This proves that math is music and music is math

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

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

SOON to move to Bandcamp as an album - in the meantime, you can also listen in sequence here: soundcloud.com/sonic-textures/sets/scalar-fields
instruments: Jeff Gardiner and jo richter
voice: Jeff Gardiner
modulation, arrangement and all that: jo richter
lyrics: Jeff Gardiner in collaboration with jo richter
image: Jeff Gardiner in collaboration with jo richter

when the waves of silence break
on the dharma nomad's sitting mind
the daylight sleeper's nagual wakes
for a body of the idle kind

that speaks in riddles, muted rhymes
and labrythene paths of jesting Joyce
where assonance stands guard and chimes
in onomatopoeic trysts

for rivers run and waters foam
and dogs are dogs of many heads
no dwellings last where prospects roam
and liquid hope permits no rest

and the knowledge of the miserly
cannot withstand the yearly flooding force
and can and will and is not allowed to atrophy
as long as eight blurred legs still make a horse

ah burning barnacles to make light
of tonal drylands and bold vertebrates
bridges? yes, bridges burn for ashes' flight
and sparks and bioluminosity and microtonal clefs

for those who walk the open scalar fields
have guns that glow and minds to blow
for those who raft down rapid vector creeks
having dreams to grow and seeds to sow -

Vector Creek

scalar fields pt 10 vector creek - with stonerjazz

Voice of Arka

Instrumental music for video from a visit to the Ozeaneum, SeaAquarium in Stralsund / Deutsches Meeresmuseum - see https://en.wikipedia.org/wiki/Ozeaneum

Ozeaneum

"Vikings of the Sunrise"

Fr0ggy5 posted an awesome drum solo and the voices in my head just wanted add a little something something.
https://soundcloud.com/fr0ggy5/bass-drum-solo-in-my-room

Piano to fr0ggy5 drums

Music by Nick Rangel.
©Nick Rangel 2025
From the album - Mirrors With Eyes

Bounce Back (edit)

You have a free afternoon. You open the windows and sit down to read your favorite book, sipping coffee.

paged.

2025-09-22

between the waves

Herzi

A spontaneous jam on a prefixed drum track.
Feddo: Piano, Soundpaint PR-S Guitar, Zebra synth
Peter: Drum, Bass and production

Artwork: unknown

Dig This

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

Please don't spam this account!

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

David Victor Feldman

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