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

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

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, my second mixed Trio in poly, features 17-19-21-23-25-28-31 seven note scale in the cello, that loops 6X against two polyrhythmic contrapuntal 24 unequal Pythagorean scale of al Farabi in the piano, and Ben Johnston's 6th Prime row of 25 unequal per octave, in the Clarinet. The interplay is finely thought out and responsorial. The poly trio is a new form I am experimenting with for more transparency for the contrapuntal interplay between the instruments in polymicrotonality. 2025

Mixed Trio in Polytempic Polymicrotonality #2

To all concerned: 
I am announcing publication of my new book, NOW AVAILABLE:
https://www.routledge.com/Maqam-Melodies-Pitches-Patterns-and-Developments-of-Music-in-the-Middle-East-and-other-Microtonal-Writings/Thoegersen/p/book/9789815129052
My book, Polytempic Polymicrotonal Music, is available on Amazon. 
https://www.routledge.com/Polytempic-Polymicrotonal-Music-The-Road-Less-Traveled/Thoegersen/p/book/9789814968294

I have a new album out through Magic&Unique Group Records on Spotify and other digital streaming platforms:
https://open.spotify.com/album/5qWHmcvyAemvVn2TJH1bio?si=OD5TmPptQ-ypr8WOI0EZKA&nd=1

https://whitecentipedenoise.com/products/peter-thoegersen-facebook-cantos-cdr-flea
My CD via Flea label has been out since April, 2021, featuring the wonderful Lore Lixenberg singing over my microtonal piano accompaniment
Hope you enjoy it!

https://www.newworldrecords.org/collections/vendors?q=Peter%20Thoegersen

From the liner notes on my CD now available on AMAZON and New World Records:
"The author of these notes has spent his life explaining radical music, and the music on this disc may be the most radical I've ever written about. Peter Thoegersen (b. 1967) is not yet a name known to the music world; not for any lack of connection to other famous musicians, but because he came to composition late, and because his artistic aims are so broad and complex that they have taken years to evolve. His aesthetic is well defined, and he is upfront about having a name for it: "Polytempic Polymicrotonal Music." That means he has musical layers simultaneously moving in different tempos, plus, even more radically, that he has different microtonal scales playing at the same time. Few composers have aimed at such fervent and heterogeneous multidimensionality. And yet, while Thoegersen's music can seem overwhelming at first, running through it is a level of charm and simplicity that belies the initial impression.

In total, these three pieces emphasize a particular side of Thoegersen's output. The image his music creates of ornately mathematical structures filled with intuitively friendly melodies maps on to the personality of the man himself: outspoken and acerbically ruthless in his musical opinions, Thoegersen is also marked by a jolly sense of humor and an inexhaustible love of cats and other animals that make him ultimately less imposing and more personable than he might seem at first. The same can be said for his music. And by going further than anyone else has gone in terms of this particularly American concept of fusing the polytempic with the polymicrotonal - extending and combining the conceptual worlds of Ives, Cowell, Nancarrow, Partch, Carter, even Frank Zappa and many others - he has created a special place for himself within American music."                                                             

-Kyle Gann (from the liner notes)

Mixed Trio in Polytempic Polymicrotonality #2

pthoege

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