[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]
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Olivier Messiaen
Performed by Yvonne Loriod
Descriptive Analysis (C1)
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).
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| Example 1. Olivier Messiaen, Île de feu 2, mm.70–75 |
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 [5, 10, 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).
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| 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, [5, 10, 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).
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, [5, 10, 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 = [5, 10, 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∪B ← I7 → C∪D
A ← I4 → D
B ← I10 → C
A∪B = {4,5,7,8,10,1} = O1–{11,2}
C∪D = {9,11,0,2,3,6} = O2–{5,8}
A∪B = {4,5,7,8,10,1} = O1–{11,2}
C∪D = {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}⊂(C∪D) and {5,8}⊂(A∪B), 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, 0, 1, 9, 10, 3, 4, 11, 7, 6, 5, 2]. 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).
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.
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, 0, 1, 9, 10, 3, 4, 11, 7, 6, 5, 2]. 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)
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>.)
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| 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>?
or,
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).
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| 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.)
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| 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.
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.
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:
____________________________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.
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| 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 http://plato.stanford.edu/entries/symmetry-breaking/#4. 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."
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."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."
[5.1] The theory/theoria distinction I am using is from David L. Hall, Eros and Irony (SUNY Press, c1982):
[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."[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)
[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.
