Warp Weft (1998)

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

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

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

Composed in honor of Tom Johnson's 60th birthday and premiered in Paris and Amsterdam by ENSEMBLE DÉCADANSE at a two-part festival in celebration thereof.  

Music based on a finite-state automaton devised by the composer;  Tom Johnson himself was pursuing musical applications of finite-state automata around that time.

T'homage (1999) for flute, violin and harp (for Tom Johnson on his 60th birthday)

Inuksuit, for Piano and Celesta

Concertino for Vibraphone, Marimba, and Orchestra

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)

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

When living things gain the ability to extract information from their environment and react to it, they can have a survival advantage.   So did our senses evolved;e.g. an animal might "hear that" a predator approaches and evade it.  

Complex beings process complex stimuli for all manner of information, so the cognitive effort after the raw sensation can take time.   This necessary delay, for human beings at least, means experiencing sense-data-in-itself even while reflecting on the question "What is it?"  For music this gives rise to a duality: even as we surrender to an immersive experience we might parse it and deduce some of its structure (or at least experience the mere presence of structure through its artifacts).  

Perhaps "duality" make this too simple: we can also reflect on the feeling of surrender, we can also reflect on the feeling of deductive experience.  We can "hear that" a sound got made in a certain way, we can hear that a sound is similar to different from some other sound, and we can "hear that" a variety of sounds enjoy some order or pattern.   Music is not just how it sounds, it is at least(!) all that the stratified listening mind does.

To Hear and Hear-That, as a kind hypercanon (a canon on a canon on canon on a canon) depends on "transposition up the steps of the overtone series (as opposed to up or down a fixed interval).  

From another but equivalent viewpoint, the many chords here correspond to solutions of the problem of expressing any given beat number, if possible, as the sum of four distinct fourth powers (a.k.a. biquadrates).   From the choice of that problem, and for mathematical reasons, a kind of rhythmic steady state emerges, a flow that neither thins nor thickens on the average, but nevertheless projects predictable/unpredictable variation (depending on the computational power of the agent who would predict).   In contrast to the rhythmic steady state, as the piece progresses the larger beat numbers require solutions with larger biquadrates, so the timbre of the piece slowly evolves from consonance to some kind of resonant dissonance.

For an ending, both a bit arbitrary but also a bit natural, the piece goes on only as long as 32 harmonics suffice.  



To Hear and Hear-That

Another hypercanon piece.  

This time, from the mathematical perspective, beats get filled when their beat number equals the sum of a square, a positive cube and 6th power.  

Because the powers vary, the music keeps the counts on different instruments. 

The numbers to square get heard on the piano (with 1 the top key and then descending chromatically).

The numbers to cube get heard on pizzicato strings tuned in 19-TET, using an ascending scale that divides three octaves into 19 equal intervals which sound as nearly 10:9 rations.

The numbers to raise to the 6th power, never more than 4, map to a small gamut of tuned drums.

The complexities of the sets of sums of just two kinds of powers asserts itself as the irregular staggered entrances of the canon on the instrument associated with other power.  

If this is too much information for you, the bottom line is an intense listen which, white it has a hypersimple underlying concept, also feature a rather complex surface regardless, and because of the density of the beat numbers with a "yes" answer, intense but also unfamiliar syncopation. 

Big thank you to Peter Thoegersen, who literally wrote the book on polymicrotonal (and polytempic) music.  He's a genius and you should check out his music.  While his music and also his working methods are quite different than mine, he has opened me up to one of his great discoveries, namely that the listening mind can instinctively decouple the layers of a complex music according to the tuning systems girding each layer (at a given moment).  The effect is like passing to an experience in a higher dimension, akin to the discovery of perspective in visual art.   I'm not sure Peter would even count this as "poly" since two of the layers are 12TET and my treatment of 19TET avoids actual microintervals.  But I'd never have come to this way of projecting my ideas without his influence.

Higher Powers, for Piano, Percussion and Pizzicato Strings

This is a kind of polytempic work in the sense of Peter Thoegersen (well, if he thinks so).  The whol time span gets divided into 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 equal parts with each tempo supporting a layer consisting of only 7th chords that have one particular root.  Since I make chords in the slower tempi sound louder, the interaction of dynamics and periodicity tends to balance the importance of all the different roots.  With the next slower layer always carrying the given layers dominant, a certain amount of root progression emerges.  But it's complicated!

Those Fibonacci Fractal Four-Chord False Relation Blues, for piano four hands

Said the poet: "Euclid alone has looked on Beauty bare."

Push back on "alone" if you like, but beauty may dwell in the immaterial, and from thus the possibility of art formed purely out of ideas. Or not purely: a sensual experience may, by its structure, simultaneously carry ideas. Two art objects, one immersive, the other cognitive, can simultaneously inhabit the same experience.

Varèse framed music as "structured sound," and I would add, alternatively, "sonified structure." Mathematics is the science of structure, so, if you like, the part of music theory robust (to some extent) against the contingencies of sonic materials. I have a long interest both in bringing to my composition ready-made mathematical objects and mathematical phenomena, and in letting the same emerge from sonic considerations themselves. But truly to infuse music with at least the ghost of "Beauty bare" requires evoking mathematical argument.

So consider communicating something by drumming. Fix a small unit of time. Now say you decide to work with only two durations between strikes, namely 2 units and 3 units. Even with the best coding, that choice limits your bandwidth—limits the rate at which you can transmit information.  Now one can quantify this in a sophisticated way with information theory, but we don't need to get into that now. But here's an interesting and not quite trivial fact: if your durations are 1 unit and 5 units, then you have exactly the same bandwidth as with 2 and 3 units! 

A long and short duration might remind you of Morse code. Well, if you "glue" the "dash", the long duration, of the 2–3 system to the following duration, you will get a drumming pattern in a 2–5–6 system. And that translation is reversible. But if you glue the "dot" of the 1–5 system to the following duration, then you also get a pattern in the 2–5–6 system, and also in a reversible way. So you can get from 2–3 to 1–5 via the trick of interpolating between them with another system you might not originally have thought to consider. There's an episode in the piece that makes precisely this transition/translation audible.

But it doesn't stop there! All the following systems also have the same bandwidth as the three mentioned above:
                                                                1–6–11–16–20, 
                                                                1–6–10, 
                                                                1–7–10–12–16, 
                                                                2–6–7–10–11, 
                                                                3–4–5, 
                                                                3–4–8–9–10, 
                                                                3–5–7–8–9, 
                                                                4–5–6–7–8.
Now hopefully the required "bridge" between, say, 1–6–11–16–20 and 3–4–8–9–10 doesn't seem obvious at all! Nevertheless one will experience it if one attends very closely to the sonic details of the piece.

Yes, you can forget all this and you still have an intense experience reminiscent of ritual drumming, albeit without any periodic underpinning. By the way, he piano part works just as the drums do, but also correlates pitches to the durations.

The title: if you wish to transmit a long random "message" in the 2-3-system, a certain ratio between the frequency of 2's and 3's will emerge.  The famous golden ratio would emerge from the 1-2-system and goes by the name "phi."  Hence "high phi" as one old and mostly forgotten proposal for a name of our ratio here.

High Phi, for drums and piano obbligato

The music can speak for itself, sonically, if you prefer.

But if you care to understand what's going on, here it is in brief.  The durations in the cello are 2, 4, and 8 8th notes; the durations in the bass line of the piano part are 3,5,6, and 7 8th notes.  Any rhythm of any duration of the former sort admits a translation into a rhythm of the latter sort, and vice-versa, indeed a translation that preserves the overall duration (to within a fixed few beats anyway).   I expect that this fact hardly seems obvious!  But the violin part is the intermediary, along with the other voices in the piano part.   Attend to the relationship between cello and violin; and between piano and violin, and, with some concentration, the logic indeed reveals itself.

Elegy for Piano Trio (High Phi II)

In both movements of this quartet, each instrument gets assigned a repeating rhythmic pattern—sequences of notes and rests of equal length (say, all eighth notes). 

The first violin plays a pattern of 11 beats (6 notes, 5 rests), the second violin a 7-beat pattern (4 notes, 3 rests), the viola 6 (3+3), and the cello 5 (3+2). These patterns cycle independently.

One full cycle  takes 2310 beats—the least common multiple of 11, 7, 6, and 5.

But—crucially—beat durations vary.

Indeed, the duration of a beat depends on its density—meaning how many instruments are sounding at that moment (0 through 4). Each density appears a fixed number of times within the full cycle, independent of pattern contents assuming the constraints above. 

The twist: beat durations get adjusted so that each density—whether a silence, solo, duo, trio, or a full quartet—occupies exactly one-fifth of the total time of the movement. Rare textures are stretched, common ones compressed.

Thus the interaction of the parts effects an emergent large-scale rhythm.

Out of millions of possible combinations of patterns, one movement employs the set that yields the most unpredictable overall rhythm (in a precisely defined mathematical sense). The other, the most predictable.

As for pitch: each instrument traces a fragment of the cycle of fifths—in near standard tuning. The quartet plays 17TET, so with the octave divided into 17 equal steps, which makes a "perfect" fifth span 10 steps. Each instrument repeatedly travels through this cycle fragment, alternating between octaves. Individually, the lines sound diatonic; together, they form unfamiliar and highly dissonant verticalities—often stranger than even the most dissonant music in 12-tone systems.

Despite this microtonality, no specialized technique is required. If each player tunes their instrument to a slightly different “A,” then standard fingerings suffice—as long as they resist the instinct to tune to one another.

Waves, Winds, Mind - Mvt. II

Waves, Winds, Mind - Mvt. I

Composing this piece with sonic and conceptual goals in mind, I was surprised to find I had produced something simple enough for novice piano students—or alternatively, a trio of treble instruments.

As the hands can remain stationary, the piece makes only the most modest physical demands.

That said, it poses other kinds of challenges.

The performer must read from three staves, each carrying a simple but reasonably independent voice.

The meter changes constantly, shaped by the additive logic of the underlying rhythm.

And while a beginner might ignore the dynamics and still produce an acceptable result, a finished interpretation depends on the ability to play piano and forte simultaneously, in order to bring out just one or two voices when required.

Whatever makes music simple also makes it transparent—and transparency always demands control. Put differently, the same minimalism that lightens the physical load calls on the performer to maintain connection and seek out rhythmic and tonal nuance that will sustain interest.

In one respect, however, the piece is unapologetically pedagogical.

It communicates a mathematical argument in support of an elementary but non-trivial proposition.

In more detail: the highest voice moves in durations of 1, 7, and 8 eighth notes; the lowest in durations of 1 and 5. Each voice constitutes a kind of translation of the other. Remarkably, any 1–7–8 rhythm can be transformed—without much change in length—into a corresponding 1–5 rhythm, via a completely reversible process. This is only possible for very special sets of durations. And by the way, those 1’s—the eighth notes—don’t just tag along. Their role is more complicated, and necessarily so.

As often happens in mathematics, revealing a hidden connection between two structures requires inventing a third that can serve as intermediary. The middle voice plays that role, moving only in durations of 2 and 3 eighth notes.

Analyze the piece for its patterns, and the translation principles hinted at here will become clear.

One may simply take this as an opportunity to learn some mathematics while learning music. Better still, one might strive to hear both at once—and so enjoy a kind of dual experience. That possibility of duality might even inform the interpretation.

Direct message me if you would like to see the score.

Proof Piece for piano or three treble instruments

Decades before 2001, I conceived this piece based on the polyrhythm 9:11 and the arithmetic fact that (9/11)^2 = 2/3 (very nearly).  I never realized the piece back in the 1970s when I thought it through, because I didn't have access to the technology that would make it possible.  Lejaron Hiller offered to help!  I never followed up.  So it's been on my to-do list for about half a century, no exaggeration.  

I working on an orchestrated version, but I quite like how this "piano sketch" came out, so I'm planning to release both.

The underlying structure is related to my piece Primetime for two marimbas (which itself was first conceived as a piece for two carillons, and I have never quite given up on the hope of someday hearing it like that.

I know that this is a long piece so....  Listen to it all.  Listen to part of it.  Listen to it all but not in one sitting.  Use it as background music if that suits you.  Dance to it if you feel it.  Whatever works for you.
 


9/11 for 19TET piano

The promised orchestration of the piano I recently posted.  

9/11 in 19TET for chamber orchestra

Didier Aschour is a pioneer specialist performer of microtonal guitar music.  His custom guitar allows for rapid exchange of fingerboards, and he has worked with a luthier to build a collection of fingerboards that cover a vast range of tuning systems.  

This piece exists in two versions.   

While the two versions share a common score, they differ with respect to the arrangement of the fingerboard frets (the "warp") and the tuning of the strings (the "weft").   Didier commissioned two special fingerboards for the piece.


In each version, between the tunings of frets and the strings, one constitutes a fragment of the overtone series and the other a fragment of the undertone series.   The resulting interaction provides a very rich selection of pitches: within a limited span, arbitrary numerators and arbitrary denominators arise--thinking here of the available pitches by their ratios to the guitar's lowest open string.  

"Just intonation" always means composing with intervals formed from small integer ratios, thus with some subset of the rational numbers.   The tuning here, you might call it Cartesian, means to project the structure of the rational numbers writ large, so without imposing any subjective selection process.

Performed live, the piece would have Didier sliding a glass "bottleneck"  for the warp and strumming for the weft.  Or course that timbrel conceit is lost in these MIDI realizations.  But hopefully one day I'll have a recording of an instrumental performance for sharing.  This will have to do for now.



   




Warp Weft (I), for microtonal guitar (1998), for Didier Aschour

Didier Aschour is a pioneer specialist performer of microtonal guitar music.  His custom guitar allows for rapid exchange of fingerboards, and he has worked with a luthier to build a collection of fingerboards that cover a vast range of tuning systems.  

This piece exists in two versions.   

While the two versions share a common score, they differ with respect to the arrangement of the fingerboard frets (the "warp") and the tuning of the strings (the "weft").   Didier commissioned two special fingerboards for the piece.


In each version, between the tunings of frets and the strings, one constitutes a fragment of the overtone series and the other a fragment of the undertone series.   The resulting interaction provides a very rich selection of pitches: within a limited span, arbitrary numerators and arbitrary denominators arise--thinking here of the available pitches by their ratios to the guitar's lowest open string.  

"Just intonation" always means composing with intervals formed from small integer ratios, thus with some subset of the rational numbers.   The tuning here, you might call it Cartesian, means to project the structure of the rational numbers writ large, so without imposing any subjective selection process.

Performed live, the piece would have Didier sliding a glass "bottleneck"  for the warp and strumming for the weft.  Or course that timbrel conceit is lost in these MIDI realizations.  But hopefully one day I'll have a recording of an instrumental performance for sharing.  This will have to do for now.


Warp Weft (II), for microtonal guitar (1998), for Didier Aschour

The Thing With Feathers, for flute, oboe and clarinet

This is a companion piece to The Thing With Feathers

on the strangest Sea  - for violin, viola, cello, celesta, harp and marimba

Even though atonal music and athematic music represent, in principle, quite distinct 20th-century developments, most well-known examples of the one also represent they other.   On the one hand, classic twelve-tone technique frowned on parading the row as a theme or motive, so twelve-tone composers have tended to hide repetitions of the row through continuous variation of contour, register, rhythm and dynamics.   On the other hand basic aleatoric procedures, while surely not generating tonal centers, also don't turn up by chance repetitions of large configurations of notes. 

Here aleatoric materials get subjected to a simple sort of aleatoric editing involving repetition and limited transposition.  A rudimentary sonicgrammar emerges.  Constellations of notes acquire identity.  Even without triads or voice leading, the radically atonal materials begin to project, if just barely, functional relationships such as one associates with common practice and not atonality.   This music thus takes the listener to places where neither conventional tonal music nor conventional atonal music can go.

The Thunder-like Percussion of Thy Sounds, 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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