Ernst Bacon Redux (1)

Gold is rarely discovered

by one who has not got

the lay of the land.[1]

– Norwood Russell Hanson

Patterns of Discovery

In 1916, Ernst Lecher Bacon, at the age of 18, submitted a technical article for publication to The Musical Quarterly. The editor of MQ at the time, Oscar Sonneck, declined to publish Bacon's article, but not because it was untutored or incompetent or poorly written or that it offered nothing original. While he may not have followed the math (the how of the article), Sonneck understood what Bacon had done and knew quite well that it was an astonishing accomplishment.

Still, he rejected the article on the grounds that MQ's primary readership – historians, analysts and others in the musicology orbit – even if they could follow the math (a big leap in the music world, even today), would find it irrelevant (again, even today). But, recognizing the genius of this accomplishment, Sonneck did not simply write a brief rejection. He wrote back an encouraging letter (via Bacon's mentor, Glenn Dillard Gunn) suggesting that Bacon submit the article to the philosophy journal The Monist. It appeared in the October 1917 issue of that journal under the title "Our Musical Idiom."[2] Unlike events in the political world of October 1917, Bacon's groundbreaking work inspired neither a revolution nor a war. Here's what happened:

It was virtually ignored.

What Bacon had done was create a combinatorial algorithm for listing all the sonorities – represented as  transposition classes – in the 12-note chromatic scale.[3] Then, using common music notation, he proceeded to list all 350 "prime-position" sonorities (he omitted the empty set and the singleton which he thought of as trivial). It was a bit like snatching the gold ring, when hardly anyone else on the merry-go-round knew there was a gold ring to snatch. Here's how he did it.

First, he stopped thinking in terms of notes or pitches (or pitch-classes) as the primary musical objects  and began thinking in terms of the intervals between notes. This is the single move that takes the task from nearly impossible to tractable. So chords or scales were not his quarry, but abstract strings of intervals.

It's the difference between, say, a C-major triad (a concrete structure which consists of the three notes/pitch-clases C and E and G) and an icon representing any triad in the same relationship as the notes in the set {C,E,G}. This unique icon simply names the chromatic intervals between notes in some circular permutation. Using Howard Hanson's notation from Harmonic Materials of Modern Music, we might represent a C-major triad within an octave span as:

C4E3G5(C).

Dropping the referential note-names, the collection of all major triads can be described abstractly by a circular permutation of the interval string 4-3-5. (This relationship must hold (mod12) if you are to respond correctly to the question, "Go to the piano keyboard and play any major triad.") This interval string notation is precisely how Bacon calculates and names all distinct transposition classes mod 12. Here is his table listing all the trichords[4]:

Note that he uses an abbreviated "name" for each unique "harmony." This is a convention that he uses throughout for sonorities of any cardinality. E.g., here the name for C.1 (Combination 1) is given as 1-1 instead of the complete 1-1-10. Since all the intervals in a string in 12tET must sum to 12, the final interval can be left off and still identify the same unique transposition class.[5] Working backward, we now ask, just how did he generate these 19 trichords? Take a look at his chart summarizing his derivation of trichords (ignore the column "Calculations of Harmonies" for the moment):

This chart for trichords represents the solutions to the three equations to the left of the arrows:

(1)     2a + b = 12    →   a-a-b (1-1-10, 2-2-8, 3-3-6, 5-5-2)

(2) c + d + e = 12   →   c-d-e (1-2-9, 1-3-8, 1-4-7, 1-5-6, 2-3-7, 2-4-6, 3-4-5)

(3)            3f = 12     →   f-f-f (4-4-4)

with the stipulation that all the solutions are positive integers less than 12. The mathematician will immediately recognize these equations as simple linear Diophantine equations, and the triples to the right of the arrows (the exact solutions) represent the 3-partitions of 12 (all the triples of integers that add up to 12).[6]

Translating to music, all the integers a,b,c,d,e,f represent intervals less than an octave (measured in number of chromatic steps). To the right of the arrows, the triples represent interval strings. If we stopped here we would have the familiar list of 12 trichord set classes, but like most musicians Bacon wants to distinguish between inversions – e.g., major and minor triads are different animals. So all these results are permuted until all the unique permutations have been discovered. This final process sorts the trichords into symmetric and asymmetric (among other things – see the next post). Permutations of the asymmetric sets (C.2,3,4,5,7,8,11) add the respective inversions, so 1-2-9 is joined by its inversion 2-1-9 (or, what's the same, 1-9-2 using Bacon's preferred ordering). So the complete list of 19 trichords Bacon generated (in integer notation) is:

(1') a-a-b: 1-1-10;  2-2-8;  3-3-6;  5-5-2

(2') c-d-e: 1-2-9, 2-1-9;  1-3-8, 3-1-8;  1-4-7, 4-1-7;  1-5-6, 5-1-6;  2-3-7, 3-2-7;

                2-4-6, 4-2-6;  3-4-5, 4-3-5

(3') f-f-f:   4-4-4

which (for those who don't recognize them) appear in abbreviated notation in his list of "harmonies" (see above). Using Bacon's ordering:

1-1    1-2    1-9    1-3    1-8    1-4

1-7    1-5    1-6    2-2    2-3    2-7

2-4  2-6   2-5   3-3  3-4  3-5   4-4

(To be continued)

________________________

[1] The context for the quote can be found online (Ch.1 of Patterns of Discovery). But the context is important enough to quote a bit more here: "It is the sense in which Tycho and Kepler do not observe the same thing which must be grasped if one is to understand disagreements within microphysics. Fundamental physics is primarily a search for intelligibility – it is philosophy of matter. Only secondarily is it a search for objects and facts (though the two endeavors are as hand and glove). Microphysicists seek new modes of conceptual organization.  If that can be done the finding of new entities will follow. Gold is rarely discovered by one who has not got the lay of the land. . . . It is important to realize ... that sorting out differences about data, evidence, observation, may require more than simply gesturing at observable objects. It may require a comprehensive reappraisal of one's subject matter. This may be difficult, but it should not obscure the fact that nothing less than this may do." [my emphases] Does this apply to music theory? Recall Lewin: "This is the methodological point: We must conceive the formal space of a GIS as a space of theoretical potentialities, rather than as a compendium of musical practicalities." (GMIT 2.3.2)

[2] I'm going from memory in this entire account, having last read the source material 4-5 years ago. I no longer have direct access to the correspondence in the Bacon, Gunn and Sonneck papers in the Music Division at the Library of Congress which are the basis for the story of the publication of "Our Musical Idiom." I should have made private copies or taken extensive notes, but I didn't.

[3] Catherine Nolan has written extensively on the history of this particular aspect of the connection between music and mathematics. Available on line, as one example, is her 2000 Bridges paper "On Musical Space and Combinatorics."

[4] I will henceforth try to remember to use "triad" when referring to the familiar, historically-conceived major, minor, diminished and augmented triads. I will use "trichord" to refer to the (ahistorically-conceived) set of any 3-voice sonority, usually the list of 12 trichord set classes or the list of 19 trichord transposition classes in 12tET. This is exactly backward from how I would prefer to use these terms since trichord suggests sonority and leaves no room for extensions of the concept such as 3-point rhythmic structures; and triad, a more abstract term that could refer to any set of 3 things, has been historically usurped within music to refer to traditional Western 3-voice chords built of superimposed thirds (or however you wish to construe/generate them). So any triad is also a trichord, but not all trichords are triads. No, this does not make any sense, but I didn't create this particular mess.

[5] This shorthand is useful but can easily result in errors such as mistaking a symmetry as an asymmetry leading one to think there is a distinct inversion where there is none. E.g., listing the tetrachords 1-1-5 and 5-1-1 as distinct inversions when citing the "full" name, 1-5-5-1, reveals it is a symmetry & has no distinct abstract inversion. So reader beware.

[6] Readers who have not previously encountered the idea of using numerical partitions to generate chord lists may wish to go to the interactive page for partitions at the Combinatorial Object Server and plug in random values for k and n (ignore m) to quickly get an idea of how important this simple idea is. Start with n=12 then increase n to get partitions (the math term for what Bacon calls "Combinations") for larger musical universes. E.g., for partitions leading to listing hexachords in 12tET set n=12 & k=6 to get 11 partitions to work from; for hexachords in quartertone space set n=24 & k=6 to get 199 partitions to work from.

Procrustean Intonations

While packing up my library yesterday, I remembered a connection that came to me a couple of years ago.

There is an undercurrent I have observed within the music theory community (ever since I realized there was such an unlikely community) that I have come to call, in my inimitably bland way, "feature cognition." I mean by this the tendency in many an eager scholar to hear (or, more likely, view) a feature or set of related features in a musical work such that, once one is made aware of the feature, it threatens to obscure the work-as-a-whole (the "Music") – a musical version of missing the forrest for the trees. I hide behind this term because, while it is not altogether accurate for what I wish to express, still it points in the right direction and feels reasonably inoffensive.

But then one day, while reading through the David Lewin correspondence, I was a bit surprised to read my thoughts on this put in a much less polite way by David:

Too many analyses I have read (or performances I have heard!) proceed on the pattern: listen to the opening of the piece until you get an idea that interests you; then ignore everything else and plow through the rest of the way, trying to make the rest of the piece fit your idea. (This I call to myself "Procrustean intonation.")[1]

The first time I read those words, while I admired the way they cut right to the quick much better than my too-polite descriptive term, I thought "procrustean" a bit over the top coming from the gentleman I thought I knew – and David was no longer around to ask about whether he would still use the word or whether his judgement about the state of professional analysis had softened significantly over the intervening 30 years. His observation came, after all, in private correspondence. But still, "procrustean" is a violent word. Or do I read more into this than David intended?

Procrustean: in the figurative sense, "violently making conformable to standard," from Procrustes, mythical robber of Attica who seized travelers, tied them to his bed, and either stretched their limbs or lopped off their legs to make them fit it. The name is Prokroustes "one who stretches," from prokrouein "to beat out, stretch out," from pro- "before" and krouein "to strike."[2]

Arguably, David Lewin was contemporary music theory's Theseus, founder of our little Athens. So feature-cognition analysts, take care! Remember how Procrustes met his end.

__________________________
[1] Letter to Oliver Neighbour, May 8, 1973. Correspondence, David Lewin Collection, Music Division, Library of Congress.
[2] Douglas Harper's Online Etymology Dictionary.

The Glass Bead Game: The New Abnormal.

First, an announcement . . . .


Mary Ann and I, along with our two golden retrievers, Sally and Sundance, are moving from the Washington metro area to Charlottesville VA in November (actually, just outside Charlottesville in Keswick). I will likely not be adding many (or any) new posts to this blog until the dust settles. But when I return to Essays & Endnotes in November-December, I have plenty of material I am planning to cover: not only the music theory and deep connections that I have been slowly trying to develop, but interesting offshoots such as an educated guess about Milton Babbitt's role in WW2 and its connection to his subsequent music & music-theoretic contributions. And what may appear to be my running in meaningless circles around elementary neo-Riemannian theory & attempts to find a way out of what appears to me to be contemporary music's death spiral, will finally focus on, among other things, Roger Reynolds' bizarre "editorial" compositional techniques as he has consistently applied them in the creation of virtually all his music. (I say "bizarre" not as a pejorative, but because these techniques, applied to what begins life as a 12-tone work, have the effect of totally destroying any reference whatsoever to the original row.)


Until I'm back, here is a restatement of what has been driving me for so many years now.


A little over fifteen years ago I wrote a short essay, "Riemannian Variations on a Theme by Milton Babbitt." The theme running through this essay was something I called a "Babbittian Question" which I defined as a problem whose solution is likely to result in further questions. Nowadays I think of it more cryptically as a question whose only correct answer is another question. (Milton indicated his approval of the idea as originally stated, but I'm not so sure he would have approved of this Cagean twist.)

Here is a slightly updated version of the theme of that essay.

Around 1900, give or take a quarter century, Western music's common practice died. But the hole it left was almost immediately filled with a different commonality that survives with a vengeance to this day.

Here's the situation:

On the one hand it is nearly impossible to imagine Mozart sitting down before a blank sheet of manuscript paper and asking himself, "How shall I arrange the twelve notes this time?" On the other hand it is equally dificult to imagine any composer of our present age sitting down in front of a blank sheet of virtual manuscript paper and not asking some version of that very question. Even the "neo-tonalist," simply trying (intuitively?) to write some of that good music left to be written in C major, at least feels its ominous presence.

Our current language points to the problem. If we were to utter words like "precompositional design" or "compositional algorithm" to Mozart, he would no doubt stare blankly at us. And if we were to mention "compositional theory" he might respond, "What other kind is there?" Borrowing a phrase from Michael Colgrass, "Instead he just wrote music. Poor soul."

So here it is. Laid bare. The truly radical core of the twentieth-century revolution in music. The single thing binding together the most antagonistically disparate minds.

We have become self–conscious.

Now, after the Great Demise, we must think about it – theorize if you will – before we compose. Whether this is a one-time event beginning our career or a re-evaluation mid-career or, literally, every time we sit down to make a new piece of music.

Whether we choose to be serialists, atonalists, diatonicists, minimalists, maximalists, spectralists, microtonalists, fractalists, math rockers, quasi-anarchists (even John Cage chose to use the I Ching), or proud naifs – before we get down to work – before we can create – before we can compose, perform, listen, judge and bloviate – there are decisions to be made and questions to be answered.

But what questions?

And do they have a common source or thread?

Perhaps they are all models of the same quest–

ion.

And, a bit out of sequence, here are two more sentences from that essay.

A sampling of current terminology is revealing: "original instruments," "authentic performance," "re-releases," "revivals," "restorations," "golden oldies," "creative programming," "cross-over." These all represent pleasant and often worthwhile and sometimes important ways to rearrange the musical furniture provided by our heritage; but they also represent convenient strategies for turning our backs on invention and innovation.

So now, here is the puzzle. Given that the common practice died, i.e., used up all the harmonic resources based on the ubiquitous """consonant""" major and minor triads such that they could no longer express anything truly new, and given that there are seventeen other perfectly good triads in the twelve tone chromatic that might expand the composer's fundamental material, why hasn't this expansion taken place in such a way that a new common practice has begun to appear?

On the Verge of Non-Riemannian Musical Geometries

Its pleasure is not the comfort of the safe harbor,

but the thrill of the reaching sail.

– Robert Grudin

In the previous two posts I used the example of a passage from the Scherzo of Beethoven's 9th Symphony purportedly to introduce an alternate nomenclature for neo-Riemannian (NR) theory's three fundamental triad progressions. Here is a complete comparison of the Riemannian moves with the same moves notated as triples (a,b,c) from what I call an extensible Riemannian (ER) system.

NR progressions

[M=major triad, m=minor triad, r=root, f=fifth, t=third, s=semitone, w=whole tone]

ER moves for SC3-11

[Numerical values indicate number of semitones in 12tET]

Given that the objects being operated on in both cases are major and minor triads, there is no real difference between the two abstract voice leading systems. They are, at this point, merely different expressions for the same thing. When two moves are combined such as R and L* to create a chain as in the Beethoven example, NR analysis must first locate the fifth of the triad for each move, alternating moving it up by whole and half steps, whereas ER, for each move, shuffles a mask by a T* permutation and multiplies it by the same formula each time. The result from ER is the same as that from NR, as I intended it to be. Obviously, in this context, there is no gain. And admittedly, the NR version of "locate a note and move it up or down by x" is musically more intuitive than "shuffle, multiply and add" – given that the objects being operated on are major and minor triads.

Let's try yet another way of viewing these three progression types by focussing on common tones rather than the tones that move:

The P [parallel] transformation exchanges triads that differ in modality but have the same root; whence the pcs common to both triads are ic5-related. The L [leading tone] transformation exchanges triads of opposite modality with common pcs related by ic3, and R [relative] exchanges triads of opposite modality that have common pcs related by ic4. (Jack Douthett. "Filtered Point-Symmetry and Dynamical Voice-Leading" in Music Theory and Mathematics: Chords, Collections, and Transformations, ed. Douthett, Hyde, Smith)

Formulations of this sort that are based on common tones (certainly Douthett was just paraphrasing older formulations) make even more clear the notion that the diatonic (usual major/minor) triad is axiomatic in Riemannian and NR-based theories. The triad – the object – is ultimately in control. The operations are valid or not as they are found to conform or not to the "shape" of the object. To wit: the assumption is that if you hold any two tones of the triad constant, there is only one place for the free tone to go to create another consonant (read "valid") chord – up or down a minor or major second.

Here is the crux of the issue. Riemann & company arrived historically at a time when at least one axiom – one unexamined, unquestioned premise – was still agreed upon by all. Riemann, like Schenker, begins with the assumption that the usual triad, however you want to tune it, is the basis for Western common-practice music. And this is true. But what this means is that all (common practice) theories, whether harmonic or contrapuntal,  revolve around the irreplaceable object and not, as "tonal" and "neo-tonal" theories seem to want us to believe, its transformations .

In geometry, Euclid takes us a long way by accepting the common sense notion that parallel lines never meet. But there are other geometries that begin by denying that axiom. There are other worlds to explore. An ever-present question for us in music ought to be: what happens if we deny axiomatic status to the earth-bound triad?

This puts us inexorably back on a path to places dreamed by poets. Stefan George, among others.

"I feel wind from other planets."

The Art of Parsimonious Orchestration

"It's not the arrival, it's how you get there."

– Unidentified passenger on The Orient Express.

I left off the previous post with the question "What do 'we' really hear [in music]?" The reason the 19 chords of Beethoven I used as an illustration have become well-known to many (well, at least a few) music theorists is because they represent one of the longest strings of neo-Riemannian transformations ("transformations" here meaning chord progressions) identified in the music literature.

En passant:

• As in many fields, in contemporary musicology an analytical theory gains greater credibility every time a confirming instance is found in the real world (meaning, generally, music composed/performed before the theory was systematically formalized). 
• Music-analytic theories are radically retrodictive.
• When a sufficiently large number of peer-approved confirmations has been collected, the theory is recognized by a cohesive, but not necessarily universal, peer group as "true." But given the overlap of theories that are incontestably true, the turf battles fought within the musical academic community (giant egos to one side) appear, to this outsider, to be fought almost entirely over applicability, relevance, importance, significance, generalizability, etc., and rarely over facticity. 

But back to those 19 chords. Following is a reduction of the score that demonstrates a form feature that arises from Beethoven's orchestration of the passage.

While it is incontestable that the entire passage is the string of neo-Riemannian transformations[1] R,L,R,L,R,L,R,L,R,L,R,L,R,L,R,L,R,L (summarily, (RL)⁹ or, using the ad hoc computer pseudo-code notation from the previous blog entry,  X¹⁸) the orchestration of that passage indicates something more is going on – at least it was in (flagrantly flouting the intentional fallacy) Beethoven's head.

Some might say the following is an alternative reading, but I see it as a concomitant. The score reduction above shows that Beethoven was doing a game of major-minor hopscotch between the strings & horns (lower staves) and the woodwinds (upper staves). Red boxes indicate major triads and blue indicate minor triads. If you were to eliminate the blue box material, the result would be a perfectly logical sequence of major triads whose roots follow a succession of perfect fourths. The same will happen if you eliminate the red box material resulting in a sequence of fourth-related minor triads. The result of alternating the two certainly is the RL sequence. But, once again, Beethoven doesn't do a "straight" alternation. His orchestration keeps them separate creating a tension not present in a straight RL.

After beginning with the strings-horns doing the major triads with the woodwinds alternating minor triads taking us from C major to D minor, there is a pause, after which the woodwinds do the major and the strings-horns the minor, getting us from D minor to Eb major. Then, starting at Eb major after another pause, the strings-horns take up the major while the woodwinds alternate the minor sequence again, and this lasts until we arrive finally at A major, ready to make a final quick chromatic hop to E minor. All in all, a little like driving from Denver to Las Vegas by way of Atlanta. Not the shortest route, but a most interesting one. For those who like graphs and circles, here is what happens (string-horn moves are inside the circle, woodwind moves are on the outside):

So which is it? What's the right way to hear this passage? Should I follow the straight RL sequence, or should I follow the hopscotch counterpoint of major and minor triads? Can I hear both at once? Or if I'm the conductor, which should I bring out? And we haven't even brought Schenker into this wonderful mess!

Enough of Beethoven for now.


__________________
[1] In neo-Riemannian lingo, R is the "relative" transformation and L is the "Leittonwechsel." There are several ways of defining these two basic transformations (the third basic transformation is P for "parallel"), but simply here, since we begin with a major triad, R indicates "raise the fifth of the triad a whole tone" resulting in a minor triad (CEG→CEA). Then, from the minor triad, L indicates "raise the fifth a semitone" yielding a major triad (ACE→ACF).

The Grifter, the Poet & the Composer

____ The Grifter ____

The "Three Shell Game," close cousin of "Three Card Monte," ranks among history's oldest and simplest scams. Here I'm less interested in the sleight of hand aspects than the simple idea that there are only a limited number of ways for the grifter to rearrange the three shells. So let's fantasize a shell game run by Honest Grifter – no sleight of hand – every move is what it appears to be.

In such a game it would be possible, in principle, to locate the shell hiding the pea every time. If the original positions are labeled left to right (123) and the pea starts under #2, it doesn't take rocket science to see that there are only six final positions possible, (123), (132), (213), (231), (312), (321), and if you concentrate and follow where #2 goes (remember, there's no sleight of hand), you win.

But Honest Grifter has even fewer actions available to him in any single move than the six final positions would suggest. After all, he has only two hands, so he can only move two of the three shells at the same time. This means he has only three possible ways to rearrange the shells in any given move: (123)→(132); (123)→(321); (123)→(213). In cyclic notation this is: (1)(23); (2)(13); (3)(12) – reflecting whether the first, second, or third shell is left alone during the move. But wait. Honest can still win because the only change we made was to remove the possibility of a sleight of hand. Honest still has control because he can make as many moves (permutations) as he wants, and he can do this as fast as he is able, leaving you cross-eyed. And, in fact, even though he can only move two shells at once, he can still reach any of the six possible final positions even with this limitation. Here's one example:

____ The Poet ____

As any form becomes canonical,

it virtually invites

experiment, variation, violation, alteration.
(Anthony Hecht)

Apologies for repeating that quote from Anthony Hecht yet again. It's not just that I really want it to sink in as the only principle of evolution I know of in the arts, but this section is an example of at least one way that principle can work by shrinking a canonical form.

I've been writing about a family of permutations tentatively based on rearranging even-odd and left-right partitions of strings of elements (cards, pitches, words, whatever). In all of this I have tacitly assumed that the size of the string must be big enough to make it interesting, such as the "Hauptmann shuffle" which I am still pursuing. Examples have hovered around strings of 6 words (the sestina) or 7 notes (usual diatonic scale) or 8 computer processors. But something interesting actually does happen when you make these related permutations act on smaller and smaller strings: the patterns start to become indistinguishable from one another. Metaphorically, the geometry of the permutational moves "collapses" into just one or two elementary patterns.

Obviously, if you go all the way down to a one-element string (the only object in a "trivial geometry") you enter a sort of John Cage world. There are exactly no moves possible (other than identity), and once you don't make them, there isn't much else you can not-do.

A George Boole world with just two objects, on the other hand, has possibilities that will come in handy.  But my interest right now is in the Three World suggested by the shell game.

Fifteen years ago, Marie Ponsot (at age 77) won the Book Critics' Circle Award for her book of poetry, The Bird Catcher. Among the poems in that book was "Roundstone Cove"[1]

The wind rises. The sea snarls in the fog
far from the attentive beaches of childhood–
no picnic, no striped chairs, no sand, no sun.

Here even by day cliffs obstruct the sun;
moonlight miles out mocks this abyss of fog.
I walk big-bellied, lost in motherhood,

hunched in a shell of coat, a blindered hood.
Alone a long time, I remember sun–
poor magic effort to undo the fog.

Fog hoods me. But the hood of fog is sun.

– three verses of three lines each followed by a one-line envoi. Let a=fog, b=hood, c=sun. Then go to the end of each line before the envoi and note the familiar spiral pattern abc, cab, bca:

V1: fog – childhood – sun

V2: sun – fog – motherhood
V3: hood – sun – fog

this is capped by the inspired envoi that uses the form itself to turn the tables on the fog:

fog – hood – me[!] ... hood – fog – sun[2]

At this point, the sestina has de-generated into what Ponsot, and many other poets since she developed the form, call the tritina[3].  The characteristic sestina permutation (615243) has shrunk to the tritina permutation (312).

The forms create an almost bodily pleasure in a poet.

What you're doing is trying to discover.

They are not restrictive.

They pull things out of you.

They help you remember.

(Marie Ponsot)

____ Entr'acte ____

So let's review what we have so far by referring to the following diagram (see A Few Notes about Notations about cyclic notation vs Cauchy one-line notation, the latter of which relates to the red arrows).

In the yellow square at the upper left is the identity permutation. Like the Cagean one-element world, taken by itself the identity permutation just sits there. If this were the only permutation allowed, it would describe a valid but trivial system, useful perhaps for navel-gazing. I think of the identity as mathematics' glue – without it, no groups, no geometries, the world itself falls apart. Still, a tube of glue just sitting on a shelf is not very interesting. So we look further.

The three gray squares on the right side represent the three valid moves available to Honest Grifter in the shell game. They are related because each of them has a fixed point, i.e., each move leaves one shell untouched, while the other two shells are reversed. Notice that the shell game arrows (which represent the Cauchy notation) do not all share the same arrow pattern in two dimensions. The shell game path at the top, 3-2-1, like the identity, is a wave. But the arrow paths for the other two shell game permutations both form spirals 1-3-2 and 2-1-3.

If we now create a mirror image of all the shell game paths on the right, we not only get the identity which we noted above, we also get two new permutations. This will then account for all of the six possible permutations of three elements.[4] One of these is the tritina. The other we have not come across before. I will label this hypothetical poetic form "Itritina" for inverse-of-tritina.[5] As far as I know, no one has every written a poem using the Itritina as a form, however composers have made good use of it.

Notice that all three shell game permutations (as well as the identity) are their own inverse. And this is where the identity stops sitting around twiddling its thumbs and begins to come into play. "Inverse" essentially means that if you apply the permutation for two iterations you end up back where you started. E.g., as in the lower right (shell), 1-2-3 → 2-1-3 → 1-2-3. If we call the identity permutation I and any one of the shell permutations S, then S repeated twice has the same result as I, or: SS=I (S²=I).

The tritina and Itritina, on the other hand, are each other's inverse. If you stick to the tritina permutation T, as we saw in the tritina form, it takes three iterations to return: 1-2-3 → 3-1-2 → 2-3-1 → 1-2-3. So TTT=I (T³=I). But after arriving at 3-1-2 from 1-2-3 by a tritina permutation, you can immediately return to 1-2-3 by applying an Itritina (inverse tritina) permutation T*. Another way of looking at this is that the order of the permutations generated by iterating the sestina is the reverse of that for Itritina, and vice versa. So we also have TT*=T*T=I.

And this now gives us a hook for making an unlikely connection between a poetry form and a well known musical chord progression.

____ The Composer ____

First, another reminder that mathematics, being what it is, means that the real world "objects" it may or may not represent need not be noun-like entities, but may also be verb-like entities. We came across this first in this series of posts with the essay on parallel processing where the objects being shuffled were instructions. The example used there came from a paper by Harold S. Stone, "Parallel Processing with the Perfect Shuffle." In that paper, Stone summarizes the process in a nutshell:

          1) shuffle;
          2) multiply–add;
          3) transfer results back to input of shuffle network.

Let's adapt this iterative procedure as Stone stated it, with a musical chord as the initial input. (Those with some background in neo-Riemannian music theory will recognize the result and wonder what's gained here. A more complete comparison of the two procedures will have to wait for now.) The procedure will stop when an iteration returns the initial chord. Here we go.

• The shuffle, performed on the masks, will be the T* (Itritina) permutation described above: (123) in cyclic notation, (123)→(231) in Cauchy notation.
• The "multiply" operation will be to multiply all three elements of the shuffled mask by (2mod3)[6] 
• The "add" operation will be to add the new mask to the previous triad.

It's easier to "watch" this happening than it is to describe it. The initial input for this example will be a C-major triad in root position. I will use integer notation with parallel letter notation for root-position triads, e.g., 0,4,7 with the corresponding traditional notation CeG. The initial "mask" (or "sieve") will be the triple (0,0,2).

If we call the procedure described for this example X (which starts with chord 0,4,7 and mask (0,0,2)) we see that it returns the initial chord after 24 iterations of the same procedure X, or, X²⁴=I

In music notation, it looks like this:

. . . and it sounds like this:



Pretty boring, eh? So let's jazz it up just a bit by adding a simple repeating rhythm:

Starting to sound familiar? Still, what composer in his or her right mind would seriously try something like this at any length? Hmmm, let's try not using the whole cycle, but using, say, 19 chords (18 iterations). It's an unusual way of getting from C major to A major – it "works" but it's still pretty dull. So let's break it up with a couple of pauses. Then mess around with that too-perfect voice-leading. Then we might get something like this:

This is a piano reduction of bars 143–76 from the second movement (scherzo) of Beethoven's Ninth Symphony.

It's tempting to say that only a master grifter like Beethoven could have pulled off a trick like this – turning theory's lock-step march in a circle into an exciting race to the edge of a cliff.[7]

Now, with a tip-o-the-hat to Richard Cohn who first brought those 19 chords to the music theory community's attention in 1991[8] (more to come on Neo-Riemannian theory's potential, including turning its dead ends into back doors, in future posts), here is a performance of the 19 chords in context. It took me a while searching YouTube to find a couple of good performances that take all the scherzo  repeats marked in the score, which I consider essential, especially here. (More another time on the subject of repeats, a favorite topic of another friend of mine, Mark Doran.) Measures 142-76 of the Scherzo movement can be heard starting at approximately 18:53 and (the repeat) at 21:00 in this recording of the entire Beethoven Ninth Symphony performed by Mariss Jansons and the Symphonieorchester des Bayerischen Rundfunks. Go ahead and listen to the whole thing for the 250th time in your life. When did you ever regret hearing it again?

I'll be back shortly with a few more brief comments on these measures. The question is still open: even given that the analysis above is "true" (which it is), is that what Beethoven heard? – Or the more accessible: Is that what  I  hear?

_____________________

[1] Image with the Ponsot poem reproduced here was found on the web site of the Poetry Society of America:

"Launched in 1992 by the Poetry Society of America and the Metropolitan Transportation Authority, Poetry in Motion® is today one of the most popular public literary programs in American history. Poetry in Motion® places poetry in the transit systems of cities throughout the country, helping to create a national readership for both emerging and established poets."

[2] Maybe this is a little far fetched, but I read c'=me; then in the envoi we have abc'→bac, one of the grifter moves but not one of Honest Grifter's since it reintroduces a sleight of hand. In one brief line Ponsot has flipped fog and hood & discovered me as the sun. The poem (to me) presents a Russian doll effect: Sun(fog(coat(mother(child)))), a prison, but one whose walls are dissolved by the identification sun↔me.

[3] A quick web search will turn up numerous examples, both professional and amateur, showing that the form was quickly added to the poet's tool kit of forms. One nice example is "Tritina for Susannah" by David Yezzi.

[4] Since we only have three objects to permute here, an equivalent way of viewing the family of permutations of (123) in Cauchy 1-line

123   321

312   213

231   132

with up-down as simple rotations, and left-right as mirrors. 

[5] I was tempted to call this permutation "Son of Tritina" but, as you will see in the next couple paragraphs, this would set up a situation where the son is also the father of his mother – an intriguing  possibility I would rather leave to be sorted out by an expert such as one of my favorite mathematician-pianist-magicians,  Raymond Smullyan.

[6] Reminder: (0×2)mod3=0; (1×2)mod3=2; (2×2)mod3=4mod3=1.
[7] It's difficult to leave this passage so abruptly. We need to hear what comes next after the example ends up in the air – no matter how often we've heard this old war horse before. In mere words, what my ear hears next is: That A-major triad collapses into a single note, A#, and then on to a solitary B, a false summit. No. Better: that B leaves an open question. Is it a root? a third? a fifth? And, given what just went before, there is no way to figure out what it's "function" might turn out to be until the bassoons start up and we (O.K., _I_) "hear" [?? – what's the aural equivalent of hindsight?] the context-less B as the dominant for the real destination of the past 34 bars: E minor. Then begins a fugato which Beethoven (not given to fully trusting conductors who are not him) marks in the score "Ritmo di tre battute"which signals that  the previous four-square "Über-rhythm" is now a three-beat "Über-rhythm."  (This is the equivalent, in this case, of going from time signature 12/4 to 9/4 keeping the quarter note constant – an early and tamer version of metric modulation.) The effect, to my ears, is (due to a horizontal superposition of 3 in 4), a piling up of the head of the opening theme that resembles the effect of an actual stretto. This, especially when we reach the call-response of the timpani "solos" with the woodwinds' replies, propels the music even faster and prepares for the still far-off presto trio. At least, that's how I hear it.

[8] Richard Cohn: 1991, "Properties and Generability of Transpositionally Invariant Sets," Journal of Music Theory 35(1-2) and 1992, "Dramatization of Hypermetric Conflicts in the Scherzo of Beethoven's Ninth Symphony," 19-th Century Music 15(3).

The Hauptmann Shuffle (2) – Abduction from the diatonic seraglio

     Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea; for induction does nothing but determine a value, and deduction merely evolves the necessary consequences of a pure hypothesis.

     Deduction proves that something must be; Induction shows that something actually is operative; Abduction merely suggests that something may be.

     [Abduction's] only justification is that from its suggestion deduction can draw a prediction which can be tested by induction, and that, if we are ever to learn anything or to understand phenomena at all, it must be by abduction that this is to be brought about.

     No reason whatsoever can be given for it, as far as I can discover; and it needs no reason, since it merely offers suggestions.

–C.S. Peirce. Collected Papers. V.171

A theory is a cluster of conclusions in search of a premiss.

– Norwood Russell Hanson, Patterns of Discovery (1958)

Stay loose until rigor counts.

 – George M. Prince (co-founder of Synectics)

Don't rush to proof.  Certainty is overrated.

Ignis fatuus: "foolish fire." This will be messy. Terminology will be loose and confusing and inconsistent and contradictory. Ill-defined thoughts will skip from one connection to another with little or no justification outside of serendipity. I am following my nose here and made the conscious decision not to "clean it up" in order to "prove," "make sense," "tell a story," or "convince the reader." There's a kind of dishonesty when you read A→B→C→D→E and you know full well that the way creativity/discovery works is that, in a flash of insight, the author started with C and worked his way out, or with E and worked backward. You read the arrows and assume they were always there to read, as any fool can see.

So my expository model here is closer to Joyce's Ulysses than to Euclid's Elements, but devoid of the genius of either. This is, in my opinion from experience, the way the mind (any mind) works in pursuit of an idea when it has no idea what that idea will turn out to be, and is open to any outcome. For anyone puzzled to know just what a Hauptmann shuffle is: right now I'm still just as puzzled as you are. I simply have (...this is weird...) "blind faith" that there is such a thing. [Cue Monk theme song: "I could be wrong now, but I don't think so."]

❐

I concluded the previous post by saying "a perfect shuffle of a maximally even set will not necessarily result in another max even set (try shuffling the octatonic as one counter example)." This is true, however there may be another reason that the usual 7-note diatonic cycles through its three characteristic forms via the perfect shuffle or its inverse. Such a reason could lead to a conjecture that predicts which structures demonstrate the same "shuffle behavior" as the usual diatonic, and which do not. To explore which other scale structures exhibit the same behavior, we'll start with the 7-note diatonic as a model. First we need to convert the traditional letter notation to integers in the usual way (C=0, C#=1, D=2, ..., B=11)

Then we record the characteristic "signatures" of each shuffle (the cyclic string of chromatic steps between the notes shown as smaller numbers) giving the three forms of the diatonic: a generating stack of six perfect fifths (7) plus one diminished fifth (6): (7777776⤸; overlapping triads (43, 34, 33): (4343433⤸;  and the diatonic scale: (2221221⤸.[1]

Note that each of the three signature strings is maximally even, meaning that

     (a) each is a mirror-symmetric string of integers (to see this, rotate each string to more easily see the symmetry: (7776777⤸, (3434343⤸, (2122212⤸),
     (b) each integer in a given string is either x or y=x+1, and
     (c) the x's and y's are distributed throughout the string as evenly as possible (e.g., the string (21122⤸ fulfills criteria (a) and (b), but the 1's are not distributed evenly with respect to the 2's (they are not as far apart as possible), so it fails criterion (c)).


Note also, since this is a permutation of integers related mod 12, that the sum of all the integers in the interval string in each of the shuffles (48, 24, 12) is a multiple of 12, the "base modulus."[2] So when we substitute other integers/notes/cards to shuffle we expect the three characteristic interval patterns that appear to remain the same, but any interval string sum, while it may change with the shuffle, will always be a multiple of m.[3]

I am assuming that any perfect out shuffle of a deck (string of integers, notes) that produces a maximally even pattern as just described will be an instance of a "Hauptmann Shuffle" (whatever that turns out to be). Symbolically:

shuffle with maximal evenness  ⊂  ?  ⊂  . . .   ⊂  ?  ⊂  Hauptmann shuffle

The next step is to generalize the special case. First, assign some dummy letters so that the pattern of intervals we are looking for is no longer married to the special case.

What remains is the pattern alone without stipulation (b) above, generalizing the pattern to one which is distributionally even, for which maximal evenness is a special case.[4] So now, a "Hauptmann Shuffle" is the family of perfect shuffles that includes any shuffle that cycles patterns that are all distributionally even.

          shuffle with maximal evenness

               ⊂  shuffle with distributional evenness

                       ⊂  ?  ⊂  . . .  ⊂  ?

                             ⊂  Hauptmann shuffle

But we're not quite there yet. A new term would not be called for if all we were talking about was "distributional evenness." And there are intriguing complications ahead.

Next we try a simple test case. Staying with a modulus of 12 (12-tone equal temperament), suppose a=1 and b=6. This will produce a seven-integer string shuffle based on (0123456⤸ which represents the first 7 notes of the 12-tone chromatic scale. Since the perfect shuffle should be familiar by now, let's skip the arrows. Here is how this entire shuffle looks:

Leaving the 12-tone chromatic behind, another interesting shuffle is found by using the interval string (3323332⤸

Some readers will recognize this as the basic 19-tone equally tempered scale system which brings the traditional triad closer to a just tuning by dividing the octave into 19 equal parts. This refinement is important to some ears and has real world form-inducing ramifications as well as expanding tonal material, but is irrelevant to the relationships in the present context since it's still just another example of the basic shuffle pattern. A similar shuffle (moving even closer to approximating just intonation) will result from starting with the interval string (5535553⤸ in 31-tone equal temperament. Other suggestive structures result from swapping string integers such as a "swapped out diatonic" (1121112⤸ (mod 9) or "swapped out 19-TET" (2232223⤸ (mod 16).

Next the question arises, are there strings of length other than seven that produce the same or similar shuffle relationships? We can fully generalize those relationships we have been seeing (so far) with strings of length 7 by first removing elements by pairs as shown in the following diagram to reveal a skeletal structure of strings of length 3.

We can then build back out from the skeleton to interval strings of any odd length 5, 7, 9, 11, 13, etc. using the following instructions that "clone" the red elements:

     – insert "aa" n times at the ellipsis in the first string, or

     – insert "cd" n times at the ellipsis in the second string, or

     – insert two "e"s, one at each ellipsis, n times in the third string

So if we create the pattern, say,  (cdcdcdcdcdcdd⤸ for any intervals c and d by the second insertion rule, we guess that perfect shuffles of the resulting integer string will give us the other related patterns, (eeeeeefeeeeef⤸ & (aaaaaaaaaaab⤸. The next example appears to confirm this but creates a new wrinkle.

We said that building out from any one of the skeleton patterns will give us the other related patterns, but we didn't say how many copies of those patterns.  When dealing with 7-note scales, the perfect shuffle permutation in cyclic notation, using (0123456⤸ from the example above, is (0)(142)(356), which means it will return to original order in just three shuffles. But the initial pattern here, (012345678⤸, uses 9 integers (or notes or cards). The perfect shuffle permutation for that is (0)(157842)(36) which means it takes six shuffles before returning to original order. The pattern holds in this example, but it is doubled. One other interesting thing happens in this example. After three shuffles, the initial cyclic order is reversed (clockwise becomes counterclockwise). So three perfect shuffles starting from (012345678⤸ result in (087654321⤸ and vice versa. And the same for (051627384⤸ ↔ (048372615⤸ and (075318642⤸ ↔ (024681357⤸. Also note that for each pair of strings of elements related by a (cyclical) retrograde, their respective interval strings are related by "inversion" (in music theory terminology); so for the first string above, a=1 and b=3 in the interval pattern, and for its reverse string, a'=10 and b'=8; and a+a'=b+b'=11, the base modulus here.

But now we have to ask, is this cycle of six shuffles actually not a "doubling" of a basic 3-shuffle circuit, but rather a complete "normal" cycle and the 3-shuffle cycle is a "short circuit"? We started by looking at the circle of fifths in 12-TET, but neglected to note that everything that happens there is mirrored in the circle of fourths (or, alternatively, moving counter-clockwise on the circle of fifths). If we begin with a string generated by fourths with the pattern (5555556⤸, and compare it to the string with the same integers, but generated by fifths, with the pattern (7777776⤸, it's apparent that you can't get from one pattern to the other by continuing across the broken line with a perfect shuffle as you can with the (012345678⤸ mod 11 example given above . . .

But we have been assuming that the only permutation to produce these related patterns is a perfect out shuffle. If we investigate the other three permutations in the COIL set for 7-strings, we find the reverse ("inside-out" – not to be confused with the inverse) of the perfect out shuffle.

Its cyclic notation is (3)(146527) so we know it will take 6 shuffles before repeating. Applying it to the perfect-fourth generated (0,5,10,3,8,1,6⤸, it reads (10)(560318) with the fixed point 10 remaining in the "third position" throughout. It produces the following shuffle cycle:

So we now move on to testing that a Hauptmann shuffle is any shuffle that cycles through six iterations – two triples whose interval strings are related by retrograde and all of which are distributionally even. But getting to this point, we've lost the perfect shuffle as the characteristic permutation holding these patterns together. Gaining another shuffle, we're now wondering if there are others:

• perfect out shuffle  ⊂  Hauptmann shuffle
• reverse perfect out shuffle  ⊂  Hauptmann shuffle
• . . .
• ?  ⊂  Hauptmann shuffle

On the other hand, this complication makes chasing this ghost all the more challenging – and possibly more rewarding. We'll see.


___________________

[1] The notation (abcd⤸ indicates that the elements a,b,c,d are to be thought of cyclically, situated clockwise on a circle. Thus there is no "first element" – (abcd⤸ = (bcda⤸ = (cdab⤸ = (dabc⤸. So, for example, if you are measuring the consecutive intervals in the C-major triad, (CEG⤸, the result will be three intervals: C-to-E (4), E-to-G (3), and "around the corner" G-to-C (5, continuing clockwise on the circle).

[2] What I am tentatively calling the "base modulus" here is assigned intuitively (awaiting formalization?) and mostly for convenience. While it is clear that selection of such a base modulus will affect the specific elements in an interval string, it is not yet clear to me whether/how this choice will affect the patterns resulting from consecutive shuffles. Interpreted "musically," the "base modulus" is the actual size of the underlying horizontal-chromatic/vertical-pulse (equivalence class) universe a composer may have chosen to work in – 12TET, 19TET, 24TET, etc., or metric lengths ("measures") such as 12/4, 9/8, 4/4+1/16, etc.

[3] Taking a foray into geometry, these sorts of structures are said to be conformal and related as homothetic transformations of one another. One of these transformations can be found in 12-tone atonal/serial music theory as the M5/M7 transformation, multiplication of pitch classes by 5 or 7. For example, multiplication by 5 mod 12 maps C-E-G (0-4-7) to C-G#-B (0-20-35 mod 12 = 0-8-11). Considered as a permutation, M5 can be expressed (0)(3)(6)(9)(1 5)(2 10)(4 8)(7 11); homothetically what happens is that the mod 12 chromatic circle (111111111111⤸ explodes into the circle of perfect fourths (555555555555⤸, a mod 60 structure whose base modulus, to preserve octave equivalence, is usually taken as 12.

[4] As far as I know, the term "maximal evenness" was first coined by John Clough and Jack Douthett in their seminal 1991 article, "Maximally Even Sets." It was suggested by a definition of the diatonic scale  by William Drabkin in the New Grove Dictionary of Music as a scale which divides the octave into five whole steps and two half steps with 'maximal separation' between the half steps. Eight years later, Clough and others found it necessary to find a different term for a further generalization of the pattern as they traced it ever deeper into the underworld of musical abstraction. They came up with "distributionally even" as an unavoidable terminological back-fill. So now we have the diatonic as a special case of maximal evenness which itself is a special case of distributional evenness – where we are now finding it meeting up with the perfect shuffle. The field of music scale theory (unlike the subject at its core) is close to impossibly complex due in large part to specialized vocabularies full of jargon that can be confusing and counterintuitive, especially to any beginner. While it is on the technical side, about the best resource that can be accessed on-line for anyone who wants to navigate the labyrinth of scale characteristics and distinctions that have been identified and catalogued over the past half-century is the article by John Clough, Nora Engebretsen & Jonathan Kochavi, "Scales, Sets, and Interval Cycles: A Taxonomy" (even if you only work through the first section).