My electronic music

My microtonal music realized by MIDI

2025

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)

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

These five pieces feel like rags to me despite their strange tuning, irrational rhythms and athematic construction.  They all follow the same progression of 73 special chords derived from division of the octave into 34 parts, special because each also closely approximates a selection of overtones from a common fundamental.  Two common tones link each chord to the next giving an effect different from but still something like functional harmony.  One might call the movements mutually isomorphic, even though they differ in all their lapidary details.   Having listened to one or two, the others will sound familiar one a first hearing, but only in a very abstract way.

Elliott Carter explored an all-interval tetrachord in his first string quartet and an all-trichord-class hexachord in his Symphonia.  Teemu Heinonen, a brilliant young Finnish composer, asked if similar resources were available in quartertones.   I answered Teemu's question completely for all equal temperaments up to quartertones and a bit beyond.   I was struck by the all-tetrachord-class 10-note chord (or scale) (0 1 2 3 5 7 12 13 15 16) in 19TET.   The existence of even one such 10-note chord is still about 10,000,000,000 times as many as probabalistic heuristics (that work very well in most tunings) would predict.  So this chord strikes me a kind of miracle.  

Here I arrange all 210 tetrachords  drawn out of the miracle chord in a Hamiltonian cycle with each tetrachord disjoint from the tetrachord before and after.   Of course I needed some design to project the individual tetrachords rhythmically, so I've settled on a structure that echos the pitch structure.   Thus I've married each pitch class is a self-similar rhythmic/dynamic pattern that remains the same same if you double the tempo(save for getting a tad louder).  So these rhythmic patterns form a kind of circle-under-octave-equivalence just as pitch classes do.   Though the rhythmic/dynamic patterns for individual pitch classes have a metric/binary character (like standard subdivisions of common time), their tempos relate by the frequency ratios so 19TET (so powers of 2^(1/19)), creating great complexity in the interaction.  

You will hear that my choice here to make each tetrachord disjoint from its neighbors induces a slow breathing in the music; it suggests locally an inhale-exhale harmonic periodicity that actually dissolves as you follow the larger composition.  The self-similar dynamics provide both the shimmering background, the delicate middle ground, and the occasional seeming interruptions (which are actually simply organic to the whole plan).

Miracle I (My Travels Through 19TET-Tetrachord-Space, by leaps)

While the midpoint of this 6 minute work seems to feature a dramatic and unexpected gesture, that moment actually arises as the inevitable consequence of the very, very simple logic in operation from the beginning of the piece to the end.  Careful listening may reveal this logic.  (Hint: pick one particular pitch to follow.)

While  just intonation informs the  pitch structure, the inharmonic timbres only suggest the pitches that they theoretically project.  In fact no (synthesized) timbre here ever gets used more than once, thus evoking a gamut of thousands of distinct but interrelated pitched percussion instruments.

The Inevitable Unexpected

Antichain

Many people associate serialism with what late composer Humprey Evans III used to call "twitchy music."   Here I explore a novel serial technique for a music I might describe as protean or fluid.   

Rows control configurations of attributes rather than pitches or other simple parameters.   In particular they control substructural justly tuned harmonies and the evolving mixes of outputs of six different filters (related related to difference tones) applied to those harmonies.  The pitches matter...but the effects also relate to some of the more purely "sonic" music of our day.

Let the title invoke not only the various meanings of the English word, but the rather different sense it has particularly in Dutch.

Drift

The following may seem like a mathematical paradox: if you start with one dollar and, at a constant rate, gamble one dollar double or nothing, you will almost surely lose all your money sooner or later.  But the average time till ruin is infinite.

Here it's notes not dollars.  A bet either generates another pitch in the same overtone series as the original note (the fundamental of a note a silent parameter) or extinguishes the note.  If the bet is won, the new note gets its own fundamental, and in this way the music, in a sense, modulates.

If you listen I hope you will hear that music is more than the sum of its simple description.

I have realized this process many times, and each realization has its own distinctive character.  If there's interest, I'll add "variations."

Technical note...  Most realization come out very short or very long.  Using some mathematical ingenuity, I have jimmy-rigged the time till ruin.  The result is provably mathematically indistinguishable from running the process till arriving by chance at a realization of the desired length, but is also in a way a fiction, since the code does not actually simulate the narrative described above.

Gambler's Ruin (electronic)

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

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Elegy for Tom Johnson (in 37TET)

David Victor Feldman