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).