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 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

Another snippet from Caligo Paean - beware of gifts when chasing gods : ) 

Caligo Paean is a 1 hour long sound piece installed on river front property in the town of Franklin, Tasmania as part of the exhibition "When The Fog Clears". It formed part of the Ten Days on The Island Festival March 2025.
The work was installed in several locations across the property including a stack of subs under an old bus, in the main exhibition space and in an abandoned shed some 40 metres from the main area . The speakers were 'tuned' quite differently depending on their location and the sound was not delay corrected with the result of each mini-installation merging with each other in odd sonic ways - kind of calling the listener from one spot to another. The the work features field recordings, percussion, piano, bass guitar and Intonarumori as well as this special one off choir : )

Caligo is the Ancient goddess of Chaos. Born of the fog she controls the darkness, dwells in the void and is as old as the primordial matter that created the universe. This work is a tribute to the goddess as she continues to influence our lives and, in the case of this installation, the site and concept informing the show.

Official press release below for The Fog Clears

Every day, the mists of the Huon Valley perform a seductive dance, gently obscuring and revealing contours of the fertile land and waters. Situated in The Watershed Franklin, When The Fog Clears features six artists from lutruwita / Tasmania exploring the region’s iconic mists through a variety of mediums including installation, sculpture, illustration, sound, and video.

Developed exclusively for Franklin, this exhibit extends from the waterfront itself into the chambers, kilns, and rafters of the shed, inviting visitors to consider fog as an atmosphere that can cloak us in unexpected ways.



Caligo Paean (A gift, a choice)

A polycyclic piece with 5 looping cyclic parts not in a neat order, but not polytempic nor polymetric, either. Formerly a string quartet, but with added synth, is essentially a quintet. 2025

String Quintet in standard tuning with synth pad

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)

String Quintet in standard tuning with synth pad

pthoege