With 12 pitch classes one has 144 2-note sequences or just 132 upon insisting on 2 distinct notes.  

Dividing these 132 sequences randomly into 3 "buckets" of about 44 sequences each one can seek out, for each bucket, all the 12-tone cycles that each bucket affords: so each adjacent pair of notes in each cycle must constitute a 2-note sequence in the affiliated bucket.

Now repeating this recipe millions of times, I eventually discover three instances where each bucket affords, in total, exactly 30 conforming cycles each.

Musically speaking, all the cycles associated to a particular bucket, seeing as how they got cobbled out of the same 2-note sequences, they all turn out to share a common feeling.   Moving from one bucket to the next that feeling changes.  Indeed I feel a distinct but subtle shift, akin, something to something like a key change or modulation and this occurs despite how composing with 12-tone cycles promotes consistent pitch saturation, and thus the classic "atonal" sound you will hear.

But honestly, even having composed the piece myself, and thus knowing in advance where to expect these quasi key changes, only after repeated hearings did I suddenly find these shifts, these reconfiguations, obvious.  

With three 30-30-30's, I had one to use to govern the pitch classes, but also one for durations, and one for orchestration.

Think of this as very unconventional 12-tone-music: the rows never get transposed or inverted or retrograded, but rather variety arises from employing multiple rows that happen, by design, to share common "DNA."  Rather than burying this "row technique" under some sort of subjective rhetoric, as traditional 12-tone composers feel they must, I organize all these materials for transparency, hoping for listeners to follow the goings on.  One usually views "complexity" and "minimalism" as opposite poles, but both attitudes inform every moment of this piece.

Fritillary, for small orchestra

This work wouldn't count as "12-tone music" even though it is based on an extraordinary tone row recently discovered by the composer.  And it also wouldn't count as "tonal" even though it unabashedly deploys some triadic harmonies.  And it's also not one of those pieces that pushes to the boundary of atonality or, though primarily atonal, flirts with suggestions of root progressions.  

To explain: there are about half-a-billion 12-tone rows total, and still about  10 million when you count only up to the usual transformations.  But this row
[ 0 3 1 9 2 8 10 11 6 5 4 7 ]
enjoys a unique, 1-in-10,000,000 property.  Consider the ten internal 3-note sequences 
[ 0 3 1 ], [ 3 1 9 ], [ 1 9 2 ], [ 9 2 8 ], [ 2 8 10 ],  [ 8 10 11 ], [ 10 11 6 ], [ 11 6 5 ], [ 6 5 4 ],[ 5 4 7 ].   They all turn out mutually
so different each-to-each that you can't mistake one sequence for a transposition of another even if some or all of the notes get nudged up or down a semitone!  So that means, in principle, the listener can know where she is inside the row even with only dodgy relative pitch!

Multiply by 7 and semitones turn into 5ths.  So instead of using the original row to control pitches, this piece uses the related row [ 0 9 7 3 2 8 10  5 6 11 4 1 ] roughly speaking, to control key centers.  Since this new row stands to the cycle-of-fifths as the original one does the chromatic scale, now instead of inflecting up or down a semitone,  you get to replace any chord note with its dominant or subdominant.   The result: no collision between the 10 3-chord progressions can occur, no matter what the substitutions. 

The hidden skeleton of this work comprises four statements of the second row (but overlapping, and with the middle two transposed). Thus one may think of the work as 40 variations, each exploring one of the 3-chord sequences, but with constant substitution as described above.  Each variation consists of 15 phrases (for no particular reason).  But the phrases move from low to high,  making them easy to count off when you first try to follow the structure.  Eventually your ear will just "get it" and after a few listens you'll find you don't need to count to hear the harmonic changes that distinguish each variation from the next.

Altocumulus stratiformis translucidus undulatus, for solo piano

First of all, what this isn't: it's not any sort of improvisation, and it has nothing to do with artificial intelligence.

While one might naturally feel curious about the rhythmic surface here, harmonic actually supplies the original impulse.  To wit, equal temperament affords twelve major 7th chords and twelve dominant 7th chords.  Composition started with the discovery that forming all 24 of those chords into a cycle whose consecutive chords have no common tones only ever happens in 70 essentially different ways.
Curiously the same applies if we replace the major 7th chords with a certain kind of augmented 7th chord.  So these 70 "solutions," in a certain natural order, occur here as 70 low-contrast "variations" (though it takes counting or a good ear to pinpoint the boundaries between the variations).

(Cover image credit: https://commons.wikimedia.org/wiki/File:Laberinto_Oxkintok.jpg)

Swing By And Come Inside If Chance Allows It, for piano

The same generative process underlying the music also generates the images in this video:

https://www.youtube.com/watch?v=QeZaBFeIY4Q&list=PL5XFDmREe2DrLvQoTwiH2kTKzEj6CoIPr&index=2

This is one of a series of pieces all built on the same principle but sounding quite differently one from another.   More to come.

Double Shuffle, for piano

A chamber concerto for piano and twelve players

The title is from a poem of Sara Teasdale

There will come soft rains and the smell of the ground,
And swallows circling with their shimmering sound;

And frogs in the pools singing at night,
And wild plum trees in tremulous white,

Robins will wear their feathery fire
Whistling their whims on a low fence-wire;

And not one will know of the war, not one
Will care at last when it is done.

Not one would mind, neither bird nor tree
If mankind perished utterly;

And Spring herself, when she woke at dawn,
Would scarcely know that we were gone.

The piece is dedicate to the people of Ukraine

swallows circling with their shimmering sound and frogs in the pools singing at night (pno & orch)

Concerning most musical works, one can reasonably say that they unfold within some preexisting musical language (so "language" at least in the sense of a vocabulary governed by a syntax governed by stylistic norms).  On one hand, many conservatives contend that a common language constitutes a minimum prerequisite for "communication," and that that principle makes any fiat language (e.g. Schoenberg's twelve-tone system) fundamentally problematic; on the other, hand many radicals advocate for language-free music grounded only in visceral sensation, music that explores the lives of individual sounds with no "glue" at all.  Of course myriad intermediate positions exist as well.

This piece indeed projects a fiat language (not Schoenberg's!), but by design allows listeners to internalize the workings that language gradually.  Now the following effect may be lost on certain experts possessed of excellent universal ear training and audacious on-the-fly analytical ability: for many listeners, jumping right to the middle of the piece may make the music there seem quite "random" and difficult to parse, while listening from the beginning renders the whole work quite legible on account of how the piece naturally trains the ear to hear what's coming next.

Apsides, an atonal passacaglia for piano

Tessellations, for orchestra (1976)

Music for Marimbas and Harps

Up to transposition, there are 250 ways that an interval can transition to another interval without any note repeated (all notes within the chromatic scale).  The basic material here is a cycle of intervals that survey all 250 transitions.   Constant variations within the dimensions of timbre and articulation, though they might at first seem to work distractions away from the harmonic idea, actually mitigate exhaustion of the ear and so keep fresh the perception of all the intervals, and the intervals between intervals.

Every Bridge, for nine instruments

Enrobe 50percent No Reverb

Inuksuit, for Piano and Celesta

Dedicated to Christopher Maher, who urged me to take a serious interest in Cuban timba, the band Klimax in particular, a type of music far removed from my usual listening habits, to say the least

Johann Wolfgang von Goethe said “Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.”

Composer that I am, I am also a mathematician and that is exactly what happened.

Harvesting Timba

Given four distinct pitches (not pitch classes!), you will have available 24 distinct orders to perform them in.   Even without any ear training, most any listener can easily distinguish, say, 
highest - next_to_lowest - lowest - next_to_highest  
from
lowest - next_to_lowest - next_to_highest - highest.
This makes for the basis of a highly audible, psycho-acoustically available variant of serialism.

As such, the present piece is based on a sort of abstract 24 tone cyclic row, abstract because, the row lacks the usual information concerning precise pitch content.  Instead the row structure specifies only, for each note, whether it sits lower or higher relative to the three notes before it and after it in the cycle.  

The existence of such a row that exhausts exactly the 24 contour possibilities is a mathematically non-trivial fact.

Less than two minutes at full tempo, the piece might service as a virtuosic concert encore.





The distinction between these 24 order 

A Study in Contour,  for viola (1975)

Concertino for Vibraphone, Marimba, and Orchestra

Berceuse[BeSecure] for octet

I was barely out of my teens when I met Tom Johnson in NYC and we started exchanging ideas about music and mathematics.  We continued doing that for more than four decades.  He left us on the last day of 2024 and I miss him greatly.  

One of his enthusiasms was "logical music."   Now this might seem a category error - after all, shouldn't it be be arguments and proofs, and not works of art, that are logical or not?  

Actually it turns out that formal logics in themselves just don't care about truth and correctness. Rather they're just oblivious rule-based systems that produce new thing from old things.  Well, sometimes they always produce new true statements from old true statements, and then that's very useful. So those kind of  logics are what people think of when they hear the word. but really the realm of logic is actually much larger.

So I understand Tom's program thusly: instead of having rules bound and bind our art, and also instead of attempting to reject rules altogether (if that's even possible), how about making works of art that, by design, faithfully project faithfully their own sui generis and hopefully intrinsically interesting rules?  With music, that means that both the sounds and the rules, so the logic, become potential objects of contemplation.  

Consider the duality of mind and brain, two distinct things with nevertheless thoroughly entangled existences.  So likewise the sound and structure of music.  By itself that's not new of course, music has had structure for millennia.  But treating musical structure as a first-class aesthetic object for contemplation by the thinking/listening subject, that was new.   Now structure could be more than a mere package or organization for the sonic experience, more than the mere bottle for the wine.  

My Elegy here is based on the fact that (11/8)^37 = 2^17 very nearly (the discrepancy is about .07%).  Both the rhythm and the harmony of this piece project this coincidence.  All the details emerge logically from the effort to project this little arithmetic coincidence.  

While the generating interval 11/8 is approximately 5.5 half-steps (actually more like 5.513), so roughly midway between a perfect 4th and a tritone (11/8 is the freshman sum of 4/3 and 7/5), if you pay close attention you will hear that the *logic* of this piece is actually not the logic of quarter-tone music, even though moment-to-moment it might have a sonic quality similar to quarter-tone music.  The faint, high-pitched wood-block type sound keeps track of this.  As Tom would say, "Just count the notes!"

Cover art from Tom's great book, Imaginary Music: 
https://www.editions75.com/pdf/books/Tom_Johnson_-_Imaginary_Music.pdf

Elegy for Tom Johnson (in 37TET)

méson kaí ákron

Lately I'm making pieces for a synthetic percussion orchestra with a virtually infinite number of instruments.

This is a proof of concept piece.

Perkodhuskurunbarggruauyagokgorlayorgromgremmitghundhurthrumathunaradidillifaititillibumullunukkunun

This is the soundtrack to an abstract video work called Grow.  

The cover image is one frame thereof.

The music and visuals are equal partners though, so "soundtrack" is misleading.  But so is "music video."  The music can stand on its own.

Contact me for a link to the video.

Two Forests

The most basic rhythmic transformation consists in subdividing a single unit of time into two equal halves.  Repeating this operation produces an infinitude of patterns of which this piece might be considered a partial catalog.  The tempo is chosen so that the beginning sounds like some sort of virtuoso drumming (but these are my own computer generated sounds, and no sampling was involved), yet gradually, as the patterns grow more complex, execution would require superhuman ability and the result verges on the biological.  The catalog has its own emergent shape, so there is rhythmic interest too on larger time scales.

Halve and Whole

A companion piece to Halve and Whole, but where that piece takes a deep dive into duple rhythm, here (often nested) triplets balance the story.  If the basic concept seems very similar, well, arguably, the overall effect not so much.

Heart and Mind

Branches

Leaves

Sound component of an abstract film of mine (the cover art is the first frame).  Please inquire if interested.

My Heart Is Beating Fast And My Thoughts Are Racing

There is a slender stream in my output, but extending back five decades now, of music based on scales repeating, not at the octave, but at the fifth (or fourth).  In particular, while the effect here is always "locally pentatonic," as the piece progresses it grows increasing pantonal.  By the end all the pitch classes get equal treatment.  

By a "hypercanon," I mean a canon whose theme is itself a canon, and perhaps, and so on.  

Here the  ultimate theme and patterns of canonic repetitions have been chosen very carefully so that music has an overall constant density.  Get the math wrong and the music will just grow muddy or thin out prematurely.

A pulse unifies the rhythmic surface, but it beats so fast, on the order of 50 clicks per second, that the audible rhythm will not imply it even the pulse arguably still informs the rhythm.  So just as the tonality here doesn't count as either diatonic or traditionally atonal, the rhythmic profile is neither entirely "free" nor based on a perceptible count.

Another quite different interpretation of the underlying structure is forthcoming.  Watch this space.

Sometimes the New Place to Go Sits In Between the Places You Know, for piano

A fact of nature: one and the same thing may admit both the simplest of specifications, yet also exhibit rich behavior verging on the quasi-random.  

If a 3-note chord has pitches sitting x, y and z semitones below C7, that chord occurs here at time x^3 + y^3 + z^3.

That's the whole score of the work (modulo a simple orchestration strategy).

So all the rhythmic complexity, and all the counterpoint and voice leading simply emerge from the implications of that short sentence. 

I thus invite you to listen for the structure, the generating intention.
But I invite you, equally, to listen for the phenomena, the temporal extension.  

To my ears what unfolds has something subjective, even expressive about it, even though I know it's all implicit already in that simple formula.

The Treachery of Simple Arithmetic, for 10 instruments

In an article I published decades ago in Xenharmonikon, I proposed strategies for composing computer based microtonal music with no fiat scale.  Composed notes thus get their intonation on the fly based on music context rather than as members of a fixed but expanded gamut.

My Fugue and Postlude puts those theoretical principles into practice.   The form makes a twisted reference to Baroque music, a(nother) time when musical values intertwined closely with the theoretical expansion of music language. 

Fugue And Postlude

The full title (shortened here only because it exceeds SoundCloud's character limit) should read:

I scissored into slender strips the wings of butterflies first one wing then the other sometimes for a change the two abreast never so good since

All the notes derive by manipulating a few measure taken from Chopin.  When I composed the work back around 1977, modal-rhythmic minimalist music still seemed quite a new thing.

This MIDI realization represents the middle movement, but that said, the framing outer movements only last a few seconds each.  Those movements depends on (psycho)acoustical effects related to masking and sympathetic vibration, so do not lend themselves to MIDI realization.

Over the years, four pianists have taken up the piece, but not for a couple of decades now.

Material Press, Frankfurt Germany distributes the score.

I scissored into slender strips... for piano (full title in Description)

Enrobe the roaring waters with my silks, for piano

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

Please don't spam this account!

David Victor Feldman

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