Museum, for chamber orchestra

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

This work for cello is derived from ideas associated with change ringing.

If one has, say, 6 bells, and one wishes to ring them in some sequence, then one has 720 possible sequences ("rows") to choose from.    Performing all the possible "rows" in some sequence constitutes an "extent" or "full peal."  A traditional desire consists of have every two consecutive "rows"  (or "change") in an "extent" differ only by swapping order of two consecutive bells.   So give any particular starting "row" of 6 bells, only 5 "rows" can legally complete the "change." 

That "extents," legal in this sense, even exist follows from non-trivial mathematical constructions, but these constructed extents tend to have a very mechanical character (and advantage for "method ringers").  

I (re?)discovered that one can make new extents from old by exchanging two consecutive segments AND time-reversing just one of them.  Call this a "twist."  "Twisting" one of the mechanical extents 2000 times (for an extent of 720 "rows") seems to produce reasonably random or organic "extents."   Philosophically these "twisted extents" lead double lives, being both highly structured and constrained, but given that, also as capricious as possible.

Four random "extents" govern this work.  Most simply, one "extent" controls the 6 dynamic levels in each row statement, one "extent" controls the 6 durations, and one "extent" controls the sequence within whatever the current collection of pitches.  

The final "extent" gets used go generate the pitch content.  Instead of ordering events, the permutations get used to order  the intervals 2/1, 3/2, 4/3,  5/4, 6/5 and 7/6, thus building up a scale (or perhaps a chord, depending on how you care to think about it).  So unlike actual bells that each carry a fixed pitch and collectively establish a fixed key, here each "row" of the 720 has its own "key."  Thus the music constantly modulates, and since each "change" modestly changes just one note in the scale (because consecutive intervals swap), the music manifests an emergent sort of voice leading, albeit a very distant relative from the voice leading of common practice music.

Simultaneously Follow Four Twisted Paths, for cello

Untitled Work for microtonal 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

Inuksuit, for Piano and Celesta

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

This piece is simply a love song to a tone row that enjoys a unique mathematical property. Let me put that in some context.

When people think of "mathematical music," three names tend to come up: Milton Babbitt, Iannis Xenakis, and Tom Johnson. For the point I want to make, let me concentrate on Babbitt and Johnson.

With his musical career well underway, Tom fell in love with mathematics writ large, despite having no formal training in the subject. He relied on professional mathematicians (myself included) to tell him about significant ideas, and he was a genius at finding ways to project mathematical concepts through music in the most natural possible ways.

In the preface to the score of Networks, Tom writes: “Composers should not tell their music what to do.” (This reflects Tom’s willingness to call himself a “minimalist,” even as the most famous minimalists rejected the label.)

I suspect Milton would have dismissed Tom’s sentiment as nonsense, holding that telling music what to do is composition perforce.

Milton did have mathematical training, and it expresses itself in his work as the rigor with which he approaches the intrinsically musical problems posed by twelve-tone practice. And unlike Tom, Milton actually fathered some original, albeit modest, mathematical discoveries. Still, his relation to mathematics resembled that of a typical engineer: formal methods helped him do what he wanted—and needed—to do, but his interest didn’t extend much beyond that.

Tom, for his part, had a studied disinterest in twelve-tone music, precisely because its practitioners often buried structural principles under richly “improvised” surfaces that attracted listeners’ attention. The common notion was that the underlying theory—the mathematics—was the exclusive domain of experts.

My piece here is twelve-tone music, but philosophically close to Tom’s minimalist spirit.

Indeed, I recently discovered this twelve-tone row:

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

It enjoys a remarkable property that makes it essentially unique.

By way of explanation: only about 1.5% of all possible tone rows allow you to combine the row with two rotations of two other forms (of the same row) so that the resulting three rows, when sounded together, produce twelve distinct trichords that exhaust all twelve trichord classes.

That said, most rows that allow this at all allow it in just a single way. But this row allows four ways, yielding four essentially different cycles of trichords.

I have never met two stools that I didn’t want to fall between. There is some simple surface detail here, and I can imagine Tom urging me to eliminate any competing idea. But that said, I am neither “improvising” here nor treating the row as mere “pre-compositional” material.

===================

I have chosen a tempo here for the web.  In performance, the player decides, and I enjoy the piece even at half the speed.

Let Me Make This As Clear As I Can, for solo piano

"Tightness" is a composer's value, a quality of music that more often revealed by score study than by casual listening.  Composers might call a score "tight" if its events and parameters turn out overdetermined, as do the answers to a crossword puzzle.  That means the little details all satisfy multiple constraints and the overall fit-together quality verges on the uncanny.

Here I have combined the serial logic with a minimalist's sense of decorum, to keep the listening experience as close as possible to a deep dive into the score.  Indeed, in the second and third sections the rhythmic invention helps you out if your relative pitch lets you down.

In particular, this piece is a love song to the only three tone rows that enjoy a property similar to but distinct from the row that forms the basis of my recent "Let Me Be As Clear As I Can."  

As with the row in that previous piece, each row, joined with certain rotations of two alternative forms of that same row, produces a succession of all 12 distinct trichord classes — and in each case, one can choose the alternative forms in four essentially distinct ways!  Where these rows differ, the succession of trichords always turns out essentially the same across all four solutions.  

Think of a jigsaw puzzle whose pieces can fit together in very different ways and yet strangely still form the same picture.

The three rows and all the solutions receive three different compositional treatments, each forming one of the sections of this work. 

I consider the work indeterminate in a limited way: other versions might replace any or all of the three rows with one of their 2304 = 48² forms (taking into account multiplications of pitch classes and position numbers, as well as transpositions and rotations), and versions can also shuffle, in six ways, which row receives which compositional treatment.




Music Tighter Than A Gnat's Ass Stretched Over A Rain Barrel, for solo piano

The ingredients:
A length-21 sequence of JI-chords every 13 seconds;
A length-13 sequence of JI-chords every 8 seconds;
A length-8 sequence of JI-chords every 5 seconds.
All these are run through a digital ring modulator.  
This results in constantly changing timbre, and a slowly morphing rhythm
(because 13/21, 8/13 and 5/8 all quite close).

At the start, all the chords share a common root/fundamental, 60 cps.  

The roots gradually drift apart to 60-60.75-61.5 and this causes the overall timbral profile to grow gradually more gnarly.

Needless to say, this track is purely synthetic and so not derived from any sampled sound.

art thou but a didgeridoo of the mind, a false creation?

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   11    1    2    8    9    3    6    7   10    5

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

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

Space Filling Curve, in 31TET,  for Marimba, Harp, Harpsichord and Organ

Enrobe the roaring waters with my silks, for piano

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)

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

Please don't spam this account!

David Victor Feldman

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