21st Century Gamuts
I think of the 3rd structural key, as the one the composer elects, while the 2nd structural key (dominant) is "natural", because it is obvious, built into every structural level. Examples: In Haydn Ab piano sonata, the 3rd structural key is f minor, and every detail from small to large points to f minor, just as, invevitably, the dominant is thoroughly imbricated on every structural level; In Bach's suite in E minor for Lautenwerk, G major is the 3rd structural key, with likewise stunning details to make that *right*.
Today, I am coming to think of "natural" (3 or 4 in number) and "custom" gamuts. Kupferman was actually on to this, and today, Joan Tower does it, and, I think Robert Pollock might also think along these lines.
Schoenberg in op. 9 is up to something like this, but in the end his Brahmsian sensibility demanded an approach that was more "individuated". Yes, in the background of any of his works, the principle is operative, have no doubt. The very whole-tony woodwind quintet contrapuntally derives moments of vertically stacked 4rths or 5ths. Of course!
Kupferman's "custom" gamut was his "infinities row", which he used to bridge into whole-tone, octotonic, and diatonic harmonies, because his "custom gamut" could link into any of those "natural gamuts". The "natural" gamuts are the obvious ones. All tonal music bridges through diminished 7 chords, and that will inevitably yield all kinds of octotonic tunes.
The "natural" gamuts are interval-cycle-derived. 4th (5th) cycle generates the diatonic gamut. Minor 3rd cycle generates the octotonic gamut. Major 2nds create the whole tone scale. The 1-gamut (chromatic scale) is nasty, but useful.
Why do I call these, "gamuts"? They are big- interval-cycle-defined sets, however, because the octotonic scale, for example, has subsets that link into the diatonic and whole tone cycles--octotonic scale contains two tetrachords consisting of tritones a step apart; octotonic scale contains two minor 7 chords. These things invite extensions into the entirety of the whole tone cycle and the entirety of the 5th cycle (diatonic). In this sense the octotonic scale is a gamut. It's best used as a gamut, not as spray paint.
The whole tone scale is a gamut. It has subsets that are also subsets of the diatonic scale. It has subsets that are also subsets of the octotonic.
The diatonic scale is a gamut. For centuries the diatonic scale was linking into the octotonic scale, as in the slow movement of Beethoven's Les Adieux. In tonal music, octotonic scales are transitional bridges between transpositions or modal shifts.
Even before Debussy's *Syrinx*, the French augmented 6 chord and the augmented triad (especially in Haydn) was probing into the whole-tone extensions of the diatonic collection. Jazz goes nuts with this.
A "custom gamut" is a tune. Schoenberg thought of his rows as thematic material, tunes. A tune is more specific than an interval cycle, more "individuated". At certain points in music history a tune was understood as a more highly developed entity than an interval cycle, it was felt to be more "mature", *knowing its relationships to all the interval cycles*.
At the moment there is so much hatred for Schoenberg and Babbitt, that these issues can only be discussed underground. So be it.
It is abundantly clear, to me, that these issues are alive in our music today, in all of it from David Lang to Paul Lansky, from Joan Tower and Meyer Kupferman to Frank Brickle and Jonathan Dawe; from Paul Moravec to Akemi Naito; and certainly Harold Meltzer.....
There are new ways to achieve "individuated" approaches to these linkages. This is where we are now. If one resorts to something like a row, the music will sound too 20th Century. (While we love 20th C. music, we are nevertheless interested in moving on.) Less "individuated" approaches to these problems can end up sounding like the composer sprayed octotonic paint all over his/her canvas, and there's plenty of this, for sure, and it must be discouraged.
How to avoid the trap of falling into a surface that is dominated by the big "natural" sets (octotonic, whole tone, diatonic)? Tunes do this. (A row is a tune.) That's the out of fashion way.
Babbitt, Pollock, and Peter Schatt experimented with trichords. The trichords bridge into and out of the bigger "natural" sets in a terribly subtle way. And the best all-trichord work is, as far as I know, Babbitt's Swan-song No. 1.
It hearkens back to Schoenberg Op. 9 in that respect. More on this....
I've been talking about the big "obvious" sets. I called them, "natural". I contrasted those with a "custom" set (like a 12-tone row).
I'm now thnking about odd beasts like the B hexachord. I approve of the B hexachord not because it is combinatorial. I love it as the bluesy diatonic hexachord. I hear weird hexachords in relation to less weird hexachords.
Arrays are fabulous, a long story, needing a terrible amount of explaining. I should love those hexachords lying vertically across an array because of their combinatorial qualities. I think Babbitt and a few others could do that. For most, it's too big a stretch. I need to reread Babbitt's piece about the inversional pairs in Dallapicola.