My microtonal music realized by MIDI

Recently I rediscovered the fact that 171 equal divisions of the octave will afford one very accurate representations of the four consecutive overtones 4:5:6:7.  Justly tuned, this "dominant 7th chord" sounds as a consonance.  

At any moment in this piece, one hears this chord present in four forms, namely the original, its inversion preserving the perfect 5th, and then some collective transposition of these two.  That interval of transposition gradually rises through an octave as the piece progresses; that's the main process.  

This results in stretches of bitonalities separated by moments where the distinct keys lock on account of common tones.

Social Dance Under The Light of Two Moons On Planet 171-TET

Untitled Work for microtonal piano

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)

méson kaí ákron

In an article I published decades ago in Xenharmonikon, I proposed strategies for composing computer based microtonal music with no fiat scale.  Composed notes thus get their intonation on the fly based on music context rather than as members of a fixed but expanded gamut.

My Fugue and Postlude puts those theoretical principles into practice.   The form makes a twisted reference to Baroque music, a(nother) time when musical values intertwined closely with the theoretical expansion of music language. 

Fugue And Postlude

Though stylistically quite different, this work emerges from the same theoretical investigation that produced _My Travels in Euler Space_, as I recently posted.  https://soundcloud.com/david-victor-feldman/my-travels-in-euler-space

This time 3-dimensional space is filled with rectilinearly organized "stars" representing pitches, with one direction corresponding to the interval of an octave, another to twelfths and another to 17ths (major 3rds plus two octaves).  All pitches within the range of human hearing thus lie near a certain plane.  Now a planet traveling a circular orbit moves through this plane, and spins so that its "year" lasts 60 of its "days."   Pitches sound when the stars that symbolize them rise over the horizon as viewed from a fixed location on the planet.

The timbre of the music morphs slowly, for no deep conceptual reason, but simply to avoid fatiguing the ear.

Dark sky, for strings and harps (MIDI)

Yet another trip through an Euler Lattice.  Primes 3,5, and 7, so no pitches related by octaves.  A given pitch will only sound three or four times in the whole work.   Each interval thus contains lots of information and that piece unfolds rather slowly, giving the ear time to take it all in.  At first the field of pitches sound (not unpleasantly?!?) random, but careful listening will reveal both harmonic structure, complex modulations and something like counterpoint, but all in a highly unusual language (even by the standards of microtonalists who usually settle on a scale upon which to build a composition).  

Odd Sounds(MIDI)

Here I try to imagine a sort of extreme "new complexity" with distant roots in common practice (sonorities informed by the overtone series, form emerging from root progression) and uninformed by any serial practice.  Complexity can bring one to the edge of legibility or past it.  I believe this music remains legible if attended.   Admittedly I speak as the composer, but I literally hear new and different things going on with each listen.

Imposto (MIDI)

Suppose that a musical monochord sounds at frequency F.  Stopping the string at two places, so dividing it into three segments, causes those segments to sound at frequencies A,B and C satisfying 1/A + 1/B + 1/C = 1/F.  

Now if one plucks the stopped monochord at a random place, the lowest of the three frequencies (attached to the longest segment) will sound with the highest probability, and vice versa.

As this piece develops, the stopping of an imaginary such string evolves so that the mathematical entropy of the random process stays approximately constant but gradually increases (until the stopping points divide the string into three equal parts).  A graph of this looks like a tight triangular spiral.

Finally the durations of the notes correspond, by design, to the probabilities at any given moment of the three frequencies --- but in a different way in each of the three voices.  A high frequency (unlikely note) in the first part grants the next note a long duration; in the second part a high itself gets a long duration; and finally in the third part a high note grants the previous note a long duration (so, from a listener's perspective, a long duration in that part heralds a high frequency coming).   This presents the listener with a counterpoint contrasting three different but related music logics.  

Another version of the piece works almost the same way, except that low frequencies correspond to long durations.

A score for an 12-tone equal temperament approximate version dates back to the late 1970's, but never received a performance.   This MIDI realization realizes the pitch relationships quite precisely, and perhaps supersedes the piece originally conceived for human players.   


Trio Entropicum (first version)



Suppose that a musical monochord sounds at frequency F. Stopping the string at two places, so dividing it into three segments, causes those segments to sound at frequencies A,B and C satisfying 1/A + 1/B + 1/C = 1/F.

Now if one plucks the stopped monochord at a random place, the lowest of the three frequencies (attached to the longest segment) will sound with the highest probability, and vice versa.

As this piece develops, the stopping of an imaginary such string evolves so that the mathematical entropy of the random process stays approximately constant but gradually increases (until the stopping points divide the string into three equal parts). A graph of this looks like a tight triangular spiral.

Finally the durations of the notes correspond, by design, to the probabilities at any given moment of the three frequencies --- but in a different way in each of the three voices. A high frequency (unlikely note) in the first part grants the next note a short duration; in the second part a high itself gets a short duration; and finally in the third part a high note grants the previous note a short duration (so, from a listener's perspective, a short duration in that part heralds a high frequency coming). This presents the listener with a counterpoint contrasting three different but related music logics.

Another version of the piece (https://soundcloud.com/david-victor-feldman/trio-entropicum-first-version) works almost the same way, except that low frequencies correspond to short durations.

A score for an 12-tone equal temperament approximate version dates back to the late 1970's, but never received a performance. This MIDI realization realizes the pitch relationships quite precisely, and perhaps supersedes the piece originally conceived for human players.


Trio Entropicum (2nd Version)

Where most microtonal music starts with the construction of a novel but fixed scale,  this music finds all its pitches "on the fly."  (MIDI piano makes this possible.)  Despite a density of diverse pitches extreme even compared with complex atonal music, a  ghostly, and yet oddly familiar, sense of root progression emerges here, and propels the music through time.   

Rhythmically, the initial relentless pulse gradually gives way to angular syncopations, but based on the golden ratio and quite impossible to notate in conventional meters.  Despite a fixed "tempo," the music seems to accelerate:   the twisted rhythms draw on a gamut of just three durations, but the mix of these three changes from beginning to end for a spectrum of feeling that gives the work shape and direction.

Milky Way Boogie Woogie

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.

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

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My microtonal music realized by MIDI

David Victor Feldman