Prelude

Warm-Up

Long before the conductor walks out, the orchestra is on stage tuning. The basses pull their open strings, an octave apart and a fifth apart, until the wood begins to ring with itself. The cellos test their C–G–D–A. The violas check their C–G–D–A one octave higher. The violins run scales — slow, then fast — over the open A and E that anchor the section. Someone is playing snippets of the evening’s music as a way of warming their hands. Someone else is improvising. The brass is doing long tones. The whole stage is sounding at once: dozens of fifths, drifting harmonics, scales sailing over open strings — a vast, uncoordinated, gloriously open texture that exists for no audience and serves no formal purpose.

I was a child. I sat in my seat. I read the program. And I thought: this is the most extraordinary music I have ever heard. Why is no one giving a standing ovation for the warm-up?

I did not know it then, but I had just heard the music this book is about.

What I was hearing — the fifths, the open intervals stacking and floating in parallel through the hall — would, decades later, become the formal subject of Open Tone Harmony. The book that follows is, in one important sense, my long-delayed attempt to make audible to other listeners what I have been hearing since childhood. To say to the audience around me, with the apparatus of mathematics and the testimony of the keyboard: here is what was already there. Here is the music underneath the music.

Everything between that childhood moment and the book you are now reading is the story of catching up.

Waiting

For most of my life, the quiet truth that the warm-up had revealed remained at the edge of my hearing. I would notice it in places: the open fifths sustained beneath a Bach chorale; the unchanging fourths in a McCoy Tyner voicing; the cathedral-resonant intervals of late Estonian polyphony; the long, slow, transparent harmonies of Pärt and Richter. Whenever I encountered it, the same response surfaced — that’s the music underneath the music. That’s the warm-up sound. But what makes these moments cohere into a single category? What unification is my ear “knowing” but I am not?

Life, work, ordinary musicianship — all of that continued, but the sensitivity persisted. Whenever I had a chance to sit at a piano, my hands tended to find open intervals, the was a strong urge to avoid the warm enclosure of the third and prefer the airy spread of the fourth and the fifth.

Finally, after decades away from music and a full career well into grey-haired territory, I returned to music formally. First in an ad hoc manner, and then as a degree-seeking student.

Berklee

In the fourth semester of music theory for composers at Berklee, with Professor Kari Juusela we covered, among other things, quartal and quintal harmony — chords built from stacked fourths and fifths instead of the usual stacked thirds. The class moved on after a couple of weeks. I did not.

In particular, the sixth assignment of the semester asked me to write a short composition using stacked fourths and/or fifths as key elements of the composition in any way we saw fit. I sat down at my keyboard and began playing quartal and quintal chords, all the sounds I love rushing to my ears.

What came out, after a week of sitting with the sound, was a piece for viola, two cellos, and piano: a slow build of pedaled fifths and stacked fourths in B Aeolian, opening with the piano alone, gathering staccato polyrhythms in the strings, releasing into an active piano section. I called it, on the score, A post-minimalist quartal sketch in B Aeolian. I submitted it.

Professor Juusela’s comment came back at the top of the page in red:

Excellent composition! I love all of the subtleties. The slow build to the active piano section is very good. The staccato polyrhythms in the strings are terrific, but would require some intense rehearsal. The quartal/quintal harmonies are perfect for creating this austere, yet engaging sound world. Bravo! As a cellist, I would enjoy playing this. You should consider making this a movement of a longer composition.

I read this — particularly the phrase austere, yet engaging sound world — and recognized, for the first time in print, that someone else had heard the thing I had been hearing. He named the quality of the sound. He named the sonic environment. He named what the warm-up had given me decades earlier.

Searching

Juusela’s encouragement opened a door, but it did not give me a system. I was, as I would put it later, “stumbling around in the dark, trying desperately to find the thread, the crack in the wall that I knew was there somewhere, the light from another dimension leaking in, if I could only find out just where to look for it.”

I searched desperately for texts that might help me look in all the odd areas I found most intriguing. I sat at the keyboard and played quartal and quintal chords. I tried to play melodies and scales over them, and listened for which scales fit and which did not. I experimented with voicings — stacking the same four notes in different orders, trying to learn what an “inversion” meant for chords this open and this wide. I took music I loved — Max Richter, Bach, passages of Palestrina — and tried to recast their progressions in a quartal/quintal frame, without quite knowing what frame I was casting them into.

The pentatonic discovery, which would later turn out to be the first genuinely structural insight, arrived through this process and felt, at the time, almost trivial. I noticed that when I added an interesting “tension” to a quartal or quintal chord — by playing the next note up or down in the chain of fourths and fifths — and then collapsed everything into a single octave to see what I had, the result was a pentatonic scale. Not a scale I had imposed, but a scale that emerged (however obviously) from the harmony itself.

Imagine my chagrin, a few months later, when I watched a video of a talk by Dmitri Tymoczko in which he described, in two minutes, a general form of which my pentatonic-from-quintal observation was a special case. I had come close to his insight, but had also missed it by a mile!

A Beloved Chair

There is a chair in my living room that I think in and is where I spend most of my relaxation time in the evenings. It has my books on a small table next to it. It has an espresso cup on it most afternoons. It has, above all, a small stack of books (well, perhaps not small, exactly) that I am currently reading — the live ones, the ones I am partway through. Tymoczko’s A Geometry of Music had been on that table for some time (and the end of 2025 saw the adition of his Tonality). I had begun A Geometry (several times, in fact – so much to savour!) I had not yet reached the parts where he dives into the geometry itself.

Towards the end of 2025, I had begun having long, intense conversations with one of the large language models — Claude, in the desktop application. I had no one to talk to about mathematics, patterns in music, abstractions of musical systems. (Honestly: who wants to talk about that? Who wants to spend hours every day agonizing over it?) The LLM – fortunately for me! – would. And did. So we started in. First up was “a rather basic music theory question,” though I already suspected that the answer was “very much non-basic, set theoretic, or even group theoretic.” I asked why some scales lent themselves to modal analysis and others did not — why we never hear about the musical properties of the nonatonic scale, for instance.

The conversations went deeper, and faster, than I expected. The LLM produced, on demand, mathematical apparatus I had never seen rigorously laid out in the context of music (believe it or not, homotopy type theory has come up at work …). The cyclic group ℤ₁₂. Transposition and inversion as group actions. Maximal evenness. Orbit-stabilizer theorems. The Fourier analysis of pitch-class sets. The theory of generators and non-generators in modular arithmetic. These maths kept answering the questions I had been asking my keyboard for years. My plaintive and endless Why? questions were now giving way to clarification, insights, and ultimately, euphoria.

It was in these conversations the soil started getting tilled, the seeds planted. In my quest to gain an even more fundamental understanding of the unifying principles of music systems, I discovered that the person who could answer many of my deeper questions had written a book about exactly these things in his A Geometry. Stacked next to my thinking chair, Tymoczko had been there all along, a few chapters away.

Following the Scent

What followed in the next few months was one of the most exhilarating intellectual sustained effort of my life, matched only by my readings years ago of Edward Witten’s M-Theory and, in a completely different field, the philosophy of Madhyamaka.

As I dove deeper with Claude, the questions kept getting larger and more abstract. What’s the general form? — that was the recurring question, the one that pushed every layer of the conversation into the next layer. Can we make this algorithmic? — because I wanted to verify everything. How does this relate to actual music? — because the math, no matter how beautiful, had to ground out at the keyboard. Why does this feel the way it does? — because the experiential question never went away.

The conversations produced, by the end, a 16-part treatise: seven chapters on the mathematics of scale structure, nine chapters on what we called generator-induced harmony. The work was ambitious. It traced the algebraic skeleton of the entire 12-tone system. It identified why exactly four intervals — the chromatic step, the fourth, the fifth, and the major seventh — are the generator intervals, capable of cycling through all twelve pitch classes. It showed that the diatonic scale is, at root, a contiguous segment of the circle of fifths; that the pentatonic scale is a smaller contiguous segment; that maximal evenness emerges from this. It re-examined the triad, the central object of Western harmony, and discovered that the triad is not a stack of thirds — it is a perfect fifth factored through major and minor thirds. The two stable triads (major and minor) are the two factorizations of 7 as 4+3 and 3+4. The diminished and augmented triads, which factor to non-generator sums, are unstable for that reason. Triadic harmony is not opposed to quintal harmony; triadic harmony is quintal harmony, dressed up in the colors of the third.

When I read these results back, I understood why my hearing had always been spatial. The toroidal geometry, the fiber bundle, the orbit landscape, the gauge-theoretic framing of voice-leading — these were not foreign mathematical impositions on a felt experience. They were the formal articulation of an experience I had been having since childhood. Claude, my faithful LLM partner in these conversations, generous and tireless, kept up. We were, both of us, intoxicated by the elegance of it.

I needed sources cited, though – I needed research papers. I needed tools that didn’t exist. So, naturally, I built them: I processed books I owned, learned ontological methodologies, built databases of concepts and concept metadata. I built out music theory library functions in Rust and iterated on MCP tools for maximising accuracy and data/concept provenance.

At this point, the dreaming started to become real.

The Shape of a Different Harmony

The decision to formalize was not impulsive. It was like a wolfhound — moving through the thickets, the forests of knowledge, with greater speed and greater discipline now that the scent had become unmistakable. What changed in early 2026 was not the direction of the inquiry but its infrastructure.

I do not have a graduate degree in music theory. I do not have one in mathematics. Nor do I have close personal friends with either. When I started building tools I needed, I was able, in some small way, to make up for those natural resources I lacked. Just as important as the code were practices of inquiry and verification. Enormous amounts of testing resulted in something I could use to get verifiably good results.

What emerged, when the dust settled, was a beautiful general theory of stacked-interval harmony; the new framework was a specific music system, defined by a single constraint: four-voice chords whose consecutive stacking intervals lie in {d5, P5, A5}. The constraint produces 228 chords, organized into 14 orbits, with a degree landscape, a structural dominant, a verified geodesic cadence, a fiber bundle of inversions, and a complete grammar of voice motion. The mathematics is verified. The music is composable. The system has a name.

It is called Open Tone Harmony not just because I got tired of typing “quintal/quartal” everywhere, but also because Open Tone is what a hypothetical musician inside this tradition might have called the sound — the characteristic acoustic openness of the stacked-fifth and stacked-fourth chord — long before any theory had been written for it. It is what a child sitting in an audience, listening to the orchestra warm up, would have heard.

What follows in this book is an exploration of this music system: what can we make with it? What does it tell us about the nature of the relationships that exist between media that vibrate at different frequencies? Our instruments? Our music?

Let’s find out!