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

Some friends felt that my cover image for the piano and organ version wasn't a visual analog of the piece.  So here's a klangfarben version hopefully as colorful as the graphic (the new graphic is just a color morph of the old one - they're both visual representations of the score).

Rational Exuberance, twelve instrument version

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

The conventional wisdom holds that chance operations abolish "glue" (Henry Cowell's term, but often invoked by Cage).  That idea rests on presuming that statistical independence governs how  the operations place musical events within the music framework.  Well, not here.

As x and y coordinatize the Euclidean plane, time and pitch coordinatize musical space as constructed say on a piano roll.  But there's a difference: the plane enjoys rotational symmetry, but its far from clear how one would even compare the magnitude of a time duration with musical interval.

Take a circle and randomly choose 41 chords (meaning, sorry, line segments that connect two points on the circumference).  That makes 820 pairs of chords, but only above one third of the pairs will intersect (meaning, inside the circle).  Those roughly 270 points of intersection, plotted on a virtual piano roll, become the notes of a variation.  The piece consists of 10 variations.  Each note/point sits on two lines, so each line will contain an average of about 13 notes.  This is the emergent clue.  Depending on the slope of the line segment, it will sound as a chord, a broken chord, an arpeggio, a line slowly ascending or descending, or even a pedal tone.  Of course the points/notes on a line don't sit evenly spaced - their position gets determined by the random dispositions of the other chords.  But every note is overdetermined -- by design it sits on two lines and thus helps describe two different musical figures.

Empirically, the 88 note space of the piano (so 352 distinct pitches by the use of eighth-tones) feels like one minute.  Too much faster and the emergent shapes get compressed into illegibility, too much slower and the notes get unglued by the strain on short-term memory.

The accompanying graphic explores a similar construction, but in the visual realm.

Time-Space-Circle-Line-Point #7, for four pianos tuned in eighth tones

The first and last movements deploy triadic harmony and local root progression yet an overall design with an atonal logic.

The middle movements hybridize the outer movements, and produce a kind of evolving local bitonality.

Larry once told me that my stuff made great driving music.  I admit that for a few seconds I did feel put in my place, but almost immediately I embraced this.  Yes, I do have a taste for compressed dynamics, so indeed you can crank much of my music and it won't get drowned out by your engine.   

And speaking of music for a journey, Larry's own masterpiece is his Lonesome Road (The Crawford Variations), and you can hear it here 
https://www.youtube.com/watch?v=Mj2LK-sfqFk&list=OLAK5uy_lh4EjWmhWFIFlMEvEW9XaedZSrcZaRWzI
and I urge everyone to give it a spin.

Tapping The Brakes On The Highway To Eternity (in memory of Larry Polansky)

12  = 
   12 = 11+1 = 10+2 = 9+3 = 9+2+1 = 8+4 = 8+3+1 = 7+5 = 7+4+1 = 7+3+2 = 6+5+1 = 6+4+2 = 6+3+2+1 = 5+4+3 = 5+4+2+1
Here each sum accumulates mutually different magnitudes; these 15 exhaust all the possibilities (when the summand order doesn't count). 

12  = 
  11+1 = 9+3 = 9+1+1+1 = 7+5 = 7+3+1+1 = 7+1+1+1+1+1 = 5+5+1 = 5+3+3+1 = 5+3+1+1+1+1 = 5+1+1+1+1+1+1+1
   = 3+3+3+3 = 3+3+3+1+1+1 = 3+3+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1
Now each sum accumulates odd magnitudes, and likewise these 15 exhaust all possibilities. 

15 possibilities both times; this is not a coincidence nor is it a special property of 12.  Start with any whole number and it turns out that the two tasks always lead to the same numbers of solutions.  

Now a claim like that requires proof; just testing finitely many cases does not suffice.  One sort of proof involves inventing some way of seeing an instance of one sort sum as a secret code for an instance of the other sort of sum (and vice-versa).  Then the implicit codebook matches up the two sets of solutions.  

Suprisingly, mathematicians know more than one recipe here for doing the coding and decoding.

And starting specifically with 12, coding by one recipe and decoding by the other happens to walk you back and forth between the all the solutions of both problems.  The structure of this piece is that process, with all the intermediate steps. The rhythms alone sonify the sums, but the pitches will guide you through the execution of the coding.

Different and Odd, for alto[aka tenor] horn and piano

Realization #82 of an open form work. 

Details for those who want to know *what* they're hearing:

Some unheard but endlessly repeating  rhythmicized tone-row girds each realization of this work.

Any tone-row contains 220 trichords.  Graphed in time and interval space, each trichord  (as a triangle now) has a geometrical (signed) area.  While the area of any one triangle depends on the units chosen for both time and interval width, the ratio of two triangle areas is invariant of the units.  So ordering the 220 trichords by increasing area makes intrinsic sense.  
It also produces a distinct musical effect, you will hear it, suggestive of careful and gradual development.  The reasons why the mathematical strategy has this musical effect are complex and perhaps hard to foresee in advance.

Now transposing the row up a semitone but also forcing all the notes back  into the original octave changes the area of 1/4 of the trichord/triangles. This shuffles the trichord order and produces a kind of variation.  This work, as a whole, proceeds through all 12 of these transposition-derived variations.

Well that describe the timpani part anyway.  

The flute doubles the timpani always at some number of octave, and climbs in register during the stretch of each variation, flute register roughly correlated with trichord/triangle area.  Octave transpositions always apply to all the notes in a trichord in order to preserve the shape, which of course is the chief musical determinant here.

The flute part can be performed alone.  It can also be performed with the timpani part on fixed media.  But live performance of the timpani part would require 12 drums, and perhaps two players, and perhaps some technology for coordination (since the rhythms will be hard to count in the usual way).

If there's interest, I'll upload other realizations.  All realizations are both recognizably "the same piece" yet also reflective of the idiosyncrasies of the basic rhythmicized tone-row.

Sharp Shadows Shadowed, for Flute and Timpani

From a set of 100.  
I'll post others if this gets some love.

Songs Sisyphus Sings The Stone, no. 98, for solo piano

Inuksuit, for Piano and Celesta

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

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2024

David Victor Feldman