My piano music

2025

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

Multiple hands keep all registers of the piano engaged nearly constantly with contrast made in other ways.  A type of "moment form" makes disjunct, atonal material surprising legible even on a first hearing.  The music seems to tell a story propulsively, but a story with no beginning, middle or end. 

When the Clouds in Autumn Crack (2016) for piano four hands

From a set that includes
https://soundcloud.com/david-victor-feldman/our-ringlets-to-the-whistling-wind-2016-for-piano-four-hands 
and
https://soundcloud.com/david-victor-feldman/when-the-clouds-in-autumn-crack-2016-for-piano-4-hands



there's no clock in the forest (2016) for piano four hands

You will hear one structure filtered through another.   Leaving aside here any description of the complex underlying structure, the filter simply quantizes (or patterns or metricizes) the otherwise very free rhythms, but on and off every few seconds.  This makes an audible yet subtle contrast, but does not jar the larger musical flow.   

Having finalized the composition,  I abruptly realized its conceptual similarity with a beautiful series of drawings by my friend Jessica Rosner.   In this series she conceives symmetric geometrical structures and then executes half the design freehand, half with straightedge.   The contrast in techniques makes a structure that coexists with a structure leaving the viewer freedom where to direct her mind.

Jessica has generously lent me her running title and permission to feature a very recent addition to this body of work as the cover image for this track.
Find more of her work at http://jessicadeanerosner.com .

Ruled Unruled for piano four hands

The algorithm is simple:  note  n  appears on beats f_n + f_m where
the sequence f_n runs over Fibonacci numbers (positive and negative): ...-377 -233 -144  -89 -55  -34 -21 -13  -8  -5  -3  -2  -1  0  1 2  3  5  8  13 21 34  55 89 144 233 377...  The notes themselves come from a scale that repeats, not at the octave, but at the fifth: 
(b$-3) (c-2) (e$-2) (f-2) (g-2) (b$-2) (c-1) (d-1) (f-1) (g-1) (a-1) (c1) (d1) (e1) (g1) (a1) (b1) (d2) (e2) (f#2) (a2) (b2) (c#3), so the emergent structure is naturally polytonal and  seems to modulate from flat to sharp keys.

Music For Piano Four Hands (The Sums Of Two Fibonacci Numbers) (1978)

We live in strange, sad times.

Recently, for an utterly practical purpose, someone asked me the following question:  

12 people who need to meet by Zoom (due to the pandemic) would all like to get to know each other, one-on-one.  But to save time, they'd like to break out repeatedly into 4 groups of 3 (rather than into groups of 2).  How many times will they have to do so for every pair of people to meet at least once in some small group?

Turns out they will have to break out 7 times, but this means 18 pairs of people will meet twice, so, while best possible, not a very tight solution.  

I sent them such a plan.

Then it occurred to me that if they'd allowed one break out into 3 groups of 4, things can work out perfectly tightly with just 5(=4+1) break outs total..

And then I thought, what if I treated pitch (classes) as the people?  

That makes the groups of 3 into trichords, 4 each from the 4 breakouts into 4 groups of 3.  So then it would seem nice to have all  16 trichords harmonically distinct (even up to transposition, which makes trichords come in 19 possible flavors total).

It turns out that there exist only 20 basic ways (of assigning pitches to people) that accomplish this. 

One of those forms the floor plan of this piece.

But also on my mind: a friend recently lost a close family member. I felt very sad to learn that the pandemic will prevent his relatives from actually gathering, so they have planned a Zoom memorial.   

And so, as I worked on this piece, I began to conceive it as a strange kind of funeral march, albeit without meter or tonality.  Thus  I dedicate it to the memory of all we have lost and will lose in this worldwide catastrophe.

Zoom Memorial, for piano

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

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

David Victor Feldman