Monday, November 9, 2015

Broken Symmetries 2

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 – Les Yeux dans les roues, Les Mains de l'abíme, Ile de feu.
– Pierre Boulez *

Olivier Messiaen
Performed by Yvonne Loriod
( A score may be located through OCLC WorldCat )

Diagram 1
Compositional scheme of Île de feu 2
(Timings refer to the Loriod recording)

Descriptive Analysis (C2)

The previous post, Broken Symmetries 1, concentrated on the analytical dilemmas posed by the brief passage labelled C1 (mm. 70–75) in Diagram 1. C1 appears, as if out of nowhere, almost exactly half way through the piece as a stream of even pp legato 16th-note triplets. Architectonically, C1 only makes sense to me as a prefiguration of C2, the lengthy section I'm calling the 'Magma Dance' (this begins around 2:55 in the Loriod recording). Like C1, C2 is a double stream of 16th notes, one stream running in the right hand, the other in the left hand.

When I referred to C1 in the previous post as a passage of even triplets in two-voice 'first species counterpoint,' I wasn't entirely joking. Both C1 and C2 are instances of note-against-note counterpoint, but this is 'counterpoint-by-other-means.' 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 counterset. 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.

But now to the story of Messiaen's notes and his (and my) 'conflict between spontaneity and organization.'

C1 characteristic that I saved to discuss for comparison to C2 is the disjunct, narrow ranges occupied by each voice in C1. There the left hand spans the range C3–B3 while the right hand stays within the range C4–B4, disjunct from the left hand (Example 1a).[1]

Example 1a                                                   Example 1b

The C2 passage on the other hand was described by Messiaen as 'a cross-handed perpetual motion in the depths of the keyboard.'[2] Example 1b illustrates that the left hand's range in C2 is again bounded within a single octave. Down a fifth from the left hand in C1, it spans F2–E3. The ceiling of the right hand in C2 also drops a fifth from what it was in C1, taking the r.h. ceiling down from B4 to E4; but the r.h. floor drops from C4 down more than two octaves to Bb1, a fifth below the l.h. floor.  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 Île de feu 2.

A final contrasting element is that in C1 both hands are marked pp legato – but in C2 the right hand is marked f staccato against a 'ghosting' left hand that retains the C1 character as p legato. We'll now focus on the ghost in the left hand.

.      .      .      .

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.

Diagram 2

Let's consider the left hand (bottom half of the strips) which as noted previously spans only one octave, F2–E3. 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):

9–5–4–10–6–3–11–7–2–0–8–\   /–8–0–2–7–11–3–6–10–4–5–9

This symmetry has been pointed out elsewhere[3], 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 C2 were 'serial' in the usual sense that word is taught to be taken, it would mean C2 is a simple concatenation of ten distinct 'tone rows' (which would be somewhat of a record for just one minute of music, and a triumph of compositional randomization.

So we've arrived at Île de feu 2'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.

Diagram 3

Diagram 3 shows that there is a vertical reflective symmetry accompanying the horizontal reflective symmetry, turning C2'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).

Diagram 4

This simplifies the analytical focus since it means that, to discover any further relationships within C2 (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.[4] Any relationship we can find among these ten 12-tone strings will be duplicated in the other quadrants.

Antinomous Analysis (C2)

The ten interversions in sections B1, B2 and B3 (compositional scheme, Diagram 1) have been the focus of most commentary about this work. The passage C2 (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 C2 passage is Île de feu'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 C2)? The story here is not, as in the fictive analysis of C1 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.'[5]

All that our strictly descriptive analysis of C2 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.

So let's begin by trying to find in Diagram 4 all the indisputably non-random features as well as noting the features that suggest randomness.[6] Following is a list of seven compositional features in C2. Each feature is followed by an analytical judgement:
   [] = functionally determined,
   [?] = 'non liquet' ('it is not clear'), i.e., suggestive of a functional relationship that remains undiscovered, or
   [] = tentatively random (no obvious evidence for either [] or [?]).

(1.) To reiterate from the descriptive analysis above, the entire 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. []

(2.) String 1 in Diagram 4 can be derived as a 'double retrograde' followed by a (k,n)-perfect shuffle.

Let's say we are given a 'deck' (or set) of 12 pitch classes 0 through 11, and we divide it into k=3 cuts (subsets), each containing n=4 pcs:

[1,2,3,4], [5,6,7,8], [9,10,11,0].

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


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:

[9,10,11,0], [5,6,7,8], [4,3,2,1].

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:


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:

A : B :: B' : A'

where the As are SC-015 and the Bs are SC-037. A={4,5,9}, A'={8,0,1}=I5(A), B={3,6,10}, B'={7,11,2}=I5(B) (and, of course, the two hexachords are also related by I5). Checking back to Diagram 2 (l.h.) we see that in mm. 92–93 Messiaen grouped the 16th notes in 3's. 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. []
<<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 (Messiaen, 1975, p.108, Table II)>>

(3.) 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


which is string 2. The initial chromatic tetras are rearranged internally but kept intact harmonically (SC-0123) and related by T4. Checking back to Diagram 2 (l.h.) we see that in mm. 94–95 Messiaen grouped the 16th notes in 4's. Again, it looks like he was grouping by quadruplets according to this string's generating function operating on sets of four elements. []
<<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 (Messiaen, 1975, p.108, Table II)>>

(4.) String 5 is generated by simple transposition, T5, from the seed string in passage C1 discussed in  Broken Symmetries 1.

[10,3,9,0,6,1,7,2,8,4,5,11] = T5([5,10,4,7,1,8,2,9,3,11,0,6]).

(... or was C1's seed string derived from String 5 by T7? At any rate ....)

String 5 functionally connects C2 to C1.

As shown in Diagram 2, in mm. 100-101, the 16ths are grouped in 3's, as they were generated in the entire C1 passage, mm. 70–75. []

(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.)

(5.) 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.

[9,5,4,10,6,3,11,7,2,0,8,1] → [5,4,10,6,3,11,7,2,0,8,1,9]. []

(6.) (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 C1? Trivially, any 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 Broken Symmetries 1). 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 [] as tentatively random. The reason the judgement remains tentative is due to the next feature observation. Examining the matrix columns, we find something else very suggestive that would seem to increase the odds against randomness for the matrix as a whole.

(7.) Column 5 reads down 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 vertical 4-string groupings with an extra element 6 tacked on the end: 6–9–3–5 / 6–9–3–5 / 6. Similarly, reading up 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 C1 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 (nn+4), where n=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'). [?]

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 C2 was consciously organized as C1 clearly was – to wrap it up in a neat functional package where everything relates to everything else. Neither is it possible to declare that the unexplained portions are merely random.

.      .      .      .

My conclusion concerning the entire passage C2 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 – especially 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 C2 he is conducting his earliest experiments in functional derivation/manipulation of material that go beyond the canonic T, R and I.

I invite others to take up the C2 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:



* 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 Orientations: collected writings. Edition quoted: Faber & Faber, 1990. p.414.

[1] Ranges are given in scientific notation: C4 = 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.

[2] Olivier Messiaen. 1994. Programme note in booklet accompanying Koch International Classics 3-7267-2 H1 quoted in Wikipedia article 'Quatre études de rythme'

[3] E.g., John M. Lee. 'Harmony in the Solo Works of Olivier Messiaen: The First Twenty Years.' In College Music Symposium. Vol. 23.

[4] In traditional notation, below is Diagram 4 with pitches as they appear in the score within the delimiting span F2–E3:


It was tempting at first to call this type of analysis 'inconsistency-tolerant' or even paraconsistent. 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 Île de feu 2 was more precisely seen as a reflection 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.

Boulez identified such a feature of a work as a compositional 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 analytic-decision 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.

[6] Four meditations on 'random.'

   (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] can't leave it alone.' 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.

   (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 formally determined (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 also 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 cannot be shown to be formally determined in this sense becomes a candidate for being judged analytically as a random 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.

   (3) As I use the word 'random' in a music-analytic setting, any strictly aural ('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, it doesn't matter'; but it preserves an essential place within techne for inspiration, serendipity, accident, and mistake.

   (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 must 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.

Friday, June 19, 2015

Broken Symmetries 1

[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. ... At some point we have to stop talking about decreasing symmetry and start calling it increasing complication.  –P.W. Anderson [1] 

Olivier Messiaen
Performed by Yvonne Loriod

Diagram 1
Compositional scheme of Île de feu 2
(Timings refer to the Loriod recording)

Descriptive Analysis (C1)

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 Île de feu 2 (sections B1, B2 and A4+B3 in Diagram 1). 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.

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 analytical process, the cell usually being a smaller motivic subset of the total chromatic/diatonic.  However, Messiaen's compositional 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.

Applying a permutation function p once to the chromatic scale yields the first interversion, also referred to here as the seed row. Applying p again to the result (the seed row) yields the second interversion, etc., until inevitably the seed row is returned by repeated actions of p. We'll pick up other information as we go along, but this is enough to start with.

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).
Example 1.
Olivier Messiaen,
Île de feu 2,

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 f1. In simple 2-line notation f1=

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:

f1 = ( 0  5  8  3  7  9  11  6  2  4  1  10 )

In either representation, pc0→pc5, 5→8, 8→3, ..., 1→10, and around the corner, pc10→pc0.

Since the length of the cycle is 12, after 12 actions of the permutation on the seed row [510, 4, 7, 1, 8, 2, 9, 3, 11, 0, 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).

Table 1.

So far there is nothing different in what I have presented from what the reader can find in several other more sophisticated sources.[2] From a single measure (we also could have discovered f1 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.

So let's now ask: Why 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?

Published analyses of these six measures that I have encountered so far agree that the seed row, [510, 4, 7, 1, 8, 2, 9, 3, 11, 0, 6] is arbitrary and unpatterned; or, like Messiaen himself, they make no statement at all about its structure or derivation.

In fact this row is 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).

The trichord generator built into the permutation function f1 is SC-016 (Forte 3-5 if you still insist). The resulting row S = [510, 4, 7, 1, 8, 2, 9, 3, 11, 0, 6] can be partitioned and the constituent (internally unordered) subsets labelled A, B, C, D:

{4,5,10} : {7,8,1} :: {9,2,3} : {6,11,0}
 A      :     B      ::     C      :      D

Note that the left and right hexachords and their constituent SC-016 trichords are mirror related, producing a nice set-class symmetry:

A ← I4 → D
B ← I10 → C
AB = {4,5,7,8,10,1} = O1–{11,2}
CD = {9,11,0,2,3,6} = O2–{5,8}

where O1 = {1,2,4,5,7,8,10,11} and  O2 = {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}⊂(CD) and {5,8}⊂(AB), 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.[4] 

But all of this set-class symmetry for constructing the seed row S is broken as soon as the seed row is permuted, i.e., as soon as f1 is applied to the seed row, generating the next row S' = f1(S) = [8, 0191034117652]. Parsing by consecutive trichords again, this time for S', 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).

Fictive Analysis (C1)

By choosing to use a generalized interversion technique, Messiaen certainly understood that, outside of the T, I and R 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.

Next note that, prior to any permutation, the potential interval-string 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 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.

In Example 2, the succession of the first three trichord interval strings <A>, <B>, <C> sets up an expectation for a <D> to "complete" the symmetry if <D> := <S>, so the "ideal" run to symmetry – the row Messiaen did not choose – would be <A><B><C><S>. (Angle brackets indicate interval strings: <A> = <+5,-6>, <B> = <-6,+7>, <C> = <+7,-6>, <S> ("symmetry completion") = <-6,+5>.)
Example 2.

The ordered interval strings within the row's trichords "should" be:

<+5,–6> : <–6,+7> :: <+7,–6> : <–6,+5>.

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 <S> 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.

.   .   .   .   . [5]

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 <D>  trichord ought to be in order to generate a row that retains the mirror symmetry of the corresponding harmonies. The question is: Why didn't 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 <D> := <M>?

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), and (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 f2 =


f2 = ( 0  5  8  3  7  9 ) ( 1 10 6 2 4 ) ( 11 )

The asymmetrically-derived f1 actually chosen by Messiaen is a cyclic permutation of length 12 yielding 12 interversions counting the seed row. The symmetrically-derived f2 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).

Table 2

So choosing f2 for the sake of symmetry in the seed row would have committed him to dealing with 30 interversions[6]. But a tougher problem is that fixed point in f2 – the 11 (B natural) which would have remained at the bottom of the 12-tone deck with every shuffle. If he were to use f2  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 and (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."

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:

Messiaen looked at the obvious symmetric ordering f2,
immediately noticed the fixed point,
scowled "This won't work for me here,"
rotated the last trichord to turn f2 into f1 (– aha! – a 12-cycle at that),
and "Problem solved."
He wrote it out,
went over to the piano, and
played mm. 70-75 right out.
His ear was satisfied.

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.

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 D trichord. (See Table 3.)

Table 3.

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 f1 it's a 12-cycle and so has no fixed points. Let's call it f3.

f3 can be derived in precisely the same way that we conjectured Messiaen's f1 may have been derived, except instead of rotating the final trichord of the f2 seed row one click "clockwise", [0, 6, 11] → [11, 0, 6] yielding f1, we rotate it one click "counterclockwise", [0, 6, 11] → [6, 11, 0] yielding f3. Example 3 shows the conjectural composed-out version of mm.70-75 had Messiaen discovered and chosen f3.

Example 3

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 how 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.

I think Grant Sawatzky was the first to note about these bars that the choice of the seed row dictates that the same 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.[2][7] 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 any permutation.

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 Île de feu 2 ....

Here [2, p.89] 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 f1 and f3.
[There is] an intervallic consistency 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 ...). That three of the possible seven interval classes (0-6) do not occur makes the [six]-measure passage sound quite uniform: one continuous episode, rather than six consecutive double statements of the aggregate. [My italics.]
We might resolve the  f1/f3 conundrum by dueling "sequence vectors." Referring to Table 3 for easy comparison, we see that every bar in the f1 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 f3 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 f1 interversions are replaced by two ic1's and one ic3 in f3 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: does it matter? 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?

It's time for an ear test.

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 (f1-generated) and which is the pretender (f3-generated). Because the real (analytical) question is, after all, IF Messiaen did 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 you have a choice, was Messiaen's the right one; would you prefer the other, or does it matter (which is to say, can you even hear the difference at performance tempo)? Here's the test:

And again one last time, whether your ear can distinguish between them or not: Would it have mattered if Messiaen ultimately chose f3 instead of f1?

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.

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:
[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[8]]
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."


[1] P.W. Anderson. "More Is Different: Broken Symmetry and the Nature of the Hierarchical Structure of Science." Science, New Series, Vol. 177, No. 4047. (Aug. 4, 1972), pp. 393-396.
[Added 11/24/2015:] See also The concept of 'broken symmetry' used in this and subsequent posts was first suggested 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.

[2] The best technical analytical survey I have read on Messiaen's music from 1950–1992, and which I highly recommend, is Grant Sawatzky, Olivier Messiaen's Permutations Symétriques in Theory and Practice, 2013.

[4] V(11,A)=V(2,B)=V(8,C)=V(5,D)=V(A)=V(B)=V(C)=V(D)=[100011].

[5] Here is where we enter music theory's version of Boorstin's "Fertile Verge."
"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….
"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 self-awareness. On the verge we notice more poignantly who we are, how we are thinking, what we are doing. Second, there is a special openness to novelty and change. 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 community-consciousness. 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."

Theoria [see 5.1 below] 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 mean (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, we leave the Angel out of this entirely in this case – "style" and automatic writing are not the same. Mostly the questions in the Fertile Verge are openings and revolve around a different sort of unanswerable. (Elsewhere I have called these "Babbittian questions.") 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 wherever X leads (to the extent our imagination can take us)  – with or without Messiaen or any other example of X that might be found in other musical works or even in things ostensibly unrelated to music. It beckons to take a chance, to consider absurd ideas, to make connections, to create new work.
[5.1] The theory/theoria distinction I am using is from David L. Hall, Eros and Irony (SUNY Press, c1982):
"[T]heoria is, above all, obedient to that sense of eros which lures toward completeness of understanding." (p.43)
"Strictly systematic theory [vs. theoria] is more often than not an ideological epiphenomenon functioning apologetically with respect to current modes of practice. Thus theory [vs. theoria] 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)
[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.

[7] The reader might be confused about what appear to be discrepancies between Sawatzky's notation of permutations and mine, he is mapping positions of pcs which is probably more relatable to accepted mathematical usage; I am mapping pcs directly, 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.

[8] In Iannis Xenakis, Arts/Sciences: Alloys, p.39-40.

Thursday, April 16, 2015

The Spiral Form – Developing Variations

To pick up a previous thread, we first need a quick review:

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 –
The Form
This abstract pattern led to a connection to a poetic form invented by a 12th-century troubadour –
The Sestina
The same pattern was also noted in card shuffling techniques that probably track back to the first known playing cards in the 9th century –
Jeu de Cartes
Then the form crossed paths with computer science in the 20th century  –
Parallel Processing
The pattern was also uncovered by a 19th-century musicologist as one basis for diatonic theory in music –
Hauptmann Shuffle
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 –
The Grifter, the Poet & the Composer

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?

Here is where the next post will eventually lead:

Sunday, March 29, 2015

Notes from the Pluriverse {14–16} (A mythology for music theory today)



In just seven years we will celebrate the tercentenary of Jean-Philippe Rameau's Traité de l'harmonie réduit à ses principes naturels (1722).
Three years later, we will celebrate the tercentenary of Johann-Joseph Fux' Gradus ad Parnassum (1725).

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 my 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.


Frontispiece from the 1725 edition of Fux' Gradus ad Parnasum

Josephus has completed his climb up the steps to the top of Mount Parnassus.
Surrounded by the Nine Muses, he receives the laurel wreath from Apollo.
In the background Pegasus charges over Mount Helicon
where his hoof strikes a rock, creating the Hippocrene spring,
fount of poetic inspiration.
As he bows to receive the wreath,
is Josephus holding his first musical work
or his completed counterpoint exercises?
– or –
Can you reach Helicon from Parnassus?

[added Apr 5 2015]

 "A Klee painting named Angelus Novus 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)

Wednesday, March 11, 2015

Notes from the Pluriverse {7–13 }


Seeing the nose on my face

"... to show the fly the way out of the fly-bottle." (Wittgenstein)

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 view a/the world. The model, flaunting the impossible, allows me to see into a/the world from outside – allows me to see (approximately) how that world relates itself to itself and, ideally, to other worlds (if they exist).

... but the fly is still in its bottle.


Dueling ontologies

I don't live in the world, but in the model of a world.
I don't live in a world, but in a model of the world.


Ethical dilemma

In the [music] pluriverse, models can be misinterpreted, misunderstood, misapplied.
But there are no wrong models.
This is not the case if [music] is a universe.
There the search is for the one right model.


Janus again.

Relationships inside/outside time (warped from Xenakis):

we theorize ourselves into (–do (–experience (–hypostatize)))
   a possible music-world inside time
          as composer, performer, audience;
we theorize ourselves from (–undo (–observe (–hypothesize)))
   a possible music-world outside time
          as analyst, critic, politician.

Question: Is there a corpus callosum that allows communication between the two?


Etymological shards from the Early Jargonic Period

(This started as a bit of dorky self-absorbed etymological play
– just killing time – with no intention of posting it.
Then I started to think about it.
And from deep in the recesses of
my aging brain came the whisper,
"Bosch's egg.")

θέα                                                   –  view
θεωρώ                               –  consider, speculate
θεωρός                  –  envoy sent to consult an oracle
θεωρία                           –  contemplation, speculation 
                                                             –  a looking at, viewing
                                                                –  a sight, show, spectacle, things looked at.
theory (1590s)                     – conception, mental scheme.

up, throughout + a loosening  –                     ανά + λύσης
analysis  –                                         ανάλυσης
         a breaking up, a loosening, releasing  –                                                                .
unloose, release, set free  –                               αναλύειν
 to loose a ship from its moorings  –                                                             .
resolution of anything complex into simple elements  –            analysis (1580s)  
composition (late 14c.)                               –  action of combining
                                             –  manner in which a thing is composed
compositus                       –  placed together
componere                                    –  to put together, to collect a whole from several parts
< com + ponere     .......
F. pondre                                   = lay an egg
One of the most remarkable facts in F[rench] etymology is the extraordinary substitution whereby the Low Lat. pausare came to mean 'to make to rest, to set,' and so usurped the place of the Lat. ponere, to place, set, with which it has no etymological connection. And this it did so effectually as to restrict the F. pondre, the true equivalent of Lat. ponere, to the sense of 'laying eggs;' whilst in all compounds it completely thrust it aside, so that compausare (i.e. F.composer) took the place of Lat. componere, and so on throughout. Hence the extraordinary result, that whilst the E. verbs composedeposeimposepropose, &c. exactly represent in sense the Lat. componeredeponereimponereproponere, &c., we cannot derive 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]  – From the Online Etymology Dictionary (retrieved Mar. 11, 2015)
Concert in the Egg, (follower of) Hieronymus Bosch (ca.1561)


Analytical heresy

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.

The clock's mechanism becomes important (and vitally so) only when the clock no longer tells the right time.


Back to the future: the "relevance" issue 25 years ago.

From David Lewin Collection, Administrative Papers, Library of Congress:
Lewin: The [Harvard] Graduate Program in Theory, iv/21/90 
What do ‘music theorists’ study?  Theorists’ opinions vary widely.  Mine: they study the vocabulary, concepts, and intellectual structures in general, through which people talk and have talked about the organization and coherence of music. 
How do you go about studying this?  Among other things,
1.     You explore the systematic bases for contemporary compositional methods.
2.     You study the history of music theory in our cultural tradition, and so far as possible in others.  (There is a question to what extent the whole notion of ‘music theory’ is meaningful when applied beyond our own cultural tradition.)
3.     You explore the systematic assumptions underlying received analytic methods.