My chamber music

My microtonal music realized by MIDI

My orchestral music

Klangfarben pieces

2025

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

If Schoenberg liberated the dissonance, and relieved it from the obligation to resolve, his style and the styles that followed mostly shunned consonance altogether.  

I this piece extreme dissonance and extreme consonance live together, in harmony, so to speak, but without the traditional hierarchical expectations.

Working on this, I kept thinking of the stripe painting of Gene Davis, but then music generally inspired those paintings as many of their titles make clear.

Consonant/Dissonant I, for orchestra

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

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

My first published work--the score appeared in Ear Magazine--I composed this around 1979.

In a live performance (it had one) any number of players would receive the full score and a die.  Everyone then starts playing at the measure located in the center of the score's hexagonal array.  From there, all the players meander through the material based on throws of the dice.   When the die points the way out beyond the boundary of the array, a player stops.

As composed, nearby measures contain similar material, so without too much trouble a listener could learn to hear the geometry of a realization.

Though the piece has no fiat sectional structure, by the nature of the process there emerges first a definite introduction, then a build-up, then a churning, dense, chaotic aural scene, then gradual reduction of forces reaching a more intimate, chamber music sort of texture, and finally a quiet finish as a duet or solo. 

The idea of building an extended performance on a limited amount of material owes a debt to Terry Riley's In C, and the progress for the orchestral to the (possibly) solo might recall Haydn's Farewell Symphony.  Perhaps the modal structure will distantly evoke Javanese pelog for some listeners.   Finally the piece certainly derives something from certain works by my my teacher, the late Jonathan Kramer --- perhaps particularly his Moving Music for an orchestra of clarinets.  




Hexagonal Music

Concertino for Vibraphone, Marimba, and Orchestra

Berceuse[BeSecure] for octet

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

The promised orchestration of the piano I recently posted.  

9/11 in 19TET for chamber orchestra

I composed this during my sophomore year of college in 1976.  The score received some recognition that year, making the finals of the BMI Awards to Student Composers Competition.  Nevertheless, it remains unrealized in performance more than 40 years later and this sound file represents merely a MIDI realization.

Sending a special thank you to Kenneth Rundt, author of SynthFont2, who kindly provided me a custom compile of his program with a modified feature I needed!!

Vocalissimus, for orchestra (1976)

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.

At first blush this music may seem to hearken back to the the Second Viennese School,  but stay with it for a few minutes and you will hear something altogether different.  This is actually as consonant as that music is dissonant.  The organization is more diatonic than serial though far more than twelve notes occupy each octave, and the harmonic universe slowly progresses through related keys like a symphonic development section does.

Trio for Strings

Six parts, each in a different meter, intertwining in a Brownian way.

Luctuosa desideria for piano, marimba, brass and cello

The 25 sections of this deliberately generic pointillistic music alternate between two types of pitch material:
1) atonal, actually simplistic serial using only untransposed prime form;
2) diatonic (in a sequence of keys that follows the row of 1)).

Babbitt famously described Arthur Berger's Duo for Cello and Piano as "diatonic Webern" (some say "white-note Webern").    If you have the ear for it, the "diatonic Darmstadt" sections will stand in stark contrast from the atonal sections.    Or maybe just in gentle contrast.  And doubtless some people won't detect the boundaries and others will start to hear the structure here only after repeated listenings.

My aesthetic object here really lies in the staged moment of awareness concerning the distinctions the brain can make (or not), that and awareness concerning how the meaning of a fixed sonic object varies with the listener.  "Experimental music" usually refers to an experiment designed by the composer, and executed by the performer, but here the listener experiments by reflecting inward on what they can hear and how it feels to hear it.

All that said, I hope you find the surface pleasant enough albeit, for me, not an end in itself.

Ishihara Test For The Ear (for piano and harp)

If a composer wishes to avoid tonality, wishes to avoid the monarchical tyranny of tonic, then a simple (but not sure-fire) recipe consists in working a symmetrical treatment of the totality of pitches (with said totality often going by the name "total chromatic," or more generally "aggregate," but aggregate can also mean a totality of anything).  The twelve-tone system treats pitch classes symmetrically by design, and aleatoric methods also prove quite effective.  Other tyrants can arise aside from tonics.  So if mezzo-forte, or the central octave, or the 16th-note should threaten domination, a composer can also serialize (or randomize) dynamics, tessitura, or duration, etc.  In practice, all this fair treatment of possibilities, all this fiat symmetry, can ironically make tyrants out of the relevant aggregates themselves, effectively replacing monarchy, one might say, by obsessive bureaucracy, with the ear experiencing concomitant fatigue (not necessarily a bad thing depending on the composer's artistic goal).  In any case the emergence of this irony perhaps explains the briefness for total serialism's vogue.  Long ago, atonal composers working more freely discovered the trick of "keeping a note in your pocket" only to pull it out again when you need it to sound fresh---and it will.  

This piece plies a latter-day total serialism (but a serialism not anchored in rows) which governs pitch, tessitura, rhythm, dynamics and articulation, but which has the note-in-the-pocket wisdom inscribed in its genetics.  The 81 sections survey all 80 hexachord-up-transposition classes (the final section reprising the first, a whole-tone scale), with each section signaled by the entrance of a single pocketed pitch class and less noticeably by the departure of another (so traversing a Hamiltonian circuit through hexachord space).  Likewise, a Hamiltonian circuit through time-point "hexachords" (a la Babbitt) characterizes the rhythm.  But then each hexachord gets developed by a special type of Hamiltonian circuit through all 20 of its own sub-trichords, with the instantiation of each trichord pulling a note of a shallower pocket. 

The spare texture of the present work pays homage to John Cage, specifically works from his Lou Harrison period, to Morton Feldman, specifically his late pattern pieces, to Jo Kondo, specifically his hockets, and to Jonathan Kramer, specifically his process pieces and works with restricted pitch sets.

The Note in Your Pocket (My Travels Through Hexachord Space), for two melody instruments

There are fundamentally 713 distinct ways to partition the aggregate (a.k.a chromatic scale)into four trichords.  Of these, 290 have no trichord type repeated and these I'm calling the tetra-trichords.  Call two tetra-trichords neighbors if they share at least three trichords types.  

This piece follows a loop, from neighbor-to-neighbor, through most of the space of tetra-trichords, 288 of them.  The two omitted tetra-trichords only neighbor one-another, so refuse to play along.  They deserve a piece all their own (I'll get to it!).

So this is fundamentally "twelve-tone" (but not row) music, as in all twelve tones are more or less constantly active and in equal proportions.  But thinking of the trichords as the-new-notes, the aggregate becomes a kind of super-chord, and the whole piece a super-four-part-chorale, with consecutive super-chords always sharing a common "tone" (actually, a common trichord) for good (super)-voice-leading.  

So this is twelve-tone music with no direct input from common practice, yet very audibly working with analogous ideas.

My Travels Through (Most of) Tetra-trichord Space, for flute, clarinet, bassoon and celesta

The title refers to that painting technique where pigments get mixed on the canvas.

Wet Into Wet, for orchestra

Wet Into Wet II

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

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9/11 in 19TET for chamber orchestra

David Victor Feldman