Tuesday, August 27, 2013

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: fogchildhoodsun
V2: sunfogmotherhood
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).

Saturday, August 3, 2013

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).