very absorbing, every interval, every space

2024

# 15

The aggregate contains six tritones, 0-6, 1-7, etc.
In 1-out-of-64 rows, the PCs of each tritone sound in just that order.
Then perhaps also the time-point gap between PCs 0 and 6 is smaller than the gap between 1 and 7, etc.  

This piece is made up all of the 957 rows that meet all these requirements.

Specifying the gaps between the tritones, one has a kind of jigsaw puzzle: to find a row with exactly those tritone time gaps.  

Sometimes such a puzzle has no solutions (for example when the sum of the gaps is odd).  

With gaps 1-2-3-4-5-7 one finds 38 solutions, the most, and these begin the piece; with gaps 1-3-5-7-9-11 one finds 1 solution, the least if not zero, and that row ends the piece.  Between, all the rows occur organized according to which puzzle they solve, the puzzles ordered so that the number of solutions weakly decreases.

With each statement, each tritone gets its own octave.

If one has the ear for it, one can attend to these details, or else just take in the energy of it all.  I would like this to looking at a descriptive picture either as a visit to a virtual 3D place (e.g. a fictive city street, mountainside room) or as a decorated planar region, typically bounded by a rectangle.  When musicians talk about "movement" and "line" and "implications," that's happening the musical equivalent of virtual space.  In reality, notes and sounds find all their places by compositional fiat and don't actually generate or cause one another.

The cover image more or less encodes the score.

Rational Exuberance (Not Your Grandparents' Twelve-Tone Music VI), for organ and piano

There are 362,797,056 ways to partition a three octave span into three aggregates.  Once you do so, reading each aggregate from low to high, you get three 12-tone rows.  If you further require that those rows feature at least 7 distinct interval classes, the 362,797,056 possibilities reduce to just 46.  One of those 46 possibilities plays the role here of a passacaglia-like chord progression.  

There are 144 ways to have three triads sit within the aggregate without any overlap.  Remarkably, one can cycle through those possibilities so that each step turns up just one new triad.  Such a cycle provides the figuration for the variations. 

The overall effect produces a counterpoint between two opposing notions of harmonic rhythm, one based on the cycle of aggregates, and one based on the the featured triads.

A Friendly Accord between Aggregates and Triads, for guitar

If a twelve-note chord features all the pitch classes, you can read it up to get a row. If the chord sits within three octaves, it can feature at most 7 different intervals (because 1+2+3+4+5+6+7+8 =36).  There are just 46 ways to partition the 36 pitches in a three-octave span into three such chords.  

Each "variation" projects one of the 46 possibilities.  You hear each of the rows three times, but with the notes transposed into the three different chords.  

As with much of my music, there is no literal repetition, but also no attempt at "high profile contrast."  Rather the music invites the listener to attend to small differences, and in this case, exactly the small differences imposed by the universe or by the mathematics or however you prefer to think about it.  

It's hard to come up with "questions" like this, questions that indeed turn out to have a manageable but non-zero number of solutions. But having fixed such a question, the set of solution lies beyond human control.  The universe provides what the universe provides.  
This accepting of what I get interests me aesthetically, as does the privilege of having sufficient computer power now in the 21st century actually to discover all the solutions. 

Score on request.

The 46 Possibilities (Not Your Grandparents' 12-Tone Music VII), for bass clarinet

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Composer, mathematician, visual artist. 
Seeks collaboration with performers and choreographers.

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