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).