i><span style="font-size: medium;">3. </span></i><i><span style="font-size: medium;">The Monster Problem</span></i></div>
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<i>[Previous Post: <a href="http://essaysandendnotes.blogspot.com/2014/11/desperately-seeking-relevance-music.html">2. Maverick Integration</a>]</i></div>
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<span style="font-size: large;"><b>A.</b></span><br />
<b>The choice</b></div>
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The previous post provided a slide show of stages in the life cycle of a hypothetical musical "system" which we will now continue to explore. The content (application) of this system is purposely left unspecified to suppress, to the extent possible, the aesthetic and political prejudices inherent in any specific application. It is presented as an abstract model to relate three fundamental conceptual roles: a model-defining theory, a generative techne, and a critical analysis. Theory and techne together are music's workshop: theory defines the model by providing the "allowable" materials and tools; techne offers ways to choose and purpose those materials and tools from work to work. Theory and analysis working together in this stable model provide ways for the model to evolve without jeopardizing its integrity. As long as techne works within the conceptual boundary identified by theory, there is no reason for analysis and techne to converse directly without theory being present – however, this is about to change.</div>
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From inside, this model is unaware of anything other than itself; it demonstrates (to itself) that it is capable of responding to any eventuality; it evolves over time before settling in to its steady state forever; its library, keeper of its works, is capable of infinite expansion; the model assumes it is <u>the</u> <i>uni-</i>verse. The previous diagrams traced the model's "phylogenesis" from birth (the initial state) through various stages of development until its final homeostatic state where a compliant techne, despite having exhausted its ability to provide mavericks to expand the model's theory, is free to continue providing model-consistent works till the end of time.</div>
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The question now is not whether or not the model can go on forever in its homeostatic state – it certainly can. Nor is the question whether or not the model can continue to challenge techne when acceptable variations on theory's core are exhausted – it certainly can not. Eternal life has its drawbacks, after all.</div>
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But the question is whether the model has judged correctly that it is the <i>only</i> possible model, which is to ask:</div>
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Is music a <b>universe</b> or a <b>pluriverse</b>?</div>
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If it's the former, then users (the finite actors playing the various roles in the model) are finally left with just one game to play:</div>
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The goal of the game is to create new combinations of knowledge within existing sciences and arts. This is also the very limitation of the game. No new knowledge is created, only new combinations of old knowledge. [Hermann Hesse's] "Glass Bead Game" is a symbol of culture in harmony and balance, but the price to be paid is high: "The most important consequence of this ... attitude, or rather of this ... subordination to the cultural process, [is] that men largely [cease] to produce works of art."<span style="font-size: x-small;">[1]</span></div>
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However, if it's the latter, if music is actually embedded in a theory pluriverse<span style="font-size: x-small;">,</span> then, given that as yet there is no known coherent description for this idea within music theory today<span style="font-size: x-small;">[2],</span> we must turn for help to recent conversations in philosophy animated by consideration of possible worlds and concomitant problems in modal logic. Here is arguably the most radical:</div>
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Are there other worlds that are other ways? I say there are. I advocate a thesis of plurality of worlds, or ['extreme'] <i>modal realism</i>, which holds that our world is but one among many. ... There are so many other worlds, in fact, that absolutely <i>every</i> way that a world could possibly be is a way that some world <i>is</i>. And as with worlds, so it is with parts of worlds. There are ever so many ways that a part of a world could be; and so many and so varied are the other worlds that absolutely every way that a part of a world could possibly be is a way that some part of some world is.<span style="font-size: x-small;">[3]</span></blockquote>
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The <a href="http://essaysandendnotes.blogspot.com/2013/04/arts-dueling-problematics.html">one-or-many question</a> is precisely where music theory is poised today<span style="font-size: x-small;">[4]</span> – and with it, music itself.</div>
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<span style="font-size: large;"><b>B.</b></span><br />
<b>In praise of monsters</b></div>
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Let's return now to the model and introduce Thompson's heterodox monster.<br />
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Cognition – our way of seeing and our way of not seeing – may, in the area I have termed 'overt', be subject to perfect control, but in the 'covert' area control can never be perfect since our way of seeing denies the very existence of this area. It is not possible to legislate effectively against that which it is held does not exist. The consequence of this inevitably imperfect control by the monitor [in our context, analysis] is that, despite all its efforts, some unruly elements get through into the world view domain. The arrival of such new elements is likely to mess up the ordering process, in some cases giving rise to quite serious contradictions between hitherto integrated patterns of value. If the world view domain becomes changed in this way then the operation of the monitor [analysis] will also change ....<span style="font-size: x-small;">[5]</span> </blockquote>
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The "overt" model developed in the previous post was indeed "subject to perfect control" by theory. Ignoring the implied contradiction, it even made provision for the possibility, however remote, that analysis might draw a work so deformed (incompatible with theory's core) that the model could not work with it at all. Such a work would have to be rejected as a monster. But we neglected to ask what would happen to such a work when it is rejected. Undisposed waste poses a threat to the balance of the model's ecology. So, if a monster did appear, what would analysis do with it?</div>
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<tr><td class="tr-caption" style="font-size: 13px;">Diagram 1</td></tr>
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It would make no sense to return a monster to a library that's set up for random draws because the monster would just keep popping up in future draws with no useful (model-supportive) purpose. Worse, the presence of such a monster in the library might cause embarrassing questions from neophyte users. So, in this isolated model, theory tells analysis that, if it ever does encounter a monster, it must ignore the library's requirement to return all draws and throw the monster in a rubbish bin. (Diagram 1). The onus is on analysis to get rid of monsters. Stuck on saving the theory within the model, analysis would have limited options.</div>
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The simplest of these, the passive "La-La" or "Hester Prynne" option, would be to ignore the monster – cover your ears – refuse to listen – shun it. This option would tag the monster with a scarlet M and throw it back willy nilly into the library, so that if the monster is drawn again, it could immediately be returned with no wasted analytical effort. </div>
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La-La's aggressive counterpart, the "Luddite" or "Tea Party" or "Zhdanovist" option, would be to campaign against any monster's right to exist and destroy it if possible. An urge to kill what is not immediately understood – what is perceived as prima facie malformed – would cause unending battles over the library's conservation mandate. Worst of all, it would institutionalize a policy of analytical intolerance.</div>
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And finally, the feel-good solution. The most ingenious option would be to build a separate room within the library. This is the enlightened "TBD" or "Academic Parking Lot" technique – to place all monsters into a special-access class of works within the library, in a room labelled "Unknown."</div>
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Also referred to as "Waiting for Einstein" or "Hedge Your Bets," TBD would treat the monster as an intriguing anomaly to be held in reserve for "advanced" research. Analysis would assume it may be able to deal with it some day by an approach awaiting discovery or one that is still under development. Then either the monster would turn out to be nothing more than a super maverick that the model can accommodate by making helpful extensions and adjustments to theory (as it has with other mavericks in the past), or it would remain a fading curiosity mostly out of sight in the Unknown and posing no threat to the majority of users' agenda of keeping the teachable Known in circulation. This approach is not only enlightened, it is reasonable and (above all) safe. It would protect the model without casting aspersions on TBD monsters that might later make analysis look like a fool.</div>
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These three possibilities echo Michael Thompson (quoted previously in "<a href="http://essaysandendnotes.blogspot.com/2014/10/desperately-seeking-relevance-music.html">Tripping over Rubbish</a>"):</div>
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[T]here are some who would go so far as to maintain that the proper aim and object of serious thought should be the systematic exclusion of such monsters. Monster exclusion is, at its worst, intolerant, puritanical [La-La], and repressive [Luddite]. At its best, it reveals a dubious prettifying intent that leads to the pretence that things are tidier than they really are [TBD].</blockquote>
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Of course monsters begin to appear.<br />
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Not only that, they begin to accumulate. La-La can't continue to ignore them; the Luddite can't assassinate them fast enough; and TBD's parking lot is starting to overflow.<br />
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Our stable model is forced to admit that it is not alone. Unknown choices may exist outside the box.<br />
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The problem now shifts dramatically due to the knowledge that another – overlapping or entirely separate – <i>competing</i> model exists; and (finally it dawns:) this other model is the only thing that could possibly be polluting the library. But what kind of analysis is possible without prior theory to guide it?<br />
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This question puts us up against the discovery dilemma:<br />
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<i>You can't assume nothing because then you don't know how to start looking, but if you assume too much then you're biased and you're not open to finding a lot of things that might be there.</i><span style="font-size: x-small;">[6]</span></blockquote>
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By placing a monster in TBD, analysis is waiting to see if the purple box in Diagram 2 (the question mark is some unknown theory hypothetically related to unexpected moves in techne) will eventually merge with, or at least overlay, the red box, revealing that the monster was just a particularly knotty maverick all along – and this discovery will result in another adjustment in theory.</div>
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<tr><td class="tr-caption" style="text-align: center;">Diagram 2</td></tr>
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But the only way of getting at the question mark is by first finding a path lying <u>outside</u> the model that leads from the monsterwork back to (an unknown) techne based on an unknown theory, and then describing that path. If this reverse, outside-the-box analysis has done its job, this description will trigger a conjecture – an educated guess at some aspect(s) of the purple box's theory. Let's call this conjecture the proposition <i>p</i>. Analysis can then look in the red box's theory for a proposition <i>r</i> that will match or cover <i>p</i>. Given that such a proposition <i>p</i> can be established, there are two possible outcomes.<br />
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(A) If analysis finds such a congruence between <i>p</i> and <i>r</i>, then the purple box belongs to the model, the work is a maverick, theory is adjusted accordingly, and the red box's theory core is saved. It may not be immediately evident, but this is an "honest" version of the standard model-affirmation analysis as described in the previous post which is a weak form of confirmation bias. The distinction is that the type of analysis here doesn't begin by assuming an underived theory, but waits until the last step to "re-cognize" it. In the end, both approaches produce the same result: theory within the model is either confirmed or adjusted. </blockquote>
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(B) If analysis can <i>not</i> find a match, the work is a true monster, and the purple box where it was spawned belongs to another, independent model, whose theory contains <i>p</i>. Rather than <i>saving</i> (i.e., confirming/adjusting) the theory-basis for the old model, <b><i>M1</i></b>, this procedure <i>discovers</i> a theory-basis for a new, autonomous model, <b><i>M2</i></b>. The work that is a monster to <i style="font-weight: bold;">M1</i> is a normal work with respect to the discovered model <i style="font-weight: bold;">M2</i>.</blockquote>
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<i>NB</i>: <i style="font-weight: bold;">M2</i> does <u>not</u> supersede <i style="font-weight: bold;">M1</i>. Discovery of a new model does not imply replacing the old one since the independent library tenaciously retains all <b style="font-style: italic;">M1</b> works as well as <b><i>M2</i></b> works. To avoid further confusion, the library must now expand to three rooms – <i style="font-weight: bold;">M1</i> works, <i style="font-weight: bold;">M2</i> works, and works of unknown origin.</blockquote>
Both (A) and (B) are clearly music-analytic versions of what is commonly known as "<a href="http://www.npd-solutions.com/reoverview.html">reverse engineering</a>," so we'll call this type of analysis that goes outside the model <i>reverse composition</i>. Reverse composition either re-derives previous theory – possibly in a new guise (version (A)), or discovers new theory (version (B)). This characteristic theory-discovery process (for both (A) and (B)) looks something like this (Diagram 3):<br />
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<tr><td class="tr-caption" style="text-align: center;">Diagram 3</td></tr>
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The engineering and music versions are alike in significant ways that suggest this is not simply an analogy or metaphor. First, they both begin with a known, accessible human-made work whose path from theory through techne has either been lost or hidden or purposely obliterated or was never recorded or even fully understood by techne. They are also alike in that their <i>goal</i> is to determine <u>a</u> path (not necessarily <u>the</u><i> </i>path) that may or may not be identical to the original path but that can nevertheless ideally compose an identical or acceptably similar work or portion of that work. In both, the discovered path to the object is not necessarily the original one. Knowledge of the path to an object is never complete enough to reproduce all the decisions made by techne leading to the original work.<br />
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The <i>schemata</i> for <i style="font-weight: bold;">M1</i> and <b><i>M2</i> </b>are the same, even though their respective <i>contents</i> (their core theories at the very least) are mutually independent and possibly incompatible. The discovered model, <i style="font-weight: bold;">M2</i> (Diagram 4), can now be treated as an independent solitary model just as <i style="font-weight: bold;">M1</i> was in the previous post, with analysis' job reverting to the confirmation and adjustment of <i style="font-weight: bold;">M2</i>'s theory when/if it draws an <i style="font-weight: bold;">M2</i>-compatible work from the library's Unknowns collection.<br />
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<tr><td class="tr-caption" style="text-align: center;">Diagram 4</td></tr>
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But there is no guarantee that there is no work in the library that is not a monster with respect to both <b><i>M1</i></b> and <i style="font-weight: bold;">M2</i> resulting in the discovery of <i style="font-weight: bold;">M3</i>. Then, likewise, <i style="font-weight: bold;">M4</i>. And on and on.<br />
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In principle, there is no end to the discovery of possible models.<br />
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There is no reason to believe we will not be surprised again and again by the appearance of new monsters and their models.<br />
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<span style="font-size: large;"><b>C.</b></span><br />
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<b>Defining a music theory pluriverse</b></div>
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Here are some <u>preliminary</u> observations and conjectures regarding a pluriverse (possible models) approach to music theory.<span style="font-size: x-small;">[6]</span><br />
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<li>(1) Many possible music theory models exist – they are just as real as any current model.</li>
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This is a version of "modal realism" with respect to models. Positing that possible models are real implies that techne or analysis <i>discovers</i> an existing model rather than <i>creating</i> one. Theory offers, suggests or commands but is powerless outside its own model; theory is incapable of discovering anything.</blockquote>
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<li>(2) Any possible model is the same <i>kind</i> of thing as any other possible model – i.e., all music theory models share the same schema.</li>
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The archetypal structure (schema) of any possible model looks exactly like the model posited in Diagram 4: the same three fundamental conceptual roles are present in the same internal relationship for every possible model, <i>sc</i>., a model-defining theory, a generative techne, and a critical analysis.</blockquote>
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<li>(3) Possible models are <i>independent</i> which means the <i>content</i> of any one model differs in at least one aspect from the content of any other model; if there is no difference in content, they are the same model.</li>
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While this says possible models are distinct, it also implies models may overlap; e.g., theory in one model may contain proposition <i>p</i> while theory in another model contains <i>not-p</i>, but all other propositions in the two theories are the same. This suggests a way to define <i>categories</i> of models leading to a taxonomy of music theories. </blockquote>
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<li>(4) The works library is built from the "output" from all possible models and does not properly belong to – or prefer – any one model to the exclusion of others.</li>
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Works are shared "public" objects available not only to music theory models, but to other kinds of possible music worlds as well (e.g., the class of possible music perception models).</blockquote>
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<li>(5) Possible models may be <i>compared</i>, but one model cannot be <i>evaluated</i> (value-judged) <u>in terms of another</u>.</li>
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<i><u>However</u></i>: <i>works</i>, having been "released" from their model(s), may be compared <i>and</i> value-judged, fairly or not, based on criteria from other kinds of possible worlds (e.g., the class of possible music perception models, possible political models, historical models, cultural models, etc.)</blockquote>
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<span style="font-size: large;">✽</span></div>
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This completes my thread on "music theory today." It is not intended to provide <i>the right path</i> (which I believe reasonable people know does not exist), but to lay out a very general plan for a future garden of forking paths.<br />
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I plan to return to this topic occasionally under the title "Notes from the pluriverse."<br />
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[1] Hans Hellsten, "Brief Reflections on the Organ Art, <i>The Glass Bead Game</i>, and Bengt Hambraeus." (In <i>Crosscurrents and Counterpoints: Offerings in Honor of Bengt Hambraeus at 70,</i> ed. Per F. Broman, Nora A. Engebretson and Bo Alphonse, p.35-38). Hellsten quotes a line near the beginning of Hermann Hesse's <i>The Glass Bead Game</i>. (<span style="text-align: justify;">I assume anyone reading this blog entry knows</span><span style="text-align: justify;"> </span><i style="text-align: justify;">The Glass Bead Game,</i><span style="text-align: justify;"> which ought to be required reading prior to beginning any undergraduate music curriculum.) </span><br />
[2] Jumping from my exploratory fantasy (the abstract model world) to the real world for a moment: there <i>are</i> an increasing number and variety of analyses of "post-tonal works" that continue to fall out of the great tonal/post-tonal bifurcation, but these do not cohere into a descriptive account. (And by "cohere" I do not mean to imply the absurd idea of a single musical theory of everything.) I know of no attempt as yet to deal <u><i>comprehensively</i></u> (i.e., beyond ad hoc treatments) with the reality that nearly everyone seems to admit is not about to go away.<br />
[3] David Lewis. <i><a href="http://www.worldcat.org/title/on-the-plurality-of-worlds/oclc/12236763&referer=brief_results">On the Plurality of Worlds</a> </i>(Malden, MA : Blackwell, 1986), p. 2. As I implied, I am not going to "accept" Lewis' radical position but, as with Michael Thompson's theory, I will attempt to mine it as best I can for ideas that appear to be a "fit" for music.<br />
[4] <span style="text-align: right;">(Music theory today is stuck on the currently popular distinction made between "tonal" and "post-tonal" theories. This unfortunate bifurcation appears to have arisen in part as an attempt to improve on the term "atonal." But it has mostly caught on due to its pedagogical utility. In undergraduate music curricula, traditional "tonal" theory is unquestioningly </span><i style="text-align: right;">required</i><span style="text-align: right;"> in all cases. "Post-tonal" theory – a potpourri of ideas about an increasingly large body of works that have in common only their inability (or mulish refusal) to fold into the comparatively well understood orbit of "tonal" works – is at best an elective or add-on. It is difficult to deny that this effectively ghettoizes all musics and their associated theories that lie outside the bounds of common practice "tonality" and its contemporary extensions to pop and jazz.) </span><br />
[4] Thompson. <i>Rubbish Theory</i>, p.147-8.<br />
[5] Jeffrey Scargle, NASA Ames Research Center, Moffett Field, CA, on the central dilemma of searches for extraterrestrial life (In <i>New Scientist</i>, December 13, 2014, p.41)<br />
[6] Again, the concept of possible <i>models</i> introduced here is <u>not</u> necessarily intended in any of the senses that the various concepts of a possible <i>world</i> are used in philosophy and logic. However, to develop the idea of possible models with respect to music theory today, it is helpful to adapt some concepts from David Lewis' <i>On the Plurality of Worlds</i> just as some of the ideas in Michael Thompson's <i>Rubbish Theory</i> are being adapted for the discussion here. I have borrowed shamelessly (but not always obviously) from both these sources throughout this exposition. All mistakes, misunderstandings and inappropriately applied concepts from these works are mine alone.<br />
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com1tag:blogger.com,1999:blog-2569892606315203117.post-35336905863556867862014-11-20T11:34:00.001-05:002020-11-21T14:51:49.303-05:00Desperately Seeking Relevance: Music Theory Today (6.2)<div style="text-align: center;">
<i><span style="font-size: large;"><b>Rubbish Theory</b></span></i><br />
<i><span style="font-size: large;"><b>and Music Theory Today</b></span></i></div>
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</span></i><i><span style="font-size: large;">2. </span></i><i><span style="font-size: large;">Maverick Integration</span></i></div>
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<i>[Previous Post: <a href="http://essaysandendnotes.blogspot.com/2014/10/desperately-seeking-relevance-music.html">1. Tripping over Rubbish</a>]</i></div>
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There are many ways to jump into this. The most direct way is to replace music-theory-today's presumed equal sign with a question mark and start from scratch.<br />
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<tr><td class="tr-caption" style="text-align: center;">Diagram 1</td></tr>
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Preliminarily, analysis is how theory probes the real world. As much as theory needs analysis if it wishes to stay grounded, analysis needs theory or it is nonsense. Without theory, analysis is at best a collection of disconnected hunches. The moment a connection appears, we know that theory is lurking in the background.</div>
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Theory with no view to application is similarly sense-less. (Nevertheless, theory can develop in the abstract, and this is often desirable and even necessary when navigating possibility's garden of forking paths. But for now, put this garden out of your mind. For the present, while it lasts, there is only certainty's garden path.)</div>
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Theory applied yields a necessary ground for analysis; each analysis provides theory with a confirmation. Each confirmation strengthens our confidence in theory as it edges toward "truth," i.e., appears incontestable. (Diagram 2.)</div>
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<tr><td class="tr-caption" style="text-align: center;">Diagram 2</td></tr>
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If we leave it here, we're stuck in a loop: apply–confirm–repeat. We have yet to ask: Just what is it that analysis analyzes? The theory–analysis transaction (Diagram 2) <i>assumes</i> a supply of motivating objects or artifacts we call musical works. Think of this supply as a library of objects available for analysis to draw from to perform its theory-mandated analytical operations. (Diagram 3.) This draw can be made in one of two ways, either randomly or selectively.</div>
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If the draw is selective, it can bias in favor of the theory, i.e., analysis is free to select just those works (or portions of works) that confirm the theory (cf. "<a href="http://essaysandendnotes.blogspot.com/2013/11/procrustean-intonations.html">procrustean intonations</a>"). This is always tempting, but it's cheating. We assume no user of the model would want analysis to cheat.</div>
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But a work randomly drawn (analysis is blindfolded for the draw) might not confirm the theory. In that case analysis looks for ways to "save the theory" by suggesting <i>adjustments</i> to theory in order to accommodate the new information supplied by the maverick work. The theory is not replaced – it is effectively the same theory<span style="font-size: x-small;">[*]</span> corrected, improved, expanded, with essential core invariants remaining untouched. Now on future draws the model can handle similar works previously considered as mavericks.</div>
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[*] <i>In the present context, two theories are the "same" in the sense that they are both versions of one abstract theory defined by a core set of invariant features (propositions, rules, objects, patterns, etc.) "Core feature" is left undefined at this point. Some may think of it as common sense, others a consensus of experts, others a structural sine qua non, still others audience expectation, cultural norm, etc. The important thing is that, were theory's core to be breached, </i><i>either </i><i>it would morph into something unrecognizable or unpalatable, or it would collapse entirely taking its model down with it (see previous <a href="http://essaysandendnotes.blogspot.com/2014/07/desperately-seeking-relevance-music.html">comments on Euclidean"axioms"</a>).</i></blockquote>
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This is evolutionary adaptation. Theory has built-in room to grow. A constant supply of works chosen randomly fuels (challenges) the analytical pump which in turn confirms or improves or corrects the theory which then allows analysis to encompass a wider, more varied selection of works. While the model in Diagram 2 represents a homeostatic system, the model in Diagram 3, simply by including the adjustment function, represents a <i>homeorhetic</i><span style="font-size: x-small;">[1]</span> system.</div>
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<span style="font-size: large;">☛</span> Note posted at the library's circulation desk: The works library is a public lending library. After a work is removed and analyzed it must be returned to the library. No exceptions.</div>
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Continuing to work backward, the next question emerges: <i>Where do all those works come from?</i> This consideration might appear to be redundant for the model. We sense that adding unnecessary weight is asking for trouble, and, as we shall see, adding a third role does add a big dose of complexity and discomfort. It would be easier to stop here. Using a wave-of-the-hand strategy, we could call the question an irrelevant nuisance. But relevance is the game we've decided to play. We cannot not look. So....</div>
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If we add a <i>composition</i> (synthesis, assembly) arrow to those arrows labelled application, confirmation and adjustment, we still have to decide on composition's source and that source's relationship to the model. Composition commonly implies a composer<span style="font-size: x-small;">[2]</span>, but the model encourages us to continue to focus on function and process over flesh and blood actors such as theorist and analyst. This suggests that the work source in the model ought to focus on the more technical <i>craft</i> of composition rather than how the craft is employed or who employs it or what the employer's motives might be. Drawing on a venerable tradition, let's call the source of any work <i>techne</i>.<span style="font-size: x-small;">[3]</span> (Diagram 4.)</div>
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Tracing back from theory-analysis through the work and then to techne suggests the next obvious question: What is the source of techne?<br />
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Within the confines of the model, techne has only one real option for a source: theory. This is represented by the arrows and overlapping green box in Diagram 5.<br />
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Theory might be "informed" by techne directly, short-circuiting the work→analysis→theory chain by going right to the work's source for confirmation and adjustment. This would suggest that techne in turn might apply theory as its source of acceptable patterns and objects – an authority/guide to be followed in order to compose a work that stays within the status quo and maintains the consistency of the model. </div>
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So techne might draw from theory in ways roughly parallel to the two possible ways analysis draws from the works library, but with reverse effects. On the one hand, techne might slavishly accept everything in theory's list of objects and patterns – somewhat akin to analyzing anything a random draw proffers. Or techne might pick just those items in theory's list that are "relevant" to its work-in-progress and ignore <i>or reconfigure</i> the rest in ways that still leave theory's core untouched – acceptable variation somewhat akin to a biased analytical draw. </div>
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This possibility, despite a circumscribed freedom of choice given to both analysis and techne within their domains, envisions theory as a Janus-faced overlord within the model, one face governing analysis, the other governing techne. In principle, any time analysis comes across a maverick work, it would no longer be a mystery where the maverick came from – only a puzzle whose solution lies within the model. Either techne misunderstood or unintentionally misused theory (a mistake), or theory in its navel-gazing abstract mode ("speculative theory") spontaneously expanded within its self-imposed bounds of internal consistency and directly encouraged techne to choose freely from the new menu of possibilities provided by this expansion.</div>
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Strangely though, closing the circuit – staying within the confines of the model – implies that theory and techne aren't really compelled to talk to one another directly at all (which can be helpful when they're separated by three centuries). Certainly techne can "read" theory directly, but a lecture from theory – especially one that does no more than rehearse techne's own past – is hardly conducive to discovering new procedural approaches. (For some reason, techne is averse to running in ruts.) For its part, theory can always get at techne the long way around via analysis of techne's works. (For some reason, theory finds analysis' company more amenable.) Similarly, given that techne stays within the model, there is no need for a direct path between techne and analysis. That particular transaction – if it is ever necessary at all – is adequately mediated by theory.</div>
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In another world we might expect to find a triangular model with three co-dependent and interacting roles. In music theory today the model more closely resembles the interaction between two pairs of roles that share one of those roles. Let's now separate those pairs to see if this is really the case.</div>
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With time (four or five centuries ought to be more than enough), theory reasonably comes to <i>assume</i> that it can always count on a relatively well-behaved techne to express itself within theory's rules or helpfully expand those rules via the enrichments of "clarifying violations"<span style="font-size: x-small;">[4]</span> which will sooner or later be caught by analysis as they appear in maverick works whence, it is assumed, these violations can be folded back into the model. Theory, again quite reasonably, comes to assume that its model is the only model: there are no other models that techne can escape to – indeed, why should techne even want to escape a benevolent dictator? And even if techne is unhappy with its prospects within a fully mature model, there are no other options. Resistance is futile.</div>
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Theory now feels safe in ignoring techne altogether because it can get all its developmental needs from the work via analysis, with no particular need to know anything more than it already does concerning the work's source. At this stage, the works library has become huge, and it <i>continues</i> to grow in quantity if not in quality. Analysis detects no more mavericks. The overstuffed library now appears to consist almost entirely of centuries of artifacts that conform to a maximally expanded theory that can take no more adjustments if it is to maintain an intact core (i.e., if it is to be itself). Obviation of the need for analysis' adjustment function means the model has effectively collapsed back into its archetype (Diagram 3 above minus the adjustment function).<span style="font-size: x-small;">[5]</span> Apply–confirm–repeat. There are plenty of artifacts to go around to support a homeostatic model without techne having to produce any more of them. This has made life easier for analysis whose only task now is to confirm the theory. In the now vanishing possibility that analysis comes across a stray nonconforming work, analysis will simply reject it as a monster.<span style="font-size: x-small;">[6]</span></div>
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Diagram 6 shows the situation. Rather than overlapping pairs we see one pair and a detached singleton. Techne – know-<i>how</i> – is now unemployed – or, to be accurate, can find no meaningful employment beyond music's version of a fast-food chain, which is to say that techne can continue to produce as many model-<i>conforming</i> works as it likes. New-to-my-generation is fine and may get a pat on the head from theory, but new-to-the-world is no longer in the cards for the model or, evidently, for the library it has created. A submissive techne can only look back nostalgically at the days when it was able to produce those reinvigorating mavericks – when it was relevant to the model.</div>
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Knowledge (knowing-<i>that</i>) is now encapsulated in the perfectly self-contained, teachable red box. Pedagogically at least, theory and analysis can go on forever, perennially renewed as succeeding generations are presented with the same library of new-to-them works to learn from. New-to-my-generation requires no more than a good curator to pull off the illusion of newness. And if perchance theory, in its navel-gazing mode, comes up with a new "approach" to some of the artifacts in the library, verification of that approach means no more than instructing analysis to do a "search of the literature."</div>
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The model is perfected.</div>
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The world is one.</div>
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<i>Ite, missa est?</i></div>
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<span style="font-size: large;"><i>Next Post: <a href="http://essaysandendnotes.blogspot.com/2015/01/desperately-seeking-relevance-music.html">The Monster Problem</a></i></span></div>
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[1] <i>Homeorhesis</i>: "The condition of a flow process which remains canalized within limits in a growing system. ... [A homeorhetic] system turns homeostatic (i.e., acquires dynamic stability) when it reaches its full development, in accordance with its archetype[!]" More at the on-line <i><a href="http://systemspedia.org/entry.aspx?entry=1573">Encyclopedia of Systems and Cybernetics</a>.</i><br />
[2] Since music is a performing art, a work may also be construed as a performance-of-a-work, and so a performer may be construed as a composer. This identification, which is my own position, still raises practical issues that remain open for discussion and development. Whichever position one takes on this would not significantly change the model; but such a discussion here would obscure the issue at hand by taking us off into a maze of sidebars.<br />
[3] Other candidates in naming this role are the more highfalutin <i>poiesis</i> and <i>synthesis</i>. My preference for <i>techné</i> comes from a desire to maintain the earthbound "workshop" essence of the model which is intended to emphasize the toolbox nature of theory and the tool-use nature of techne. Both art and kitsch emerge from tool-users whose activities are circumscribed by the tools available within the model. Also, identifying the work's source within the model as techne has the added advantage of interpreting the model as yet another variation on the old philosophy–etymology game played between genesis (doing, making, craft, <i>techné</i>) and knowledge (theory-analysis, <i>epistémé</i>). Knowing-<i>how</i> vs. Knowing-<i>that</i>.<br />
[4] The phrase is Charles Wuorinen's ("Toward Good Vibrations" originally pub. in <i>Prose</i>, reprinted in Elliott Schwartz' <i>Electronic Music; a listener's guide,</i> p. 257). While Wuorinen was making a point about the contribution of interpretation in the composer→score→performer context ("clarifying violations of the text"), I find it also neatly summarizes techne's contribution to enriching the closed normative model presented here.<br />
[5] "Growth of a system is normally homeorhetic, because it is the only way to maintain its identity[!] The system turns homeostatic (i.e., acquires dynamic stability) when it reaches its full development, in accordance with its archetype." (<a href="http://systemspedia.org/entry.aspx?entry=1573" style="font-style: italic;">Encyclopedia of Systems and Cybernetics</a>)<br />
[6] "Monster" is Thompson's word for such a reject. It is particularly appropriate in our present discussion given it's etymology. In the 14th century "monster" was associated with any creature afflicted with a birth [read "compositional"] defect. As the discussion proceeds it is important to keep in mind that such a work rejected by analysis as irredeemably nonconforming was nevertheless conceived in the womb of techne. The monster's story will be told in the next post.<br />
<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com2tag:blogger.com,1999:blog-2569892606315203117.post-72628204146354063932014-10-14T18:00:00.000-04:002015-01-05T18:25:30.073-05:00Desperately Seeking Relevance: Music Theory Today [6.1]<div style="text-align: center;">
<br />
<i><span style="font-size: large;"><b> Rubbish Theory</b></span></i><br />
<i><span style="font-size: large;"><b>and Music Theory Today</b></span></i></div>
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What is it to supply a theory? It is to offer an intelligible, systematic, conceptual pattern for the observed data. The value of this pattern lies in its capacity to unite phenomena which, without the theory, are either surprising, anomalous, <i>or wholly unnoticed</i>. [my italics]</blockquote>
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Norwood Russell Hanson, <i><a href="http://www.worldcat.org/title/patterns-of-discovery-an-inquiry-into-the-conceptual-foundations-of-science/oclc/522180">Patterns of Discovery</a></i> </div>
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<i><span style="font-size: large;">1. Tripping over Rubbish</span></i></div>
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I was on my way out of a used book store when I spotted a spine title that made me stop. I pulled the book off the shelf and began to thumb through <i>Rubbish Theory: The Creation and Destruction of Value</i> <span style="font-size: x-small;">[1]</span> by Michael Thompson. I had never heard of the book or the author before. It had a lengthy section on catastrophe theory and a foreword by E.C. Zeeman. The used book price of $9.50 seemed a bit steep, but how could I resist having the title <i>Rubbish Theory</i> on my book shelf?<br />
<br />
The little gray book I purchased was the 1979 first edition – actually, it remains the only edition as far as I know. Thirty years later, I still have the book. Unknown to me at the time I purchased it, the book had created somewhat of a stir among anthropologists and economists when it came out. The reviews were mixed. It clearly rattled some cages. In some cages it went unnoticed.<span style="font-size: x-small;">[2]</span> Google Scholar currently lists over 650 citations for the book. It's now a collector's item with prices varying between $250 and $500. But even if I wanted to sell it, which I don't, I could never get near those prices. The reason? The used book I bought in nearly pristine condition (a discard from the "Army Library"(?) that appeared to have never been checked out) now has my pencil and ink scribbles on half the pages. Another book ruined by my overwrought conversations and arguments with the author.<br />
<br />
Despite all those cites, I'm not certain of the status of either the book or its author within the social sciences at this point (I've only taken the time to check out a relatively small sample of the citing articles). Still, it has become clear to me that Thompson's work was among the first in cultural anthropology to recognize rubbish as an indispensable object category in the study of "the creation and destruction of value." Curiously, as I write this, neither <i>Rubbish Theory</i> nor Michael Thompson have an entry in Wikipedia, the final word on stuff to be taken seriously. The book's lack of an entry in Wikipedia (making it invisible to the bulk of on-line humanity) coupled with its increasing market value and citation history (suggesting it has a persisting value to collectors and scholars) is an ironic demonstration of Thompson's central thesis.<br />
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In this final relevance-thread entry (due to length I've broken it into <strike>two</strike> three parts) I'm not going to attempt to "apply" Thompson's rubbish theory directly to music theory today. Rather <strike>(in part 2)</strike> I am going to steal some of Thompson's ideas and adapt them into a meta-theoretic model to help focus on some significant blind spots in music theory and the wider field of musicology. My intention is to lay the foundation for an actual discussion/debate, or at the very least, to demonstrate the need for a serious reexamination of the discipline. In today's climate, I doubt any of that will happen, and my essay will be no more than a passing curiosity at best.<br />
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<span style="font-size: x-large;"><b>. . . . .</b></span></div>
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Here is what rubbish theory is all about.<br />
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On page 9 Thompson gives an introductory summary:<br />
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[The two <i>overt</i> categories,] the durable and the transient, do not exhaust the universe of objects. There are some objects (those of zero and unchanging value) which do not fall into either of these two categories and these constitute a third <i>covert</i> category: <i>rubbish</i>.<br />
My hypothesis is that this covert rubbish category is not subject to the control mechanism (which is concerned primarily with the overt part of this system, the valuable and socially significant objects) and so is able to provide a path for the seemingly impossible transfer of an object from transience to durability. What I believe happens is that a transient object gradually declining in value and in expected life-span may slide across into rubbish. In an ideal world, free of nature's negative attitude, an object would reach zero value and zero expected life-span at the same instant, and then, like Mark Twain's 'one hoss shay', disappear into dust. But, in reality, it usually does not do this; it just continues to exist in a timeless and valueless limbo where at some later date (if it has not by that time turned, or even made, into dust) it has the chance of being discovered.</blockquote>
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He then gives a diagram that represents a first approximation of his theory:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3xYbgoUSvNK4-N596EitRGGKxQfAPs3f-w_T5TNI58I0fzWWBmI7I5dHQisPmP1_m0Ico9r5uRw7rG04YjJ9ok6fmG6NRYJnGrD9Bw6URgPmzvbreuCyWZu8g_SOlM0LDJQmj-c_ck6Y/s1600/rubbish+theory.png" imageanchor="1"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3xYbgoUSvNK4-N596EitRGGKxQfAPs3f-w_T5TNI58I0fzWWBmI7I5dHQisPmP1_m0Ico9r5uRw7rG04YjJ9ok6fmG6NRYJnGrD9Bw6URgPmzvbreuCyWZu8g_SOlM0LDJQmj-c_ck6Y/s1600/rubbish+theory.png" height="35" width="400" /></a></div>
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Thompson contends that these arrows represent the only allowable transfers between the three categories –– no left-pointing arrows are valid, and no transfers between transient and durable can occur directly. At creation, objects start out in the transient category where, at varying rates of decay, they begin to lose value until they fall into the rubbish category. Here they not only have no value, they are so worthless as to be invisible. Any rubbish object will remain at zero value (invisible) theoretically forever <i>unless</i> it is "discovered" (<i>made</i> visible) in which case it pops into the durable, value-increasing category where it remains (theoretically forever).<br />
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In a 2003 article, John Frow gives a helpful summary from a different vantage point:<br />
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Rubbish is a zero-degree of value; and as such it's either the invisible limit point of social value, <u>or it's something we actively conspire not to see</u>. It is thus in an asymmetrical relation to the two major categories of value, which Thompson calls the <i>transient</i> (this is the normal state of things: a state of decreasing value) and the <i>durable</i> (an exceptional state in which objects have permanent and increasing value). Consumer goods are the paradigm case of the former, works of art, perhaps, of the latter.<br />
... [T]he corollary to this view of function as a matter of use rather than an entelechy of intrinsic properties of the object is that objects are likely, in a complex world, to have a number of actual or potential overlapping uses. No single game exhausts their function; no single description exhausts the uses to which their properties might appropriately or inappropriately lend themselves. Indeed, objects don't simply occupy a realm of objecthood over against the human: <u>they translate human interests, carry and transform desires and strategies</u>.<span style="font-size: x-small;">[3]</span> [my underlines]</blockquote>
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Before noting all the problems that immediately come to mind, and trashing the theory before it's fully unwrapped, the reader should note that the summary just given is a "naive" first-approximation which works quite well in <i>ideal</i> applications, especially for physical objects traded on the market. It's beyond these applications (where the theory at first might seem to fail) that it provides an expanding analytical framework for less tractable examples. Thompson takes most of the rest of the book exploring many theory-related complications, explanations, caveats, and implications.<br />
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When I first began to read this book 30 years ago, I started out being mildly amused and slightly bored. I have no abiding interest in snot (yes, you read that correctly) or Stevengraphs. But about 50 pages in, when he started talking about the Knockers-Through vs. the Ron-and-Cliffs, I started to perk up. (Readers wishing to decode the last two sentences will just have to read the book –– sorry.)<br />
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At Chapter 4, "From Things to Ideas," he had me. Metaphors became apparent, and I began to test rubbish theory for its applicability for music. By "applicability" I mean, first, does it conflict with any well-established historical models? It seems to pass this first test. As an obvious example, using what is known about the reception history of J.S. Bach's music from the 18th century to the present, Thompson's model doesn't yield any new facts; but it does provide a complementary interpretive framework to hang those facts from, taking Bach's music from creation as it descended quickly through the transient category down into rubbish where it remained (to general audiences) at the "invisible limit point of social value" for a century –– and then popped into the durable category where its value began to climb to its present value. But does rubbish theory add anything new to acceptable approaches in musicology? Here, it seems to me, comes the big crunch. It's time to leave musicology's –– and music theory's –– comfort zone.<br />
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There are two sides to the "canon" problem. First is the obvious: how does an object get into a "canon"? This is effectively identical to the question and example just discussed of how an object escapes the rubbish heap to become durable. But there is a dark side to selection: to select something we must <i>not</i> select other things. Selection is necessarily biased. Whether it is unintentional or an "active conspiracy not to see" (Frow), the only way to create and maintain a "canon" is by keeping out the majority of candidates according to some written or unwritten rule or constraint. This brings me to this statement at the opening of Thompson's Chapter 7, "Monster Conservation":<br />
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[T]he charm of rubbish theory is that it seems always to lead straight into illogicality, anomaly, and paradox. Regrettably, there are many who find these qualities not so much charming as monstrous, and there are some who would go so far as to maintain that the proper aim and object of serious thought should be the systematic exclusion of such monsters. Monster exclusion is, at its worst, intolerant, puritanical, and repressive. At its best, it reveals a dubious prettifying intent that leads to the pretence that things are tidier than they really are.</div>
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Monster exclusion is a distinct, and often dominant, intellectual style. (p. 131)</div>
. . . .<br />
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[M]onster exclusion can all too easily become monster extermination. Monster extermination can result in the permanent removal of the exceptions to a social theory and, in consequence, monster exterminators are particularly prevalent in the social sciences. The result is that social processes that rely on contradictions for their very existence are almost invariably described by theoretical models of impeccable internal consistency. (p. 133)</div>
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We'll meet "monster extermination" up close and personal in <strike>Part 2</strike> Part 3 which will be posted soon.<br />
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I don't particularly look forward to walking the high theory wire from one praxis to another without the aid of a net. Fortunately, I have such a net handy. I'll end this part with an observation that indicates rubbish theory is grounded in an intellectual tradition more broadly based than its limiting sphere of application in the social sciences –– a tradition that echoes throughout the entire "cultural sensorium." This net is a general strain of criticism found in the philosophy of culture. It's nicely summarized by the late David L. Hall (my underlining):<br />
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It is essential that one not succumb to the fallacy of completeness in either of its guises––namely, either in the sense that one claims completeness with respect to evidences employed, or in the sense that one <i>requires</i> completeness in the use of evidence. Some degree of specialization is essential. The question is this, however: <u>Has the specialized employment of evidences determined the omission of important areas of experience which may in fact be seasonally relevant in our period of cultural activity? To respond affirmatively to this question involves one in the criticism of the manner in which the inertial character of the past has overdetermined the nature of the cultural present.</u><span style="font-size: x-small;">[4a]</span></blockquote>
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Strictly systematic theory is more often than not an ideological epiphenomenon functioning apologetically with respect to current modes of practice. Thus <u>theory is practical by definition if one means no more by theoretical endeavor than that systematic, principled form of thinking shaped by the desire for application.</u><span style="font-size: x-small;">[4b]</span></blockquote>
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<u>[T]he attempt to avoid contradiction leads inevitably to the exclusion of experiences or claims about experience which are consistent with alternative explanations of the way of things that, by virtue of their internal consistency and applicability to the world of experience, have an equal claim to be counted as theory.</u><span style="font-size: x-small;">[4c]</span></blockquote>
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<span style="font-size: x-large;"><b>. . . . .</b></span><br />
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<i><span style="font-size: large;">Next Post: <a href="http://essaysandendnotes.blogspot.com/2014/11/desperately-seeking-relevance-music.html">Maverick Integration</a></span></i><br />
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[1] <i><a href="http://www.worldcat.org/title/rubbish-theory-the-creation-and-destruction-of-value/oclc/5578907&referer=brief_results">Rubbish Theory</a></i> <i>: The Creation and Destruction of Value</i> (Oxford UP, 1979)<i> </i>can still be found in many libraries. Those wishing to read a <a href="http://64.62.200.70/PERIODICAL/PDF/Encounter-1979jun/14-27/">condensed version of Thompson's thesis</a> as an introduction can read a pre-publication article he wrote in 1979 that was recently put on the web.<br />
[2] Michael J. Kowalski, <a href="http://www.contempaesthetics.org/newvolume/pages/article.php?articleID=585">"The Curatorial Muse"</a> (In <i>Contemporary Aesthetics</i>, vol. 8, 2010) FN 34:<br />
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The ramifications of the complex embedding of the system of aesthetic validation within a broader context of social validation are explored in Michael Thomson's brilliant and unjustly overlooked study, <i>Rubbish Theory: The Creation and Destruction of Value</i> (Oxford University Press, 1979). The fact that Peter Bürger's ponderously agued theories of avant garde literature, which appeared in English five years after the publication of Thompson's essay, should have become a canonic text for art critics, while the deft and humorous argument of <i>Rubbish Theory</i> was largely ignored, says a great deal about North American writers' knee jerk obeisance to Continental theory. I do not exempt myself from the charge.</div>
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[3] John Frow. "Invidious Distinction: Waste, Difference, and Classy Stuff." (In <a href="http://books.google.com/books/about/Culture_and_Waste.html?id=iG_Eoqo9bAoC"><i>Culture and Waste: The Creation and Destruction of Value</i>.</a> 2003. Ed. Gay Hawkins and Stephen Muecke).<br />
[4] David L. Hall. <a href="http://www.worldcat.org/title/eros-and-irony-a-prelude-to-philosophical-anarchism/oclc/42855232?referer=br&ht=edition"><i>Eros and Irony: A Prelude to Philosophical Anarchism</i>.</a> (Albany : SUNY, c1982) [a] p. 41, [b] p. 45, [c] p. 46.<br />
<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com2tag:blogger.com,1999:blog-2569892606315203117.post-88720627016600529642014-07-18T12:35:00.001-04:002014-08-31T12:30:13.932-04:00Desperately Seeking Relevance: Music Theory Today [5]<br />
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<i>Thus while it is commendable for composers to be concerned with the limitations of the senses, it is well to remember that music is directed, not <u>to</u> the senses, but <u>through</u> the senses and <u>to the mind</u>. And it might be well if more serious attention were paid to the capacity, behavior, and abilities of the human mind.</i></div>
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<i>–Leonard B. Meyer</i><i>,</i><br />
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<u> Music, the Arts, and Ideas</u></i></div>
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<span style="font-size: x-large;">▲</span></div>
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<b><span style="font-size: large;">THE UBIQUITOUS TRIAD</span></b></div>
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At conception, roughly 500 years ago, the tonal triad – barely defined, almost invisible – was all potential, a gift waiting to be unwrapped.<br />
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Then came history. A lot of history. Today we've arrived at the end of that history.<br />
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Now, the triad-as-we-know-it-today is ubiquitous, fetishized, decoupled, anthropomorphized, overused, tired.<br />
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But most of all ubiquitous. This is the perfect word for it. It's not only that it is present everywhere, having invaded and pervaded the musics of virtually every culture on the planet. It's not only that its sound has captured the ears of most children even before they begin to talk. The concept of "ubiquity" originated as the Lutheran doctrine of the omnipresence of the body of Christ. The triad came to be heard as the omnipresent body of Music in the same mysterious sense. But ultimately came the whispering voice of the eternal devil lurking just outside the door of the <i><a href="http://www.poets.org/poetsorg/poem/conundrum-workshops" target="_blank">Workshop</a>,</i> and of course the less subtle shout from the devil we invented:<br />
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An idea in music consists principally in the relation of tones to one another. But every relationship that has been used too often, no matter how extensively modified, must finally be regarded as exhausted; it ceases to have power to convey a thought worthy of expression.<span style="font-size: x-small;">[1]</span><br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcSxhk4kC01PaQbAzYRiaowrS-s6A5hesgp1zIKi0v1D0iC46kK7IQ" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcSxhk4kC01PaQbAzYRiaowrS-s6A5hesgp1zIKi0v1D0iC46kK7IQ" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">What chord is the robot playing, and why?<br />
(Image from <a href="http://www.cs4fn.org/music/8bitlarynx.php" target="_blank">CS4FN, Queen Mary, University of London</a>)<br />
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As a fundamental compositional object, our triad has had a good run. But today its theory, now capable of speaking only through the many voices of analysis<span style="font-size: x-small;">[2]</span>, has ossified into dogma. It's time to let it rest. Our composers–––not the gondoliers, but the explorers–––left it behind a century ago to map new coastlines and interiors. Now we all have to let go. But how?<br />
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<div style="text-align: center;">
<b>. . . . . . .</b></div>
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Specific justifications for asserting the preeminence of the tonal triad, correctly but misleadingly referred to as the triad's multiple "natures," continue to multiply in the academic community. All of these natures/justifications taken together are considered by many analysts to be the basis for a demonstration of the inevitability of the tonal triad as <i>the</i> foundation for "our" music, coincidentally the music most amenable to extended analyses. It's as if there is a belief floating around out there that the more natures that theorists can identify or manufacture, the more solid a case can be made that the triad is a natural object whose status can't be challenged without bringing down the entire world of music. But there does come a day ––– "The lady doth protest too much, methinks." ––– when hollow echoes from that groaning tower of justifications makes us suspect that there's nothing left to justify (if there ever was a need). Except, perhaps, the justifications.<br />
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As we shall see, other sonorities <strike>(I will work out just one in the next post to serve as an example)</strike> may also have interesting and compositionally suggestive multiple natures –– some natures will be shared with the old workhorse, others will be different. First, to know what we will be looking for and to be able to contrast and compare any new music theory with the established, we need to briefly discuss what I consider to be the closest thing music theory today has to a set of axioms. Of course, that is not precisely what they are (even less are they Euclidean requests!) but they do share the axiomatic sense of being proposition sets–––rules, if you will–––some version of which, however warped, is required for any <u>pitch-based</u> music game. A complete list of quasi-axioms for the tonal triad would be unwieldy, but I believe all of them fall into four basic categories that (unintentionally?) mix facts and claims. Remember, these refer to the <i>triad</i>, not its most compatible matrix, the diatonic <i>system</i>.<br />
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<li><u style="text-align: justify;"><b>Form inducing</b></u><span style="text-align: justify;">: The triad's structure invites characteristic compositional procedures and techniques such as "parsimonious" voice leading, modulation, chromaticization through decoupling from the diatonic, and so on. It's in this category that some of tonal music theory's biggest claims and most bewildering terminological tangles are found. The triad's form inducing properties are arguably a </span><i style="text-align: justify;">sine qua non</i><span style="text-align: justify;"> for tonal theories from Fux and Rameau through Schenker and Riemann as well as contemporary instructional manuals from Piston and the latest undergraduate harmony text to popular treatments such as those found in any guitar method book and </span><i style="text-align: justify;">Music Theory for Dummies</i><span style="text-align: justify;">.</span></li>
<li><u style="text-align: justify;"><b>Extensible</b></u><span style="text-align: justify;">: The tonal triad is capable of combinatorially generating other harmonic objects such as seventh chords, whether by adding sevenths or sixths, by triad superposition, or by third stacking. There are different opinions as to the correct analysis of the way this generation works, but the end result –– new objects that are harmonically similar to their progenitor –– significantly expands available harmonic material.</span></li>
<li><u style="text-align: justify;"><b>Matrix-defining</b></u><span style="text-align: justify;">: The triad's "shape" as a second-order maximally even structure connects it logically to the maximally even diatonic (I assume this recommends it based on our human aesthetic preference for symmetry, though I've never heard this argument specifically – only an amazement (which I share) at the triad's "fit" within a nested symmetry.) More importantly, beyond its symmetry and fit within the diatonic and other scales, the repetition of the triad shape at every level creates a defining coherence for the diatonic matrix. </span></li>
<li><u style="text-align: justify;"><b>Aurally preferable</b></u><span style="text-align: justify;"> (<i>apart</i> from any system or matrix): The major triad appears in nature in the lower partials of the harmonic series. This fact is often cited in conjunction with the questionable notion that, presented with the choice, humans have a physical or psychological preference for "natural" over "synthetic." (Unfortunately, to get at the essential <i>minor</i> triad in the harmonic series requires some intellectual juggling.) Another nature-preference argument comes from noting the relatively smaller (ergo simpler) frequency ratios of the tonal triad's constituent intervals, and relating this to humans' alleged preference (again, presented with the choice) for simple over more complex structures. Finally, there is a claim that a natural preference for the triad's sound <i>per se</i> is internal –– somehow wired into the human brain/psyche. Cognitive science has been enlisted to demonstrate this claim which, if it could be done, would lend credence to the "We <i>all</i> like it" argument, a statistical syllogism that derives first-person plural status from a sufficiently large sample of first-person singular preferences. The unacknowledged underbelly of this attempt to ally with science in order to get to <i>we</i>, is that it can be easily confused with the discredited, but often employed, rhetorical <i>argumentum ad populum</i> with a little <i>ad baculum</i> thrown in for spice. At any rate, any applicable valid science here continues to be surrounded by a lot of big <i>if</i>s. As far as I know, cognitive science is continuing to tell us that the innate preference feature will be thoroughly understood by next Sunday. So stay tuned if you believe the outcome might justify your personal listening preferences or provide rocks to throw at composers who refuse to comply with nature.</span></li>
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This list resembles Euclid's axioms at least in the sense that each of the four basic categories (as proposition sets) is independent of the others. In particular (looking ahead), categories 1, 2 and 3 all offer pragmatic <i>techne</i> suggestions and useful game rules for composers of tonal works past and present and are easily seen to have nothing to do with the preference propositions alleged in category 4. The first three can stand untouched whether or not the triad object sounds good to you or me or anybody.<br />
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If this decoupling of sound from procedure is difficult to swallow, try this old philosophers' trick (usually done as an imaginary(?) conversation with the devil).<br />
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Imagine an extraterrestrial visiting Earth who, when encountering chords as simultaneities, experiences pleasure from those intervals found consecutively in the higher overtones (the higher the overtones, the more pleasant the sensation) and excruciating pain from intervals appearing in the lower partials. Our ET's hearing is so sensitive that she can clearly distinguish overtones well over the Pythagorean comma, creating a harmonic preference that is very odd to us: the higher the partials, the closer consecutive intervals get to unison; so she loves the near-unison, but the perfect octave down at the bottom is almost unbearable to her. On her home planet they also have consonant triads as verticals, but each triad's constituent intervals are so close together we Earthlings can only hear them as a single fuzzy tone. Well, this is unusual, but at least we can relate in that some of our own musicians and theorists have been investigating microtones for a long time, albeit not this radical an upside-down harmony preference. But then it gets really weird. She tells us that their melodies are generally stepwise with occasional leaps for effect and to avoid boredom; except that by "step" she means intervals from the lowest partials and by "leap" she means intervals toward the higher end. To illustrate she takes out something she calls a jPod and plays a recording of an old accompanied folk melody from her planet. To our ears it is a random jumble of sounds jumping all over the acoustic spectrum, but she smiles as it plays. We ask her to please turn it off. Her planet's way of forming "simple," enjoyable harmonic and melodic material is precisely the <i>opposite</i> of ours. We make one more try to understand and ask her to explain how her concepts of melody relate to harmony. She produces what she calls a jPad and we scroll through a document she tells us was written by an ancient philosopher-composer from her world named I. I. Fux simply titled <i>Counterline</i>. At first it makes no sense. Then gradually we realize that if we carefully switch certain words around, step <––> leap, consonant <––> dissonant, and a few more ––– and then if we re-read the teacher-student conversation, leaving the "rules" exactly as they are, just switching a few basic definitions ....... hmmm.... Our mind wanders out of music and into math for some reason ––– we vaguely remember something about duals, dual spaces, dual theorems, switching out points for lines .... hmmmm. Our reverie is disturbed by an obnoxious sound like the rapid repetiton of the highest and lowest notes on a piano. It's our visitor's jPhone. She says she must return immediately. Her planet's North Polar Cap has declared war on the South Polar Cap again. Some things, beside the laws of physics, are the same across the universe. We ask her to accept a musical gift to remember us by –– an accordion. She politely refuses. We understand, of course, and wish her well. She steps into the old abandoned phone booth and disappears. Down on the ground we see the jPod she must have accidentally dropped. Hmmmmmmm.......</blockquote>
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[1] Arnold Schoenberg (in <i>Schoenberg</i>, ed. Merle Armitage. 1937. p. 267)</div>
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[2] It is my understanding that "music analysis" conceived as an independent discipline (and today considered as all but synonymous with "music theory") began in earnest only a couple centuries ago. This makes sense since music analysis is dependent on a sufficiently large body of artifacts that somehow managed magically to appear (as well as to be shared and enjoyed) without any independently coherent analytical theory to speak of. It took some time for these artifacts to accumulate before the professional analyst could make an appearance. This answers how (academy oriented) analysis became possible, but fails to address the question of why it was, and still is, considered indispensable. Or often why it's helpful at all. (And a host of other questions.) C.H. Langford, writing about G.E. Moore's "analytical paradox," sums up my own conundrum regarding analysis:</div>
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Let us call what is to be analysed the analysandum, and let us call that which does the analysing the analysands. The analysis then states an appropriate relation of equivalence between the analysandum and the analysands. And the paradox of analysis is to the effect that, if the verbal [music] expression representing the analysandum has the same meaning as the verbal [graphic, verbal] expression representing the analysands, the analysis states a bare identity and is trivial; but if the two ... expressions do not have the same meaning, the analysis is incorrect. (Langford, "The Notion of Analysis in Moore's Philosophy" in <i>The Philosophy of G.E. Moore</i>, ed. P.A. Schlipp, p.323)</blockquote>
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But the paradox didn't stop Moore from philosophizing. None of this is meant to say that I don't see a place for analytical work to inform theory and composition –– that would be absurd, even to me. If I were asked how I can then even make a distinction, I would be forced to admit that I see theory as the attempt to show how things <i>might</i> be (while knowing that all possible paths <i>will not</i> all be chosen) and analysis as the attempt to show how things <i>are </i>(with no attempt to provide a <i>normative</i> framework for either composer or listener). Still it's in my nature to rebel against the latter, even (or especially) when I find myself faking that pose in a group portrait.</div>
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com1tag:blogger.com,1999:blog-2569892606315203117.post-46454592417653882162014-07-11T13:58:00.001-04:002020-08-24T23:40:21.992-04:00Desperately Seeking Relevance: Music Theory Today [4]<i style="font-size: x-large;"></i><br />
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<i><i><span style="font-size: x-large;">"Modes of Imagining"</span></i></i><br />
<i><i><span style="font-size: large;">A few thoughts on rules, definitions, common notions,</span></i></i><br />
<i><i><span style="font-size: large;">requests, postulates, axioms, hypotheses,</span></i></i><br />
<i><i><span style="font-size: large;">and other stuff.</span></i></i></div>
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<i>I will not run through all the modern axioms laid down by Russian boys on the subject, which are all absolutely derived from European hypotheses; because what is a hypothesis there immediately becomes an axiom for a Russian boy, and that is true not only of boys but perhaps of their professors as well, since Russian professors today are quite often the same Russian boys. And therefore I will avoid all hypotheses.</i> – Ivan Karamazov<span style="font-size: x-small;">[1]</span></div>
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I often think of music as a game. There may be a large, even infinite, number of ways to play, but a limited, relatively small set of rules defines any game. Over a long period of time, since originality is often an important goal, some of the rules might be altered so that boredom doesn't set in (possibilities tend to get used up as the history of play is extended). But the question then arises, which rules are you allowed to change and how much can you alter them before you find you are playing a different game altogether?<br />
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It doesn't take long before tic-tac-toe becomes boring. So you change the rule that says it has to be played on a two-dimensional surface. Is tic-tac-toe in three dimensions still tic-tac-toe? I think most people would say yes. But let's say you stick to two dimensions, expand the grid, change a few other rules, expand the grid some more and so on. Over the years, possibly centuries if you live so long, you come to realize you're playing a game that looks an awful lot like a game in another culture called <i>Go</i>. But are you still playing tic-tac-toe? Does it matter?<br />
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If we ask if Wagner was playing the same game as Josquin, it seems the answer would be both no and yes, depending on which rules you look at. Some of the rules in Wagner's music game would be unrecognizable to Josquin, and some of Josquin's would be anathema to Wagner. Others (very deep ones, I think) would be unchanged between the two – if not, we couldn't say that both composers were composing music that sounds enough alike such that no one is surprised when we call them both music.<br />
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The game metaphor suggests a few related knotty issues as well. When is a rule change called for? Is it simply a matter of avoiding boredom as I suggested at first, or are there other compelling reasons for rule changes? This raises issues of authority. Who decides a rule change is called for and what rule or rules to change? The composer? The performer? Who gets to decide whether or not to accept a rule change? The audience? The critic? History?<br />
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Now I suppose I could continue in this vein to segue into a critique of musical developments and experiments in the early-to-mid 20th century. It now seems to many, not without reason, that rule-changing itself became the game in those years. I can't deny this happened, but neither can I say that many of these rule changes were not desirable or necessary. The old game was getting awfully stale and a little too easy. But that's not the turn I want this metaphor to take. At least not yet. Instead, I'd like to move away from music a bit to see how players in another game have dealt with radical rule changes.<br />
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"In the nineteenth century, geometry, like most academic disciplines, went through a period of growth verging on cataclysm."<span style="font-size: x-small;">[2]</span> (One might say that in the twentieth century, music, like most other arts, went through a period of cataclysm verging on growth.)<br />
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While there was an explosion of activity in many branches of mathematics, I think most would agree that a main cause for all the foment was a problem that kept bubbling up from just below the surface for centuries: the Fifth Postulate in Euclid's <i>Elements</i>. But the "<i>parallel</i> postulate" is not the issue here. The issue I'd like to dwell on is the word <i>postulate</i> per se, and how mathematics as a discipline learned to handle the problem of its "primitive concepts."<br />
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I just recently discovered that the Greek word Euclid used, which is generally translated as "postulates," was αιτεματα [aitemata], more accurately rendered as "requests." This is significantly different from the old, and still common, sense of postulates and axioms as self-evident propositions. It would seem that Euclid's <i>Elements</i> does not really begin with five postulates, nor with five axioms. Euclidean geometry begins with five <i>requests</i>. So then is the reader free to accept or reject any of these requests? Yes. But the <i>consequence</i> of rejecting any one of them is that Euclidean geometry doesn't work. – Well, that's not entirely true. As it turns out, the Fifth Request, concerning "parallel lines," <i>can</i> be left out, leaving much of the Euclidean edifice still standing, but the space it is left standing in need not be "flat" as common sense would dictate. And if the Euclidean Fifth Request is denied such that space is no longer "flat," some very strange objects and relations begin to appear – logically valid but playing havoc with our human sense of the way things are and the way they ought to be.<br />
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[N]o mathematician would invent something new in mathematics just to flatter the masses.... He who really uses his brain for thinking can only be possessed of one desire: to resolve his task. He cannot let external conditions exert influence upon the results of his thinking.... An idea is born; it must be moulded, formulated, developed, elaborated, carried through and pursued to its very end.<span style="font-size: x-small;">[4]</span></blockquote>
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[V]ery little of mathematics is useful practically, and ... that little is comparatively dull. The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the <i>significance</i> of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.<span style="font-size: x-small;">[5]</span> </blockquote>
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Still, it was not always so. Mathematicians are human, and tend to cling to their inherited realities like anyone else. It took centuries (and the development of a more safely liberal culture) for them to face up to the problem fully. In 1733, Giovanni Saccheri, having proven many theorems in hyperbolic geometry, dismissed his own work simply because it contradicted Euclid. To this day, <i>outside</i> the world of mathematics and science generally, the common sense (consensus: literally "feeling together") tells us that everything in Euclidean geometry is obviously true. Imagining the world otherwise, even "in theory," is nearly impossible. Outside of science fiction (<i>NB</i>!), suggesting a world where parallel lines meet is an abomination. To explain what our collective sense tells us, therefore, <i><u>we accept Euclid's request because we need it to prove (justify) what we already know is true.</u></i> We knowingly commit to this invalid argument because we need to get along in a world we can understand. "We do, doodley do, ... what we must, muddily must...." That's all. Is this beginning to sound familiar??</div>
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Despite being locked in by collective wisdom, in the nineteenth century mathematics found a way to stop begging this ancient question. Nikolai Lobachevsky devised an alternative system of geometry based on the negation of the Fifth Postulate (we may as well return to this commonly accepted translation now, my point having been made (I think)). Lobachevsky called the geometry he built an <i>"imaginary"</i> geometry.<span style="font-size: x-small;">[6]</span> Around the same time, Janos Bolyai simply deleted the Fifth Postulate and termed what could then be deduced from the definitions and the first four postulates alone the "<i>absolute</i> geometry."<span style="font-size: x-small;">[7]</span> In 1854, Bernhard Riemann (who Milton Babbitt liked to call "the <i>good</i> Riemann") built a spherical geometry where there are <i>no </i>parallel lines (paving the way to the general theory of relativity). There were now <i>two</i> non-Euclidean geometries. Mathematicians had finally freed themselves from trying to make the worlds of reality and imagination conform to the Euclidian sacred text, and they did this essentially by populating the world with other geometries – not to improve upon, and certainly not to take the place of Euclid, but to <i>join</i> him. Euclidean geometry became one among many valid geometries, and this caused a different problem. With more geometries appearing on the scene, what, if anything, was the connection between them?</div>
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Enter Felix Klein and his Erlangen Program of 1872 which was a synthesis of many of the then-existing geometries (including Cartesian, projective, and others – the exception was spherical) as models of the same "<i><u>abstract geometry</u></i>."<span style="font-size: x-small;">[8]</span> Remember that term.</div>
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Neither reverence for the past nor the common sense of reality prevailed. Euclid remained, but the Euclidean lock was broken.</div>
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I'm going to break off this <i>ultima Thule</i> of a digression here because a musical Erlangen Program is precisely the place where I would like to resume my commentary on music theory today (MTT) with a challenge that will take two more posts. In my next post I'll review a list of what I understand to be MTT's basic, mostly unexamined assumptions. These assumptions I see as tracking the early history of geometry described above. Full of potential, MTT has stopped short of Lobachevsky & company and settled into the safe holding mode of a brilliant but frightened Saccheri. Just what the Erlangen Program is in math and what it might suggest to music theory should then become clear with what I intend to be my final post in this thread. But I'm more than willing to extend it if the challenge is taken up in a meaningful way.</div>
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<i>The recognition of frontiers implies the possibility of crossing them. It is just as urgent for musical theory to reflect on its own procedures as it is for music itself. It is the bitter fate of any theory worthy of the name that it is able to think beyond its own limitations, to reach further than the end of its nose. To do this is almost the distinguishing mark of authentic thinking.</i></div>
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––Theodor Adorno (<i>Quasi una Fantasia</i>)</div>
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[1] Fyodor Dostoevskii. <i>The Brothers Karamazov</i>, Part 2, Book 5, Chapter 3. Tr. Richard Pevear and Larissa Volokhonsky. San Francisco: North Point Press, 1990. p. 235.</div>
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[2][3][6][7] (<a href="http://plato.stanford.edu/entries/geometry-19th/#KleErlPro" target="_blank">"Nineteenth Century Geometry," Stanford Encyclopedia of Philosophy</a>)</div>
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[4] Arnold Schoenberg. "New Music, Outmoded Music, Style and Idea" (1946) (In <i>Style and Idea</i>)<br />
[5] G.H. Hardy. <i>A Mathematician's Apology</i> (p.89)<br />
[8] Felix Klein. "<a href="http://math.ucr.edu/home/baez/erlangen/erlangen_tex.pdf">A Comparative Review of Recent Researches in Geometry.</a>"</div>
stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com0tag:blogger.com,1999:blog-2569892606315203117.post-25993845994751385642014-06-04T12:32:00.000-04:002014-06-04T12:32:06.643-04:00Desperately Seeking Relevance: Music Theory Today [3]The following entry was posted in Essays & Endnotes in September 2013. I am copying it here in its entirety because it really belongs in this thread.<br />
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While packing up my library yesterday, I remembered a connection that came to me a couple of years ago.</div>
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There is an undercurrent I have observed within the music theory community (ever since I realized there was such an unlikely community) that I have come to call, in my inimitably bland way, "feature cognition." I mean by this the tendency in many an eager scholar to hear (or, more likely, view) a feature or set of related features in a musical work such that, once one is made aware of the feature, it threatens to obscure the work-as-a-whole (the "Music") – a musical version of missing the forrest for the trees. I hide behind this term because, while it is not altogether accurate for what I wish to express, still it points in the right direction and feels reasonably inoffensive.</div>
<br />But then one day, while reading through the David Lewin correspondence, I was a bit surprised to read my thoughts on this put in a much less polite way by David:<br /><blockquote class="tr_bq">
Too many analyses I have read (or performances I have heard!) proceed on the pattern: listen to the opening of the piece until you get an idea that interests you; then ignore everything else and plow through the rest of the way, trying to make the rest of the piece fit your idea. (This I call to myself "Procrustean intonation.")<span style="font-size: xx-small;">[1]</span></blockquote>
The first time I read those words, while I admired the way they cut right to the quick much better than my too-polite descriptive term, I thought "procrustean" a bit over the top coming from the gentleman I thought I knew – and David was no longer around to ask about whether he would still use the word or whether his judgement about the state of professional analysis had softened significantly over the intervening 30 years. His observation came, after all, in private correspondence. But still, "procrustean" is a violent word. Or do I read more into this than David intended?<br /><blockquote class="tr_bq">
<u>Procrustean</u>: in the figurative sense, "violently making conformable to standard," from <i>Procrustes</i>, mythical robber of Attica who seized travelers, tied them to his bed, and either stretched their limbs or lopped off their legs to make them fit it. The name is <i>Prokroustes</i> "one who stretches," from <i>prokrouein</i> "to beat out, stretch out," from <i>pro-</i> "before" and <i>krouein</i> "to strike."<span style="font-size: xx-small;">[2]</span></blockquote>
Arguably, David Lewin was contemporary music theory's Theseus, founder of our little Athens. So feature-cognition analysts, take care! Remember how Procrustes met his end.</div>
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<br /><br />__________________________<br />[1] Letter to Oliver Neighbour, May 8, 1973. Correspondence, David Lewin Collection, Music Division, Library of Congress.<br />[2] Douglas Harper's <i><a href="http://www.etymonline.com/index.php?allowed_in_frame=0&search=procrustean&searchmode=none" style="color: #1c768a; text-decoration: none;">Online Etymology Dictionary</a></i>.<div style="clear: both;">
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com0tag:blogger.com,1999:blog-2569892606315203117.post-55539033211096710802014-06-02T17:57:00.000-04:002015-10-22T08:57:58.233-04:00Desperately Seeking Relevance: Music Theory Today [2]<div style="text-align: center;">
<span style="font-family: Georgia, Times New Roman, serif; font-size: large;"><i>The Possession of Doctor J</i></span></div>
<div style="text-align: center;">
<span style="font-family: Georgia, Times New Roman, serif; font-size: large;"><i>and How Ernst Krenek Saved Me from Drowning in the Devil's Triangle</i></span></div>
<div style="text-align: center;">
(Another True Story)</div>
<div style="text-align: left;">
<br /></div>
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<br /></div>
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I am sitting in a room.</div>
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<br /></div>
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Waiting for Dr. J.</div>
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<br /></div>
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It's Monday, September 24, 1962, a warm autumn morning on the campus of Augustana College in Rock Island, Illinois.</div>
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<br /></div>
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The class is Harmony I, and six or seven other students are waiting with me. We've only met a few times so far – maybe half a dozen, and Dr. J has always been punctual. He seems a happy fellow, full of enthusiasm for his work and himself, but he also seems deaf to the possibility that his enthusiasm is not all that contagious and just might be grating on others – the sort of person who forces agreement by using the first-person plural way too much. How are we today? What do we call this chord? On the first day of class he introduced himself by reciting his CV. The two points he stressed were, first, that he was a doctor, and second, more importantly, that he had "studied with Nadia Boulanger." I don't believe anyone in class, including me, had ever heard of Nadia Boulanger before. But we figured she must be important because Dr. J managed to pronounce her name with due reverence – <i>my teacher, Madame Boulanger</i> – at least once every session. I had no problem with the material, but early on Dr. J started to really get on my nerves, as he would later get on my case.</div>
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<br /></div>
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Suddenly Dr. J bursts into the room. He has lost any semblance of composure. His face is red – eyes wide – jaw locked. There is no happy nod to his students. Instead of going straight to the piano and playing his clever signature unresolved dominant 7th chord greeting, he paces. He slaps his notes down on top of the piano. Starts to sit down. Paces some more. It is frightening to watch, and no one knows quite what to make of it. Finally, he sits down. He apologizes for being late. Then the story comes out.</div>
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<br /></div>
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He had watched the Sunday, September 23, 1962 broadcast of the dedication of Lincoln Center's Philharmonic Hall featuring Leonard Bernstein and the New York Philharmonic. The centerpiece was the premiere of a new work by the great American composer Aaron Copland. It was horrible! Horrible! Horrible!! The great Aaron Copland, first among all the American pupils of <i>my teacher, Madame Boulanger</i>, had written a ... <i>twelve-tone piece</i>.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Well.</div>
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<b>. . . . .</b></div>
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<br /></div>
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Dr. J somehow managed to regain his composure and get on with his daily triads. He never mentioned it again. But things were never the same for me. If the words "twelve tone piece" elicited such a violent reaction in such a little man, I <i>had</i> to know what the fuss was all about. A couple weeks after Dr. J's tantrum, I went to the library. I don't remember if I found it in the card catalog or just by browsing the shelves – but the library had one short [37 p.] book on the subject: <i>Studies in counterpoint: based on the twelve-tone technique</i> [1940] by Ernst Krenek.</div>
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Thus began my own 50 year journey through the very strange and exciting worlds of music theories.</div>
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<br /></div>
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Now put on your seven league boots.</div>
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<b>________________________</b></div>
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After I devoured that little book I eventually got on to Schoenberg and the more usual suspects. This was a lonely project, but libraries continued to supply my teachers for this forbidden fruit during my undergraduate years, creating a schizophrenia I've never really found a cure for.</div>
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I mostly forgot about Ernst Krenek for many years. Then one day I discovered a box sitting on a shelf in the music division at the Library of Congress. It was barely taped shut, and on the top was written, "<i>Noli me tangere</i>!" [The only one in the music division I can think of who would have been capable of thinking up that little joke was Wayne Shirley – TOTH, Wayne, wherever you are!] The box contained the memoirs of Ernst Krenek and another label indicated the box was to remain sealed until 15 years after Krenek's death. Of course, the box had been opened before, so I admit I looked inside as well – but never spoke of the actual content with anyone until after December 22, 2006. The story of the writing and "publication" of the memoirs remains steeped in mystery to me. Here are a few facts.</div>
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The deposit is a typescript consisting of 1106 pages (single spaced as I recall). Beginning in 1950, it was sent to LC in six installments corresponding to six chapters. The period covered is 1900–1939, Krenek's birth until his arrival in the U.S. <span style="font-size: 16px;">Beside being Krenek's account of his music and life during those years, they are an extremely valuable and detailed record of musical and political life in Europe & particularly in Austria leading up to the Anschluss. </span>He began writing it September 6, 1942 in St. Paul & finished January 6, 1952 in Rio de Janeiro. The entire original manuscript was written <i>in English</i>. That's where the mystery lingers. Krenek died in 1991 in Palm Springs. In 1998 his widow, Gladys Krenek moved to Vienna and the Krenek Institute was founded there. I was told that Gladys Krenek has a (carbon?) copy of the English manuscript. In 1998, the English manuscript copy(?) was translated into German and published in Hamburg under the title <i>Im Atem der Zeit</i>. After going out of print, a <a href="http://www.amazon.de/Atem-Zeit-Erinnerungen-die-Moderne/dp/345511170X" target="_blank">second edition</a> was published in 2012, again in German. To my knowledge, there is no published version of the Krenek memoirs available in English, the language in which he originally intentionally wrote them. I make no guesses, and certainly no judgements, regarding these facts other than: This is a bizarre situation in so many ways. And I can only hope that the original typescript at LC has been sent for preservation, as well as, ideally, digitization. The last I saw it in 2011 the type on the onion skin paper was beginning to fade in places.</div>
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<b>________________________</b></div>
<br />
I realize that bringing up the Krenek memoirs mystery here risks drawing attention away from everything else I'm trying to say as many rush to get their personal opinions into a list serv or email or FB page or tweet. But as much as I'm concerned about the memoirs (and many other important things ignored and gathering dust and worse at LC), my purpose in lingering on Ernst Krenek, a remarkable and important composer and teacher, is two-fold.<br />
<br />
First, to make a statement to the Dr. J teachers out there. I have no desire to talk <i>with</i> you, let alone debate you, and I realize it's futile at any rate. So cherry pick history. Go ahead and ignore the repertoire and theory you consider irrelevant or dangerous. I doubt Krenek, at this point, will make it into your worthiness lists beyond a brief mention of <i>Jonny spielt auf</i>. Just know that, through the miracle of the Internet, I'm now talking to your students directly, without your presence to ameliorate my evil influence.<br />
<br />
Go for it, kids: Don't just question authority, chew it up and spit it out.<br />
<br />
Finally, my reference to the Krenek English manuscript was background to quote the following (in transcription), without being able to supply a cite that's easily available to check. Some poetry from the first two pages that says precisely how I feel right now:<br />
<blockquote class="tr_bq">
<blockquote style="text-align: justify;">
<span style="font-family: "Book Antiqua"; mso-bidi-font-family: Arial;">These pages are
dictated by the fear that, if I would not write down certain things, the memory
of them would be lost forever. It seems wise to do so now since I might be
nearer to death than ever before. When I say that the memory of things may be
extinguished, it means that the things themselves are in danger of being lost,
because things of the past<span style="mso-spacerun: yes;"> </span>do not
exist except in our memory. Only the works of men last for some time,
particularly those of the mind, which are sufficiently cleansed of perishable matter
and have a peculiarly solid construction of their own. These we call works of
art. The perishable matter of which the works of art must be free is precisely
those things which exist solely in our memory: the bewildering maze of events
which seemingly make up the reality of our lives. These events, innumerable as
they happen every second to each one of the fifteen hundred million individuals
living on earth, are nothing if we do not remember them. The work of art, in
order to be experienced by those to which it is addressed, has to enter this
process; somebody has to look at it, or to listen to it, in a certain given
moment, and this will be one of the events which we must remember so that it
would not be lost. However, the work of art will still be there, regardless of
whether or not he remembers his experience. Remembering an event is a
silent, inwardly act which in itself is an event doomed with oblivion if it is
not remembered. Thus, if we want to salvage an event, or the memory of an
event, from speedy annihilation, we must impart to it the durable quality of
the work of art. The inherent difficulty of memoires, or of history for
that matter, is that they are necessarily made up from that very perishable
matter of which the work of art should keep free. The problem is not one of
different degrees of significance. The outcome of a so-called decisive battle
is in itself no more significant than the result of a private conversation
between any two individuals, and the reader characteristically enough makes no
distinction of that kind. At any time he is ready to prefer the imaginary
quarrels of fictitious characters in a well-written novel to an allegedly
faithful, but uninspired account of Napoleon’s campaigns. Thus the value of
memoires does not rest upon their veracity (which can hardly be tested anyhow),
but upon the amount of interest which their author can arouse by his peculiar
way of remembering the facts which he relates. It is probable that this
interest is proportional to the urge that he feels for preventing his memories
from being obliterated by his silence. I feel that with me this urge has
recently become so strong that I may dare to begin writing down my memories
this Sunday, September the sixth, 1942, in St. Paul, Minnesota, where I arrived
two days ago in order to take over my new job as director of the department of
music at Hamline University, in this city.</span></blockquote>
</blockquote>
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<!--EndFragment-->stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com3tag:blogger.com,1999:blog-2569892606315203117.post-20348295384630421482014-06-01T11:47:00.000-04:002014-06-03T08:08:18.007-04:00Desperately Seeking Relevance: Music Theory Today [1]<div style="text-align: center;">
<br /></div>
<div style="text-align: center;">
A true story.</div>
<br />
Years ago, but not that many, I was having a conversation with N – a good friend, a fine pianist and outstanding chamber musician. He also, like many musicians, taught theory to undergraduates. He's retired now, but at the time we had this conversation he was a full professor at a well-respected school.<br />
<br />
It was after dinner and we were enjoying what was left of the wine. I had recently been to a music theory conference where one scholar reading a paper used a term that I hadn't heard before – theory-based performance. I was telling N that the idea was further discussed afterward on the smt list with a few participants enthusing over the idea of a theorist getting together with, say, the Juilliard Quartet in rehearsal and collaboratively preparing a "theory-based performance" of, say, a Mozart quartet. Sort of the ultimate test of a theory, although it was never clear what it would be tested against since it would still, like everything else in "applied theory," eventually run up against the wall of first person avowals. (Still, I was <strike>intrigued</strike> entertained by the image of Doctor Ruth invited into the Juilliard's bedroom to coach them on how to improve their performance. But back to my conversation with N.)<br />
<br />
Unspoken, but pretty obvious from the exchanges on the list, was the exciting notion that a theorist might be brought into the composer-performer-audience loop to provide "expert advice" prior to a specific performance that would bring that performance up a notch or two, clarify larger formal aspects or details that a performer working alone may miss, etc. [I don't recall the precise content of the smt list discussion, but enthusiasm for the idea was palpable.] I told N that this appeared to me to be an extension of the older notion of "theory-based listening" (not to be confused with "ear training," which has its own set of problems). At any rate, the conversation finally got more specific and came around to the relevance of Schenker to performance as well as (new) composition and whether Schenker training might improve a listener's experience and how could you tell if it had. Finally I asked N if Schenker played any role in his own work. (I was thinking of his work as a performer, but I wasn't clear about that.) His answer:<br />
<br />
"My God! If it wasn't for Schenker I wouldn't have anything to teach!!"<br />
<br />
We both laughed at his unguarded admission. And the wine was gone, so it was time to go home.stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com2tag:blogger.com,1999:blog-2569892606315203117.post-84952176200626814752014-05-31T16:57:00.000-04:002014-06-03T08:07:34.866-04:00Desperately Seeking Relevance: Music Theory Today [0]<div style="text-align: center;">
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<div style="text-align: start;">
The intent of the entries in this thread is to explore. This means, to me, that there will be challenges, but probably no winners. Questions, but I hope no answers. Just <i>what</i> I'm intending to explore here will have to wait to be defined over time by an accumulation of contexts. These will be my personal contexts (this is, after all, a blog), but anyone should feel free to improve on these by adding their own. But I cannot promise that I will post and reply to all comments.</div>
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stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com2tag:blogger.com,1999:blog-2569892606315203117.post-47402549401345859472014-04-10T08:16:00.000-04:002014-04-10T08:37:09.397-04:00Partition Puzzle 5: Feldman Analysis of Z-rels mod 31<span style="font-family: inherit;">Yesterday David Feldman left a comment on a previous E&EN blog entry. I felt it ought to be brought forward as a post of its own so it doesn't get overlooked. He has agreed to let me post it here as a Preliminary Report.</span><br />
<div style="text-align: left;">
<b>__________________________</b></div>
<br />
<span style="color: #262626; font-family: Arial, Helvetica, sans-serif; mso-bidi-font-family: Helvetica;"><a href="http://essaysandendnotes.blogspot.com/2014/02/observations-from-jon-wild-on-partition.html" target="_blank">[Jon Wild wrote:]</a></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; mso-bidi-font-family: Helvetica;"> "</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">There is a quintuplet of Z-related hexachords in 31-tone equal<br />
<o:p></o:p></span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">temperament whose interval vectors consist solely of 1s--they are all </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">interval hexachords, expressing 15 interval classes in just six pitches. This </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">set of 5 hexachords can be packed into the 31-tone aggregate leaving only </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">one pitch uncovered. The odds that this is by chance are astronomical-</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">there is some other principle at work, related to the observations Stephen </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">has started to make, but I don't know what that principle is."</span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><br /></span></span>
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626;"><span style="font-size: 17px;">Here is a description of [this] example which may remove some of the mystery.</span></span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><br />
</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">Modulo 31, the five sets can be written as</span><span style="color: #262626;"><span style="font-size: x-small;">[1]</span><span style="font-size: 13pt;"><o:p></o:p></span></span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span></span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><br /></span></span>
<br />
<div style="text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">{3^0 , 3^10, 3^20, 3^3,<span style="mso-spacerun: yes;"> </span>3^13, 3^23}</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span></span></span></div>
<div style="text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">{3^2 , 3^12, 3^22, 3^5,<span style="mso-spacerun: yes;"> </span>3^15, 3^25}</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span></span></span></div>
<div style="text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">{3^4 , 3^14, 3^24, 3^7,<span style="mso-spacerun: yes;"> </span>3^17, 3^27}</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span></span></span></div>
<div style="text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">{3^6 , 3^16, 3^26, 3^9,<span style="mso-spacerun: yes;"> </span>3^19, 3^29}</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span></span></span></div>
<div style="text-align: justify;">
<span style="color: #262626; font-family: Arial, Helvetica, sans-serif; font-size: 13pt;">{3^8 , 3^18, 3^28, 3^11, 3^21, 3^31} </span></div>
<div style="text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><br /></span></span></div>
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">In the language of abstract algebra,<br />
</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>{3^0 , 3^10, 3^20}<br />
</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">constitutes a subgroup of the multiplicative group of the field with 31</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">elements and<br />
</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>{3^3,<span style="mso-spacerun: yes;"> </span>3^13, 3^23}</span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">one of it's cosets. </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><o:p> </o:p></span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">Moreover</span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>{3^0 , 3^6, 3^12, 3^18, 3^24}<br />
</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">constitutes a disjoint subgroup with 5 elements.</span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><br /></span></span>
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">The five sets come from </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">the first by multiplication, as one see quickly if one keeps in mind the </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">congruence of 3^30 with 1 modulo 31.<span style="mso-spacerun: yes;"> </span>Multiplication permutes intervals,</span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">but preserves all interval hexachords.</span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><br /></span></span>
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">One could still argue that {3^0 , 3^10, </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">3^20, 3^3,<span style="mso-spacerun: yes;"> </span>3^13, 3^23} giving an all-interval hexachord seems miraculous, </span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">but one surely gets all the intervals if no interval occurs twice. So write</span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><o:p> </o:p></span><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>A = {3^0 , 3^10, 3^20}<span style="mso-spacerun: yes;"> </span>B = {3^3 ,<span style="mso-spacerun: yes;"> </span>3^13 , 3^23}</span></span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #262626; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">Obviously distinct pairs in A give distinct intervals (or the interval would have to be preserved by multiplication by 3^10); similarly distinct pairs in B; similarly a pair in A compared with a pair in B similarly two pairs each crossing A to B. Thus the only not obvious case would be a pair in A, say, and a pair crossing from A to B.<span style="mso-spacerun: yes;"> </span>One can also rule out some of these cases a priori, and reduce the total number by symmetry, but in any case one doesn't have so many cases that the non-occurrence of an equality seems miraculous anymore.</span></span><br />
<div style="text-align: right;">
<span style="font-family: Arial, Helvetica, sans-serif;">– David Victor Feldman</span></div>
<div style="text-align: right;">
<span style="font-family: Arial, Helvetica, sans-serif;">9 April 2014 </span></div>
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<span style="color: #262626; font-family: Helvetica; font-size: 13.0pt; mso-bidi-font-family: Helvetica;">_______________________</span><br />
<span style="color: #262626; font-family: Helvetica; font-size: 13.0pt; mso-bidi-font-family: Helvetica;"><br /></span></div>
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[1] For a direct relationship to these sets in more recognizable integer notation, it may be helpful to use the following distribution of 3^n as an overlay:</div>
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</div>
<blockquote class="tr_bq">
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<tr height="29"> <td class="xl24" height="29" width="63" x:num="1.0">1</td> <td class="xl24" width="63" x:num="25.0">25</td> <td class="xl24" width="63" x:num="5.0">5</td> <td class="xl24" width="63" x:num="27.0">27</td> <td class="xl24" width="63" x:num="24.0">24</td> <td class="xl24" width="63" x:num="11.0">11</td> </tr>
<tr height="29"> <td class="xl24" height="29" x:num="9.0">9</td> <td class="xl24" x:num="8.0">8</td> <td class="xl24" x:num="14.0">14</td> <td class="xl24" x:num="26.0">26</td> <td class="xl24" x:num="30.0">30</td> <td class="xl24" x:num="6.0">6</td> </tr>
<tr height="29"> <td class="xl24" height="29" x:num="19.0">19</td> <td class="xl24" x:num="10.0">10</td> <td class="xl24" x:num="2.0">2</td> <td class="xl24" x:num="17.0">17</td> <td class="xl24" x:num="22.0">22</td> <td class="xl24" x:num="23.0">23</td> </tr>
<tr height="29"> <td class="xl24" height="29" x:num="16.0">16</td> <td class="xl24" x:num="28.0">28</td> <td class="xl24" x:num="18.0">18</td> <td class="xl24" x:num="29.0">29</td> <td class="xl24" x:num="12.0">12</td> <td class="xl24" x:num="21.0">21</td> </tr>
<tr height="29"> <td class="xl24" height="29" x:num="20.0">20</td> <td class="xl24" x:num="4.0">4</td> <td class="xl24" x:num="7.0">7</td> <td class="xl24" x:num="13.0">13</td> <td class="xl24" x:num="15.0">15</td> <td class="xl24" x:num="3.0">3</td> </tr>
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</blockquote>
If X is any one of these five sets,<br />
<blockquote class="tr_bq">
V(X) = [6111111111111111]<br />
V(X, complX) = [0 a a a a a a a a a a a a a a a] (a=10)</blockquote>
remembering that complX always includes 0, the odd man out.<br />
<br />stephen soderberghttp://www.blogger.com/profile/17300056962479866094noreply@blogger.com1