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.

[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.

Thursday, February 26, 2015

Desperately Seeking Relevance: Arts Administration Today

Like 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.

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.

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:
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.
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?

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?

Daniel Wolf has also written on this in his web site, Renewable Music. Go to: "Slow Death by Administration."

Wednesday, January 14, 2015

Notes from the Pluriverse {2–6}


Why do so few people get the distinction between "rule" and "rule"? One is confining, the other is liberating. There are "laws," and then there are "laws." Some divide, others unite.

Already we behave as if we live in a world that holds only a remnant of what there actually is .... I believe the major cause of this more mental than physical rift lies less in the folly or onesidedness of our societies and educational systems, or in the historical evolution of man into a predominantly urban and industrial creature, a thinking termite, than in the way we have, during these last hundred and fifty years, devalued the kind of experience or knowledge we loosely define as art; and especially in the way we have failed to grasp its deepest difference from science.
– John Fowes, The Tree.
That there are rules is a fact of art. That the rules are immutable is not.


In a 2002 program note[1], I included a summary of that exciting/nightmarish (take your pick) period in Western music (roughly 1890–1920) and referred to it as "The Crisis" (Fragment A below). It was during this remarkable short stretch of time, that the line of European music history finally broke out into three lines of Euro-Anglo-American music history. (It's important to note that I'm using the word "crisis" here not so much in its original sense of the "turning point of a disease," but in the sense of passing through a critical point of no return. Surviving The Crisis, you can still look back and perhaps learn from the past, but any continuity, real or imagined, has been broken: you can't regain either the innocence or the ignorance of the past.[2]) At the same time that The Crisis was working out publicly in concert halls, there was a related crisis in music theory that was at work below the surface, not so much among analytical theorists but primarily among theorist-composers[3] (Fragment B). While remnants can occasionally be found, today the three lines have mostly disappeared, replaced by a riot of musics and their theories: a pluriverse of possibility.

Anyone who claims the crown is a fool.

Fragment A: Praxis
If we ignore most of the fascinating detours, forget the dead ends, remain blind to other cultures or treat them as irrelevant – in a word, if we squint hard enough at European music history – we can just barely draw a straight line that traverses a millennium and a half from the Middle Ages right down to Brahms and Wagner on the verge of the twentieth century. Unfortunately, right at that point, our straight-line project fails completely. American musicologist Charles Seeger put it this way:
Since sometime before the First World War there has been a general realization among both conservatives and radicals that the great romantic tradition of nineteenth-century Europe was in difficulties. It had become encrusted with so many bypaths that some sort of revision seemed inevitable, either to set it upon its feet again or to form from its honored remains a new style.[4]
Seeger’s student and friend, composer Henry Cowell, was not as circumspect, leaving much less room for a comfortable conservative outcome:
Let us, however, meet the question of what would result if we were frankly to shift the centre of musical gravity from consonance, on the edge of which it has long been poised, to seeming dissonance, on the edge of which it now rests.[5]
Indisputably, Schoenberg and Stravinsky at this point were at the heart of the crisis in Europe; more importantly, a musical revolution that officially began in Europe was now heavily “in the air” on both sides of the Atlantic. Seeger capsulized the beginnings with due credit to Europe and then summarized what came next with a single word:
Certainly a revolution began, but a gradual one – perhaps a series of small revolutions: first Satie, Debussy, Strauss; second Scriabin, Schönberg, and Stravinsky; then the deluge.[4]
After the “exquisite elaborations” [Seeger's words] of the nineteenth century with figures such as Debussy and Strauss, the line of history forked into three main branches. The traditionalist branch attempted to set European music “upon its feet again,” stretching the use of the old material by insisting that it wasn’t yet exhausted. As critic and scholar Eric Blom noted, Fauré, Elgar, Vaughan Williams, and Sibelius remained “fundamentally loyal to their key signatures.” The other two branches, setting off in radically new directions and causing riots in the concert halls, were the “atonal” branch – centered in Vienna and identified with Arnold Schoenberg – and the “neo-classical” branch centered in Paris and headed by Igor Stravinsky.
Fragment B: Theoria
In 1985, Schoenberg scholar Severine Neff was visiting her friend, composer Otto Leuning, then 87, in his New York apartment. While helping him sort through his papers, Neff spotted an unfamiliar journal called The Monist published in Chicago in 1917. When she picked it up it fell open to an article entitled "Our Musical Idiom" by Ernst Bacon [see also prior E&EN entries on Bacon]. Among other things, the article included what appeared to be a complete list of all the chords possible in the twelve-tone chromatic scale. But the year––1917––was all wrong. Prior to Neff's discovery, 1960 [Howard Hanson's Harmonic Materials of Modern Music] was the date generally associated with the first publication of such a list. As Neff later wrote,
Five years before the debut of twelve-tone music and over thirty years before proliferation  of mathematically based theories of non-tonal music, Bacon was working on order permutation, invariance, and symmetrical inversion of non-tonal music.[6]


A note on post-tonal voice leading. There is a vast area ripe for research re note {4} which I will not even try to pursue in the detail it deserves. Both Bacon and Luening were greatly influenced by Bernhard Ziehn's theories. Busoni called Ziehn "Die 'Gotiker' von Chicago," and was inspired by him to take up the study of counterpoint once again. John Alden Carpenter studied with him also. In Howard Pollock's bio of Carpenter, there is this intriguing passage:
Carpenter wrote hundreds of harmonic and contrapuntal exercises under Ziehn. Most of the harmonic exercises involved short progressions; starting from a given triad or seventh chord, he would quickly move to some distant triad via passing tones, a whole-tone bass, or some other designated way. Some of the results sounded like Wagner or Franck, some like Reger or Busoni, some like modern jazz, and some like nothing recognizable.
There is a growing literature on Ziehn, but Neoriemannians looking for some new connections during this stormy period (I believe Ziehn was not too keen on Riemann's work, but that's irrelevant) might start by looking at his 1911 Five and Six Part Harmonies with reference to the above passage quoted from Pollock.


Harmonices Mundi Mod XII


[1] The "program note" was an extended essay titled "Toward an American Music" included in a season (2002-3) booklet celebrating the forty-year residency of the Juilliard String Quartet at the Library of Congress (also the end of JSQ's residency at LC –– but that's another story). I chewed off way more than I could manage in anything less than a book. I said nothing in the essay I would retract even today, but overall the essay doesn't make the points I had hoped to make.
[2] There must have been a sense of fear and loathing for many people during this key period –– a fear and loathing that appears to survive to this day in some composers, performers, audiences, and musicologists. The Online Etymology Dictionary humorously notes that a German term for "mid-life crisis" is Torschlusspanik, literally "door-shut-panic," fear of being on the wrong side of a closing door.)
[3] Previous posts are predicated on my contention that some analytical theorists may be composers, but all composers are theorists.
[4][5] I have lost the exact citations for these quotes, but as I recall, the two Charles Seeger quotes are from Studies in Musicology II, 1929-1979 (ed. & intro. Ann M. Pescatello. Berkeley, UCal Press, c1994) & the Henry Cowell quote is from his New Musical Resources (orig. pub. 1930, but I probably used the 1969 repub. [NY], Something Else Press)
[6] Severine Neff. "An American Precursor of Nontonal Theory: Ernst Bacon (1900–1990)." Current Musicology 48: 5-26.

Monday, January 12, 2015

Notes from the Pluriverse {1}



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."

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.

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.

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.

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.

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.

"Steve, do you compose?"
Big G.P. while I chewed on a bite of toast.
"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 ...."

He interrupted, quietly, almost conspiratorially:

"You should make the time."

No one had ever before gotten to my well-guarded core.

Others began to straggle in and join us, and then we were all shuttled off to Buffalo (U) for the day.

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. It was much later, after his death, that I got an answer of sorts.

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.

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.
Your overriding interest is in the man [Schoenberg] 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”?
February 26, 1974