My orchestral music

Not Your Grandparents' Twelve-tone 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

Though its own kind of very strict 12-tone-music, this music has nothing to do with Schoenberg's system or with row manipulation.  And though minimalism certainly also informs this music, its atonal and non-repetitive character keep it from falling into that category.  In the end, this music's real affinities connect it to recent developments in the visual art.  There an ongoing revival of formal abstraction now receives energy and subtle inspiration from a resonance with the world's craft traditions.

Not Your Grandparents' Twelve-tone Music (for woodwind quintet)

In 1974, Robert Morris and Daniel Starr famously enumerated every all-interval twelve-tone row. 

Inspired by their work, and in around 1979 in unpublished works, I generated an analogous catalog of what one might call the all-trichord rows, meaning rows, which together with two of their untransposed rotations, and in "first-species counterpoint," produce a sequence of all twelve distinct trichord classes.  By the standards of the day, this 
involved very substantial computational search.  But lately I have vastly expanded that calculation, now allowing transposition, retrograde and inversion of the rotations. Some of these new rows provide the basis of my recent pieces.   

The catalog of all-trichord rows runs to about 300 more or less, depending upon the accounting for symmetries.  That makes these rows quite rare among all the twelve-tone rows.  This rarity suggests my original compositional conception but unrealized at the time: a museum-like installation where a listener might walk in a large space from sonic exhibit to sonic exhibit, and experience the different ways these rows fit together and do their thing.  

So here are just three typical exhibits, realized now as works for chamber orchestra.  In each, the basic row first gets a plain but lapidary statement, and then the rotated forms fade in, statistically speaking, and eventually the rows likewise fade out in the order in which they entered, leaving finally a statement of the basic row.  By design, this treatment could loop seamlessly as one "display," as per my original conception.

0    4    8   11    2    3    1    6    7    9    5   10

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)

On a popular understanding, Schoenberg devised the twelve-tone system to enforce a kind of democracy among pitch-classes.  For example, here is Milan Kundera's unsympathetic riff on this theme in The Book of Laughter and Forgetting: "One day a great man determined that after a thousand years the language of music had worn itself out and could do no more than rehash the same message. Abolishing the hierarchy of tones by revolutionary decree, he made them all equal and subjected them to a strict discipline: none was allowed to occur more often than any other in a piece, and therefore none could lay claim to its former feudal privileges. All courts were permanently abolished, and in their place arose a single empire, founded on equality and called the twelve-tone system."

One can approach Schoenberg's purported goal by means not involving rows.   One can understand this piece as twelve compositions played simultaneously, each involving a single pitch-class.  Each layer (or channel if you think of the pitch classes as akin the primary colors that make a digital picture) unfolds in a uniform rhythm, but each in a tempo different from the others.  The durations of twelve-pitch classes divide the total duration of the piece into parts as follows: 
12*79, 19*80, 14*81, 21*82, 16*83, 23*84, 
18*85, 13*86, 20*87, 15*88, 22*89, 17*90.
One can view the 12*79 notes for the first pitch-class as 12 cycles of 79 or 79 cycles of 12.  Dynamics and doubling bring out both ways to parse the whole span.  Structure emerges from the interaction of the dynamics and doubling in all twelve "channels."  The shorter cycles interact to project chromatic scales (with octave displacements) and the longer cycles to project cycles-of-fifths.   Pitch classes move from octave to octave only gradually, so the polyrhythmic structure induces little melodies that seem to repeat but actually morph constantly.

Channels (Not Your Grandparents' 12-Tone Music III)

Another piece which possibly parses as 12 simultaneous pieces, one for each pitch-class.  Unlike my previous effort along these lines, where a steady pulse characterized each layer, I've applied approximations to the mathematical object called a Cantor sets to make the rhythm of each layer clumpy.  This allows a variety of pitch-sets to emerge but with no hard boundaries between these "harmonies."

Of course a cantor is also a kind of singer...

Cantor, for piano (Not Your Grandparents' Twelve-Tone Music IV) (2019)

In a sense, this piece comprises just a single statement of one 12-tone row: 6  7  4 10  1 11  0  8  3  5  9  2.   Understanding these numbers as pitches (not pitch classes), they form a kind of all-interval row as 11 different intervals sound (ignoring here whether they rise or fall!).

But the intervals don't manifest in this piece as juxtapositions of steady tones.  Rather the pitches modulate one into the next, by means of the virtual washboard of the title.  Think of a bumpy two-dimension surface.  An object making contact with the surface and moving horizontally and sinusoidally will sound a pitch.  I let this horizontal motion morph into a vertical motion, on the way passing through Lissajous figures (where one frequency controls the vertical and another controls the horizontal).   The Lissajous sounds suggest common undertones, but with equal temperament tuning no definite fundamental emerges.

The same process, but eight octaves down creates the envelops that one hears as rhythm.  Thus the same quasi-all-interval row controls a sequence of metric modulations.  

The two channels realize the same process, but with different virtual washboards - which means somewhat different resonances and quite independent rhythm.

A personal note.  When I start to go on a bit about feeling old, I dwell on how young experimental composers seem to have lost interest in "notes" altogether.   Well, Cage liberated noise, sure.   But it's hard to develop a music theory of noise, and old fashioned pitches still lend themselves better than anything else to projecting structure.   I feel happy letting virtually any sound into my music universe, but I feel like I'd have nothing to say without pitch and the structure it affords.   

Well, in this piece I try to have to have it both ways.  So, on one hand, I deal with protean sound and I avoid the sort of discrete pure tone events one normally associates with "note."   On the other hand though, pitch logic, both 12-tone (for the overall structure), and harmonic (emergent from the logic of the Lissajous figures described above) still haunts the whole sonic profile.

The graphic image displayed visualizes the morphing of Lissajous  figures.

In the future I'll post some significant variations.   But please understand even this post as a single realization of an open form piece.  I've classified all 1328 quasi-all-interval rows and any of them would serve.  And one can swap in alternative virtual washboards for similar but still distinct timbres and rhythms.

Virtual Washboards (Not Your Grandparents' 12-Tone Music V-a)

A pair to 
https://soundcloud.com/david-victor-feldman/virtual-washboards-not-your-grandparents-12-tone-music-v
but the feeling is a bit different.

This time one of the tracks states the row or its retrograde 7 times as compared to once in the other track.   That, combined with the complexity of the morph from one pitch to then next, creates a sense of counterpoint or even emergent harmonic motion.  Both pieces best heard with headphones.

Virtual Washboards II (Not Your Grandparents' 12-Tone Music V-b)

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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David Victor Feldman