Open Tone Harmony

Open Tone Harmony

A Non-Triadic Music System

Formalisation • Exploration • Composition

[DRAFT]





by Duncan McGreggor

Copyright

Published by Cowboys ‘N’ Beans Books

https://github.com/cnbbooks ◈ http://cnbb.pub/ ◈ info@cnbb.pub

First electronic edition published: 2026

© 2026, Duncan McGreggor

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License

About the Cover

The artwork on the cover of this book, and that interspersed amongst its pages, is by Cowboys & Beans house artist Vigdís Ljósadóttir, a construct of style and aesthetic so compelling she got her own origin story. Vigdís is an Icelandic painter whose mother is a mathemticaian, botanist, creator of agent-based systems, Lisp programmer, and actor in a community theatre troupe. Her father is a physicist, scifi nut, 1970s progressive rock fan, and part-time stage manager in said community theatre troupe. Vigdís discovered her love of colour and light early in life, gazing upon the waters, stones, and space around Iceland during its long summer days and longer winter nights. She has her father’s love of scifi and 70s prog rock, and her mother’s love of botany to thank for most of of the subject matter that pushes her to stand in front of the easel.

Vigdís’ first project for Cowboys & Beans was the cover of a scifi novel. They loved her work so much, she was asked to paint for the author’s various and related world-building projects, and thanks to this work ultimately decided that the novel would, in a certain sense, be illustrated. When Open Tone Harmony was cleared for publishing, she was contacted for this book as her third project for C&B. We chose her perfect piece for the cover, but suspect (as with her other projects for C&B), more will make their way into the pages between the covers. The rest will soon be included in her yet-to-be released online gallery and exhibition site.

Dedication

To Mrs. Torno, my first and most precious music teacher. I lived much of my life before finally realising that – tucked away in Castine, Maine – she had instilled in me a deep love of music theory, the majesty behind the sound. Like great teachers everywhere and when, she was an inspiration to and example for all of her students.

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!

How to Read This Book

Prerequisites

A Note on Notation

Acknowledgments

This rather strange book rests upon the giant shoulders – not to mention mind, works, and explications – of Professor Dmitri Tymoczko (and before him, Richard Cohn and David Lewin). I would not have been able to have the fullness of love for music that I have today without those shoulders. For an introduction to 20th century music theory and encouragement for my explorations into non-standard harmony, I am in the debt of Professor Kari Juusela. For continuing that education and pushing me to be a better composer, and for humouring me so patiently while attempting to do so in a tonal harmony no one ever asked for, I am forever grateful to Professor Gabriele Vanoni. My only hope is that the gift of knowledge and pedagogy these have provided and which can never be truly repaid finds fertile soil in another mind, causing yet wilder frontiers to be explored.

Part I — The Living Tradition

An Imagined History of Open Tone Harmony

Hearing the System

The Pentatonic Root

Part II — First Harmony

The Stacking Act

The Open-Tone Field

First Compositions

Part III — Voice Leading and Motion

The Atomic Step

Passing Tones, Suspensions, Pedal Points

Counterpoint

[DRAFT]

Differences from the Counterpoint of Triadic Harmony

1. There are no forbidden parallels in the traditional sense. The chord-graph already encodes voice-leading parsimony — every legal motion is one voice × one semitone × three common tones. The Fuxian prohibition on parallel fifths/octaves does not transfer directly because every chord contains fifths by construction. The OTH analog is the orbit-persistence prohibition: avoid sustained motion within a single orbit (more than two consecutive edges in the same orbit can produce voice-fusion).

2. Consonance and dissonance live at the chord-class level, not the interval level. Within any chord, every interval is in {d5, P5, A5} — there are no “dissonant intervals” by construction. What is “dissonant” in OTH is the chord-class itself: high-degree orbits (Summit, Plateau Q787) are stable; low-degree orbits (Narrows, Precipices) are tense. The Saddle is uniquely tense-yet-connected.

3. Cadence is geodesically determined. The OTH cadential formula is Saddle → Slope → Summit — a verified geodesic of distance 2 through the chord-graph. This is not a postulated rule; it is the unique distance-2 path between the structurally extremal tension chord (Saddle) and the structurally extremal stability chord (Summit).

4. Voice motion is intrinsically four-voice. OTH chords are tetrachords by definition; there is no clean two-voice reduction. The species progression in OTH varies degrees of freedom in chord motion, not rhythmic subdivision against a cantus firmus.

5. Some devices have no triadic precedent. The multiset-collision shadow, the fiber-as-color voice, the leading-interval-pair resolution, and the wedge convergence on a shadow note are OTH-native. They are not “OTH versions of triadic devices”; they are devices that depend on OTH’s specific geometry.

The “cantus firmus”

In Fuxian counterpoint, the cantus firmus is a fixed melody in whole notes against which the student writes a counterpoint melody. In OTH, the equivalent is the chord-graph path — a sequence of base-space chords connected by legal motions. The student writes voice lines that move through this harmonic frame.

A typical exercise cantus firmus might look like:

Chord 1: C-G-D-A          Q777 Summit (home)
Chord 2: C-G-D-Bb         Q877 Plateau (one note moved: A → Bb)
Chord 3: C-G-Eb-Bb        Q787 Plateau (one note moved: D → Eb)
Chord 4: C-G-D-A          Q777 Summit (return — two single-semitone edges back)

This is a 4-chord cantus firmus traversing two chord-graph edges away from the Summit and back. Each chord transition is a single edge (verify: each adjacent pair shares 3 of 4 notes). The path is geometrically clean: Summit → Plateau → Plateau → Summit.

Throughout this guide, exercises use chord-graph paths of this kind as the cantus firmus. The student writes one or more upper voices that move melodically while the harmonic frame holds.

Setting up an exercise

To begin an OTH counterpoint exercise:

1. Choose a chord-graph path. Start short — 3 to 5 chords. Verify each transition is a single edge (use the music-theory MCP’s get_oth_neighbors if needed). Include at least one return to a starting orbit or one cadential gesture.

2. Choose a voice register and starting pitch. Decide whether the added voice will sit above (alto/soprano), below (tenor/bass), or interleaved with the cantus firmus.

3. Decide the species you are working in. This determines what motion types are available.

4. Write the line one chord-zone at a time. Each chord in the cantus firmus defines a “zone” in which the voice can move melodically; species rules determine how much motion is allowed.

5. Check legality. Voice lines should follow species rules; the harmonic transitions of the cantus firmus should be valid chord-graph edges.

Creating Counterpoint — The Five Species

1. Note Against Chord

1.1 First Species — Note-against-Chord

Definition. First species places one note in the added voice for each chord in the cantus firmus. The added voice plays a chord tone; it does not move melodically within a chord-zone. Voice motion happens only at chord-graph transitions.

What this teaches. First species teaches the student to see the chord-graph as the source of motion. Every voice movement happens because the underlying chord-graph moves, not because the voice elaborates within a static harmony. The student learns to track which chord tones are available at each step and which create good melodic lines.

Setup. A chord-graph path of 4-8 chords. The added voice begins on a chord tone of the first chord and ends on a chord tone of the last chord — preferably the cadential Summit if the path is cadential.

Step-by-step procedure

1. Identify the chord-graph path. Write out the cantus firmus chord by chord. For each chord, list the four chord tones. Note which orbit each chord belongs to.

2. Identify common tones across each transition. At every adjacent chord pair, identify which notes are held in common (typically three of four). The common tones tell you what the static options are at each transition.

3. Identify the moving voice in the cantus firmus. At each transition, exactly one voice in the cantus firmus moves by one semitone. Identify which chord tone changes. This is the cantus firmus’s “active voice” at that moment.

4. Choose the added voice’s range. Decide whether the added voice sits above the cantus firmus (in soprano range) or below (in tenor range). Above is more common for upper voice lines; below is harder because chord-graph collisions with the bass are more likely.

5. Pick the first note. Choose a chord tone of the first chord that is in the added voice’s range, not duplicated in the existing voicing. If the cantus firmus already has C-G-D-A in low register and you’re adding a soprano voice, you might pick C5 (chord tone, octave-displaced from the bass C2).

6. Move chord by chord. At each chord transition, decide:

   • Stay on the same pitch if it is still a chord tone in the new chord (a “common-tone hold”). This is the OTH analog of a pedal note — it shows that the chord-graph moved but the added voice held.
   • Move to a new chord tone that is one or two chord-graph edges away from your current pitch (a melodic step or skip).

7. Approach the cadence by contrary motion. As the cantus firmus approaches the Summit, the added voice should move in contrary motion (cantus firmus ascending → added voice descending, and vice versa). This honors the OTH “leading-interval-pair” cadence: two voices each moving by one semitone, one expanding a tritone, one contracting an A5.

8. End on a chord tone of the final chord (preferably a stable one — Summit or Plateau).

9. Check. Every note in the added voice must be a chord tone of its corresponding chord in the cantus firmus. No exceptions in first species.

Worked example

Cantus firmus (bass + held voicing):

Chord 1: C2-G2-D3-A3      Q777
Chord 2: C2-G2-D3-Bb3     Q877  (A→Bb in voice 4)
Chord 3: C2-G2-Eb3-Bb3    Q787  (D→Eb in voice 3)
Chord 4: C2-G2-D3-A3      Q777  (return — two edges back; via Q877 again)

Wait — chord 4 must connect to chord 3 by a single edge. From Q787 = {C,G,Eb,Bb}, moving to Q777 = {C,G,D,A} requires two changes (Eb→D and Bb→A). That’s two edges. Inserting an intermediate chord 3.5: Q877 = {C,G,Eb,Bb} → {C,G,D,Bb} (Q877; Eb→D) → {C,G,D,A} (Q777; Bb→A). This expands to:

Chord 1: C2-G2-D3-A3      Q777
Chord 2: C2-G2-D3-Bb3     Q877
Chord 3: C2-G2-Eb3-Bb3    Q787
Chord 4: C2-G2-D3-Bb3     Q877  (return through second Plateau)
Chord 5: C2-G2-D3-A3      Q777  (cadence)

Five chords; four single-edge transitions; cadence at chord 5.

Add a soprano voice. Rules: chord tones only; contrary motion to the cadence.

Soprano analysis:

Soprano line: G4 → Bb4 → C5 → D5 → A4.

Check. Each note is a chord tone. The line ascends G4→Bb4→C5→D5 (three stepwise moves) and descends to the cadence A4 (a structural-fifth leap downward). Contrary motion at the cadence: the cantus firmus’s last edge is Bb→A (descending semitone in voice 4); the added voice’s last move is D5→A4 (descending fifth). Both descend — so this is similar motion, not contrary. To get contrary motion, the added voice should ascend at the cadence: e.g., end on D5 → C5 (descending whole step) instead of D5 → A4. But wait — the cantus firmus’s voice 4 moves Bb→A (descending), so for contrary motion the added voice should ascend. Let me revise:

This is too static. Try again:

Hmm. The cantus firmus’s last edge Bb→A is a single voice descending; for the added voice to move in contrary motion at the cadence, it needs to ascend or stay. Let me make the soprano line:

Chord 1: D5
Chord 2: D5  (held — common tone)
Chord 3: Eb5 (step up — chord tone)
Chord 4: D5  (step down — chord tone; contrary to soprano-4-Eb5)
Chord 5: A4  (descent to cadence — structural P5 descent; cantus firmus also descends; SIMILAR motion)

The OTH cadence resolves Q877→Q777 with one voice moving Bb→A (single descending semitone). For the added voice to have contrary motion at the cadence, it must ascend or hold. Holding D5 across chord 4 → chord 5: but D is in both Q877 and Q777, so this is a common-tone hold:

Chord 1: D5
Chord 2: D5  (common-tone)
Chord 3: Eb5
Chord 4: D5
Chord 5: D5  (common-tone hold — contrary motion is *oblique*)

This is a clean first-species line. The cadence is reached via oblique motion (added voice holds D5; cantus firmus moves Bb→A). The added voice traverses D5 → D5 → Eb5 → D5 → D5 — a single neighboring excursion to Eb5 and back. Plain but legal.

Common errors in first species

1. Non-chord tones. First species requires every note to be a chord tone. Slipping in a passing tone (which is allowed in second species but not first) is the most common error.

2. Parallel motion across multiple chord-graph edges. If the added voice moves in lockstep with the cantus firmus’s moving voice across multiple transitions — both ascending or both descending by one semitone each time — you create the OTH analog of parallel-fifth motion. In first species, prefer contrary or oblique motion at every transition.

3. Failing to land the cadence. The added voice should end on a chord tone of the final chord, ideally the root of the cadential Summit (or its octave). Ending on an inner chord tone (like A4 over a C Summit) is acceptable but feels less “settled.”

4. Voice crossing. The added voice should not cross below the cantus firmus’s top voice. If your added voice is in soprano range, watch out for moments when the cantus firmus’s voice 4 ascends into your range.

5. Static line. A first-species line that holds the same pitch for all five chords is technically legal (every note is a chord tone) but pedagogically boring. Aim for at least one melodic step or skip somewhere in the line.

2. Two Notes per Chord-Zone

1.2 Second Species — Two Notes per Chord-Zone

Definition. Second species places two notes in the added voice per chord-zone. The first note is on the chord-graph “downbeat” (the moment of chord arrival); the second note is on the second half of the chord-zone. The second note may be a passing tone or a chord tone.

What this teaches. Second species introduces melodic motion within a chord-zone. The student learns to use passing tones and neighbor tones — the OTH analog of dissonance treatment — within the chord-graph framework. It also teaches the use of non-chord tones in OTH: notes that are not chord tones of the current chord but that resolve by single semitone to a chord tone of the next chord.

Setup. A chord-graph path of 4-6 chords. The added voice plays a chord tone on the first half of each chord-zone; the second half may be a chord tone or a non-chord tone (passing or neighbor).

Step-by-step procedure

1. Begin with a first-species line. Lay out the cantus firmus and an acceptable first-species added voice. This is your skeleton.

2. Identify opportunities for melodic motion. Within each chord-zone, ask: between the first-species note in this chord and the first-species note in the next chord, is there an intervening chord tone or stepwise non-chord tone that would create a smoother melodic line?

3. Insert passing tones for chord-tone-to-chord-tone leaps of 3+ semitones. A leap of a third or more between adjacent first-species notes is a candidate for a passing tone. Example: first-species line G4 → Bb4 (a minor third, 3 semitones). Insert A4 as a passing tone: G4 (chord tone) → A4 (passing) → Bb4 (chord tone of next chord). The A4 is a passing tone if it does not violate the underlying chord (it must be either a chord tone or a non-chord tone that resolves by step).

4. Use neighbor tones for variety. A neighbor tone is a non-chord tone approached and left by step in opposite directions. Example: holding C5 over chord 2 and chord 3 (a common-tone hold) becomes more interesting with C5 → D5 → C5 (upper neighbor) within the chord-zone of chord 2, before the C5 in chord 3.

5. Pay special attention to the OTH leading-interval-pair. When the added voice is moving by single semitone toward a chord tone of the next chord — and the cantus firmus’s moving voice is doing the same — this is a leading-interval-pair: two voices each moving by one semitone in coordinated motion. This is OTH’s strongest cadential gesture. In second species, exploit it at the cadence.

6. End the cadence on a chord tone of the final chord, approached by step. Second species cadences are smoother than first species because the added voice has had two notes per chord-zone leading up to the cadence.

Worked example

Cantus firmus: same five-chord progression (Q777 → Q877 → Q787 → Q877 → Q777).

First-species added voice: D5 → D5 → Eb5 → D5 → D5 (the line from §1.1).

Second-species elaboration:

Revised line, beat by beat: D5–D5 / D5–Eb5 / Eb5–D5 / D5–D5 / D5–D5.

Check.

• Every “beat 1” note is a chord tone. ✓
• The “beat 2” notes: chord 2 beat 2 = Eb5, which is not a chord tone of Q877 = {C,D,G,Bb}. But Eb5 is a chord tone of the next chord (Q787 = {C,Eb,G,Bb}). So Eb5 functions as an anticipation — it sounds the next chord’s tone before the chord arrives. This is a legitimate non-chord-tone usage. (In Fuxian terms: anticipation. In OTH terms: leading-tone-pair with the cantus firmus’s D→Eb edge, shifted earlier.)
• The chord 3 beat 2 = D5, which is a chord tone of the next chord (Q877 = {C,D,G,Bb}). This is also an anticipation.

So the second-species line uses anticipations to create melodic motion; the line is now: D5 D5 D5 Eb5 Eb5 D5 D5 D5 D5 D5. More interesting than the first-species version.

Common errors in second species

1. Unprepared non-chord tones. A non-chord tone that doesn’t resolve by step (or doesn’t anticipate a chord tone of the next chord) is not legal in second species. Example: D5 → F5 → Eb5 (where F5 is neither a chord tone nor a stepwise approach to anything). The F5 is an unprepared dissonance.

2. Too much motion. Second species is restrained — one note added per chord-zone. Adding three or four notes pushes you into third species.

3. Forgetting the contrary-motion cadence. A second-species cadence should still resolve in oblique or contrary motion. If your beat-2 of the penultimate chord moves in parallel with the cantus firmus, you’ve lost the contrary-motion feel.

4. Passing-tone collisions. Inserting a passing tone that is a chord tone of the current chord is fine; inserting one that creates a tritone with another voice is risky. Check for tritones and either resolve them properly or pick a different passing tone.

3. Four Notes per Chord-Zone

1.3 Third Species — Four Notes per Chord-Zone

Definition. Third species places four notes per chord-zone in the added voice. This is florid melodic motion within a slow-moving harmonic frame. Passing tones, neighbor tones, anticipations, and arpeggiations are all available.

What this teaches. Third species teaches motivic shaping. With four notes per chord-zone, the student starts to develop melodic figures — the small-scale rhythmic and pitch shapes that will later become the motivic cells of the piece. Third species also tests the student’s ability to control voice independence at fast rhythmic rates.

Setup. A chord-graph path of 4-6 chords. The added voice plays four notes per chord (or, in 6/8 time, six eighth notes per measure if each chord is one measure long). Most of these notes should be chord tones; non-chord tones are limited to passing/neighbor/anticipation/escape tones with stepwise treatment.

Step-by-step procedure

1. Begin with second-species line. Lay out a working second-species line. This becomes the structural skeleton.

2. Identify the underlying motion at each beat. What are the chord tones the line passes through? At third-species rhythm, you’ll have time to traverse multiple chord tones within a single chord-zone.

3. Develop motivic cells. Pick one or two short rhythmic figures and use them throughout. Two simple cells:

   • Stepwise cell: four eighth notes ascending or descending stepwise, e.g., C5 D5 Eb5 D5.
   • Turn cell: a melodic turn around a central pitch, e.g., D5 Eb5 D5 C5 (upper neighbor + lower neighbor).

4. Place cells at structurally appropriate moments. Use ascending cells at moments of rising tension (before a Saddle); descending cells at moments of resolution (after a cadence); turn cells over stable chord-zones (Summits, Plateaus).

5. Punctuate with rests where the line needs to breathe. A third-species line with continuous activity from start to finish lacks shape. Rest the line briefly between phrases.

6. Ensure non-chord tones resolve. With four notes per chord-zone, the student has more freedom but more responsibility. Every non-chord tone should resolve by step to a chord tone — either of the current chord or the next.

7. Approach the cadence with rhythmic acceleration. Speed up the surface rhythm as the cadence approaches. Hold the cadential note longer than the surrounding rhythm.

Worked example

Same cantus firmus. Develop a third-species added voice in 6/8 time with six eighth notes per measure.

The cadence held note A4 at chord 5 is approached by Bb4 → A4 (the leading-interval pair). Voice 4 of the cantus firmus does the same Bb→A; the added voice mirrors it at the appropriate octave. This is parallel motion at the cadence — but the parallelism is intentional: both voices descending by single semitone is the OTH cadential gesture, which is what we want.

Common errors in third species

1. Continuous activity without shape. Six eighths per measure for five measures = 30 notes. If they’re all the same rhythmic value, the line sounds metronomic. Vary by introducing held notes, rests, or larger rhythmic values.

2. Cell repetition without variation. Using the same rhythmic cell six times without modification creates monotony. Vary at least one element per recurrence: pitch, direction, or rhythm.

3. Unresolved non-chord tones. With four notes per chord-zone, it’s easy to slip in a non-chord tone and forget to resolve it. Check every beat.

4. Voice-crossing under high rhythmic activity. When the line is moving fast, it’s easier to cross under (or over) another voice. Watch the register carefully.

4. Suspension and Fiber Color

1.4 Fourth Species — Suspension and Fiber Color

Definition. Fourth species introduces two new resources:

• Suspensions: a note from the previous chord-zone is held into the new chord-zone, becoming a non-chord tone, then resolved by step to a chord tone.
• Fiber inversions: motion through the fiber bundle (changing inversional state without changing the orbit).

What this teaches. Fourth species teaches rhythmic displacement and registral elaboration. The suspension is the cleanest expression of the OTH leading-interval-pair: a held note becomes dissonant against the new chord, then resolves by single semitone to a chord tone. The fiber inversion teaches the student to use registral motion as a temporal event, not just a static voicing choice.

Setup. A chord-graph path of 4-8 chords, with the added voice playing one note per chord-zone but with the note tied (or repeated) across the bar line into the next chord-zone, where it becomes the suspension. Alternatively, fiber inversions are inserted between consecutive base-space chords.

Step-by-step procedure for suspensions

1. Identify candidate suspension moments. Look at each chord-graph edge in the cantus firmus. The voice that moves in the cantus firmus is changing by one semitone. If the added voice can be held on the old note of that voice into the new chord, the held note becomes a suspension (a non-chord tone in the new chord) and resolves by single semitone to the new note.

2. Write the suspension. The added voice plays the chord tone on the “downbeat” of chord N. The voice holds (ties) into chord N+1. On the second beat of chord N+1, the voice resolves by single semitone to a chord tone of chord N+1.

3. Resolve down (or up) by step. OTH suspensions can resolve in either direction. The choice depends on whether the resolving voice is following a tritone-expanding (d5 → P5) or A5-contracting (A5 → P5) motion.

4. Stack suspensions for stretto effects. Multiple voices can have suspensions in different chord-zones, creating overlapping resolutions.

Step-by-step procedure for fiber inversions

1. Identify a chord-zone in the cantus firmus. Pick a chord that you want to elaborate registrally without changing its harmonic identity.

2. Compute the chord’s fiber positions. Each chord in B has 4 inversional positions in the fiber bundle E. Use the music-theory MCP to find them, or compute by Tymoczko inversion. The four positions are spaced 12 semitones apart (the L1 law).

3. Choose a fiber position to move to. Typically, move “up” in the fiber by one inversion step — e.g., from voicing [C2-G2-D3-A3] to [G2-D3-A3-C4] (rotating the lowest voice up an octave to become the highest).

4. Notate the fiber inversion. Within the chord-zone, write a transition from the base voicing to the fiber-inverted voicing. The pitch material of the chord is unchanged; only the registral distribution changes.

5. Use fiber inversions for color, not function. A fiber inversion does not progress the harmonic narrative — it elaborates the current chord’s registral character. Use sparingly; the listener interprets fiber inversion as a texture event, not a harmonic event.

Worked example — Suspension

Cantus firmus: chord 1 (Q777, voice 4 = A3) → chord 2 (Q877, voice 4 = Bb3). Single-edge transition: A→Bb in voice 4.

Added voice (soprano, in fourth species):

Chord 1: A4 (chord tone of Q777)
Chord 2: A4 sustained (held over from chord 1, becomes non-chord tone) → Bb4 (resolution by single semitone)

The A4 across the chord boundary is the suspension: it was a chord tone of chord 1 but is non-chord-tone in chord 2 (Q877 = {C,D,G,Bb}, no A). It resolves by ascending semitone to Bb4 (a chord tone of chord 2). The resolution mirrors the cantus firmus’s own A→Bb edge — this is the leading-interval-pair in its purest form: two voices each moving A → Bb across the chord boundary, in different octaves.

Common errors in fourth species

1. Unprepared suspensions. A suspension must be a chord tone of the previous chord. If you start a held note that wasn’t a chord tone before, it’s not a suspension — it’s just a held non-chord tone.

2. Resolving in the wrong direction. Suspensions in OTH should resolve by single semitone to a chord tone of the new chord. If you resolve by a leap, you’ve abandoned the suspension figure.

3. Fiber inversion treated as harmonic event. The fiber operates on color, not function. Using a fiber inversion to “modulate” or “resolve” misuses the device. It is a textural variation, not a harmonic move.

4. Stacked suspensions creating cluster dissonance. When two voices both have suspensions over the same chord, the held non-chord tones can stack into dissonant clusters. Plan resolutions in different beats so each suspension can be heard distinctly.

5. Florid

1.5 Fifth Species — Florid

Definition. Fifth species combines all preceding species: any motion type, any rhythmic value, any number of notes per chord-zone, suspensions, fiber inversions, and the OTH-native devices (multiset-shadow, wedge, real canon, etc.) are available.

What this teaches. Fifth species teaches compositional judgment. With all resources available, the student must decide what to use when. Mastery is the ability to choose the right device for the right moment in the chord-graph trajectory.

Setup. A chord-graph path of any length, with any number of voices. The student composes freely, subject to the cumulative constraints of all preceding species.

Step-by-step procedure

There is no single procedure for fifth species. Instead, a series of compositional decisions:

1. Plan the chord-graph trajectory. Where does the path go? Where are the moments of greatest tension (Saddle, Narrows)? Where are the cadences (Summit arrivals)?

2. Plan the textural arc. When do voices enter? When do they rest? Does the texture thicken toward a structural arrival?

3. Plan the motivic identity. What rhythmic and pitch cells will recur? Where are they introduced, developed, recapitulated?

4. Plan the device usage. Which OTH-native devices are appropriate at each moment?

   • Multiset-shadow at Plateau zones where the orbit shares a multiset with a relevant Summit
   • Leading-interval-pair at every cadence (Saddle → Slope → Summit)
   • Wedge convergence at structural arrivals (especially Summit arrivals after a long traversal)
   • Real canon in fiber zones where harmonic motion pauses
   • Fiber-color voice for textural support in fiber zones
   • Wing-color contrast for mode-mixture-like effects

5. Write voice by voice, layer by layer. Start with the cantus firmus (chord-graph path). Add an upper voice. Then a second upper voice. Each new voice should have its own motivic identity but rhyme with what’s already there.

6. Listen back and revise. The most important step. Play what you’ve written. Does each phrase land? Does the cadence resolve? Are voices independent?

Common errors in fifth species

1. Device overuse. Using every device available in every measure makes the piece feel “academic” rather than “musical.” Reserve devices for moments where they earn their keep.

2. Loss of motivic coherence. With everything available, it’s easy to write material that doesn’t connect to anything else in the piece. Every new voice and every new figure should rhyme with something that’s already been heard.

3. Cadence neglect. A florid piece without clear cadences sounds aimless. Every major formal section needs a clear arrival point.

4. Voice texture imbalance. Three voices all playing eighth notes at once creates rhythmic mush. Vary the activity rates across voices: when one voice is florid, others should be sustained.

5. Skipping the planning phase. Fifth species is the most rewarding species but also the most demanding. Writing without plans 1-4 above usually produces incoherent results.

Assessing Counterpoint

2.1 The six dimensions of assessment

When evaluating a piece of OTH counterpoint, work through six dimensions in sequence. Each tests a different aspect of the music’s geometric and contrapuntal coherence.

1. Chord-graph fidelity

2.1.1 Chord-graph fidelity

Question. Does every chord transition in the harmonic backbone follow a legal motion type — single edge, compound perturbation, window slide, or fiber inversion?

How to check.

1. List every chord in the harmonic backbone. Identify the orbit of each.
2. For every adjacent pair: count the common tones. If the common tones equal 3, the transition is a single edge (legal). If 2, it’s a compound perturbation (legal but more complex). If 0-1, the transition needs justification — either a window slide (entire chord transposed) or a fiber inversion (orbit unchanged but inversional state different) or it’s an illegal jump.
3. Flag any transition that does not fit one of these motion types.

Common findings. Most beginning OTH composers stick to single edges, which is fine. Compound perturbations should be reserved for moments that warrant the extra motion (textural climaxes, structural arrivals).

2. Voice independence

2.1.2 Voice independence

Question. Do the voices move independently — different rhythms, different pitch contours, different motion types — or do they fuse into homophonic block-chord motion?

How to check.

1. Look at three or more consecutive chord transitions. Are the voices all moving in the same direction? (Bad: parallel block motion.)
2. Look at the rhythmic profiles. Are they identical across voices in many places? (Bad: homophony where polyphony was intended.)
3. Look for orbit-persistence. Are three or more consecutive chord-graph edges all within the same orbit? (Bad: voice fusion likely.)

Common findings. A common failure is “parallel-perfect-orbits” — extended sequences within the same orbit, the OTH analog of parallel fifths. Watch for it.

3. Cadential integrity

2.1.3 Cadential integrity

Question. Does the cadence resolve properly? Is the leading-interval-pair employed at the cadential moment?

How to check.

1. Identify the cadence point (the structural arrival, usually a Summit or a Plateau Q787 chord).
2. Check the chord before the cadence. Is it a Saddle, a Slope, or another tension chord?
3. Check the voice motion at the cadence. Are two voices moving by single semitone in coordinated motion (the leading-interval-pair)?
4. Are non-chord tones at the cadence resolved by step?

Common findings. Cadences that “miss” usually do so because the penultimate chord was not a tension chord (no demand for resolution) or because the leading-interval-pair was not coordinated (the two voices moved at different times rather than together).

4. Motivic coherence

2.1.4 Motivic coherence

Question. Does the piece have one or more recurring motivic cells, and are they developed coherently?

How to check.

1. Identify the cells. Look for short rhythmic-pitch units that appear more than once.
2. Track variations. Each subsequent cell statement should differ in at least one element (transposition, inversion, retrograde, augmentation, diminution, intervallic adjustment to a new chord context).
3. Look for cell distribution across voices. A motif heard only in voice 1 throughout is a missed opportunity for canonic or imitative treatment.

Common findings. Motivic coherence is the dimension most often neglected in OTH writing because the harmonic system is so rich it can carry the ear without strong motivic work. But a piece without motivic coherence quickly loses memorability.

5. Wing-color identity

2.1.5 Wing-color identity

Question. Does the piece have a coherent wing-color (d5-flavored or A5-flavored), or is mode-mixture used intentionally?

How to check.

1. Catalog the orbits used. Categorize them as d5-wing (Q676, Q686, Q767), A5-wing (Q878, Q788, Q688, Q868), or core (Summits, Plateaus).
2. Compute the relative weight (number of measures) spent in each wing.
3. Is the distribution skewed toward one wing? If so, the piece has a wing-color identity.
4. If it crosses freely between wings, is each crossing a structural moment (mode mixture, modulation), or is it incidental drift?

Common findings. Beginning OTH composers often drift unconsciously between wings, missing the opportunity to use wing-color contrast as an expressive resource.

6. Shadow-note treatment

2.1.6 Shadow-note treatment

Question. Are non-chord tones (shadow notes, suspensions, anticipations) prepared and resolved properly?

How to check.

1. Identify every non-chord tone in every voice. (A non-chord tone is a pitch not in the orbit’s pitch-class set at that moment.)
2. For each non-chord tone, determine its function: passing tone, neighbor, suspension, anticipation, escape tone, multiset-shadow note?
3. Verify proper treatment:
   • Passing tone: approached and left by step in the same direction.
   • Neighbor: approached and left by step in opposite directions, returning to the same chord tone.
   • Suspension: prepared (was chord tone in previous chord), then held into new chord, then resolved by step.
   • Anticipation: a chord tone of the next chord sounded before the chord arrives.
   • Multiset-shadow: a non-chord tone that belongs to the multiset of a sister orbit (e.g., A4 over Q877, where A is in the related Q777). These are not “errors” — they are deliberate shadows. But they should be heard as such.

Common findings. Beginners sometimes use non-chord tones without resolving them, treating them as freely available pitches. This is a violation in second-fourth species but acceptable in fifth species if the non-chord tone is doing real work (e.g., establishing a wing-color or producing a multiset-shadow).

Common Errors

2.2 Catalog of common errors

These are the most common errors in OTH counterpoint, in rough order of frequency:

1. Orbit persistence (parallel-perfect-orbits): three or more consecutive chord-graph edges within the same orbit. Fix: introduce an orbit change.

2. Cadence miss: the penultimate chord is not a tension chord, or the leading-interval-pair is not coordinated. Fix: approach the Summit through a Saddle or Slope; coordinate the two semitone resolutions to occur in the same beat or adjacent beats.

3. Unresolved non-chord tones: a passing/neighbor tone that doesn’t resolve by step. Fix: either resolve it stepwise or relabel it as a chord tone (perhaps you’ve miscounted the orbit).

4. Voice crossing: an upper voice descends below an inner voice or vice versa. Fix: re-voice; or adjust the offending pitch by an octave.

5. Static voice line: a voice that holds the same pitch for many chord-zones without melodic motion. Fix: introduce a passing tone, neighbor figure, or melodic step.

6. Rhythmic homophony: all voices moving in identical rhythm. Fix: offset entries; vary durations across voices.

7. Wing-color drift: the piece wanders between wings without intent. Fix: either commit to one wing for sustained passages, or make crossings deliberate (perhaps coinciding with structural arrivals).

8. Motivic isolation: one voice has a clear motif; other voices have nothing comparable. Fix: introduce variations of the motif in other voices, even if displaced or transposed.

9. Fiber abuse: using fiber inversions as if they were harmonic moves. Fix: recognize the fiber as color, not function. Reserve fiber inversions for textural elaboration of a single chord-zone, not for narrative motion.

10. Cadential rushing: the cadence arrives too soon, without proper preparation. Fix: prolong the penultimate chord (or its preparation) so the listener anticipates the resolution.

Worked Example

2.3 A worked critique

Consider a hypothetical short OTH counterpoint exercise:

Chord 1: C2-G2-D3-A3   (Q777 Summit)        | Soprano: C5 D5 E5 D5
Chord 2: C2-G2-D3-Bb3  (Q877 Plateau)       | Soprano: D5 E5 F5 E5
Chord 3: C2-G2-Eb3-Bb3 (Q787 Plateau)       | Soprano: Eb5 F5 G5 F5
Chord 4: C2-G2-D3-Bb3  (Q877 Plateau)       | Soprano: F5 E5 D5 C5
Chord 5: C2-G2-D3-A3   (Q777 Summit)        | Soprano: D5 (held)

Walk through the six dimensions:

1. Chord-graph fidelity. Q777 → Q877 → Q787 → Q877 → Q777. Common tones at each transition: 3, 3, 3, 3. All single edges. ✓

2. Voice independence. The soprano line moves at four notes per chord-zone (third species pace). The bass moves at one chord per measure. Different rhythmic activity rates. ✓ But the soprano’s pattern is identical in chords 1, 2, 3 — a four-note ascending shape (X X+1 X+2 X+1, where +1 is a step up). Looking voice-by-voice across the chords: soprano starts C5 → D5 → Eb5 → F5 → D5 — a stepwise ascent followed by a descent. The bass voice 4 moves A3 → Bb3 → Bb3 → Bb3 → A3 — A neighbor figure. Both voices have similar shape (ascent, then descent). This is similar motion at the phrase level — not a violation, but worth noting.

3. Cadential integrity. Chord 4 → chord 5: Q877 → Q777, single edge (Bb→A in voice 4). The soprano descends from C5 (chord 4 beat 4) to D5 (chord 5 sustained). That’s a step up (C5 to D5 is +2 semitones). At the cadence, the cantus firmus voice 4 is descending Bb→A; the soprano is ascending C5→D5. Contrary motion at the cadence. ✓ But neither voice is moving by single semitone; the soprano is moving by whole step. So we don’t have a clean leading-interval-pair. Partial credit.

4. Motivic coherence. The soprano line uses a “X X+1 X+2 X+1” shape three times (chords 1, 2, 3) — clear motivic identity. Chord 4 uses a descending variant of the same cell (F5 E5 D5 C5 — a four-note descending stepwise figure, the retrograde of the ascending cell). Chord 5 lands on a held D5. Excellent motivic coherence — one cell, used in ascending form three times, then retrograde once, then resolved. ✓

5. Wing-color identity. All five chords are in the core (Summits and Plateaus); no d5-wing or A5-wing chords involved. The piece has a “core” identity — neutral in wing-color terms. This is fine for a short exercise but limits expressive range.

6. Shadow-note treatment. The soprano line has E5 (chord 1 beat 3, chord 2 beat 2, chord 2 beat 4) — E is not a chord tone of Q777 ({C,D,G,A}) or Q877 ({C,D,G,Bb}). Is E5 a passing tone (approached/left by step in same direction)? In chord 1: D5 → E5 → D5 — that’s an upper neighbor, not a passing tone. Approach by step up, leave by step down. Neighbor treatment. ✓ In chord 2: D5 → E5 → F5 → E5 — E5 is approached by step up, left by step up, then returned to. Mixed treatment — function as passing tone going up (D→E→F) and as neighbor going down (F→E→F). Both legal. ✓ Similarly F5 is non-chord tone in chord 1 and chord 3 (Q787 = {C,Eb,G,Bb}, no F), but appears in chord 2 (Q877 = {C,D,G,Bb}, no F either). F5 is non-chord-tone throughout but always approached and left by step. ✓

Summary of critique. The exercise is well-constructed: clean chord-graph path, motivically coherent soprano line, contrary motion at the cadence, properly treated non-chord tones. The weaknesses are: (a) the cadence does not employ the cleanest leading-interval-pair (would benefit from a Bb→A and an A→Bb in coordinated motion); (b) wing-color is neutral, limiting expressive scope. Overall: a good first-species/second-species exercise stretched into third species. Grade: solid B+.

Self-Assessment

2.4 Self-assessment checklist

Use this checklist when reviewing your own work:

• Chord-graph fidelity. Every transition is a legal motion type (single edge, compound perturbation, window slide, or fiber inversion).
• No orbit persistence. No three or more consecutive edges within the same orbit.
• Cadential preparation. Every structural cadence is approached through a tension chord (Saddle, Slope, or related).
• Leading-interval-pair at cadences. At least one cadence employs two voices each moving by single semitone in coordinated motion.
• Voice independence. Voices have different rhythmic profiles at multiple points.
• Motivic coherence. At least one short rhythmic-pitch cell appears more than once with variation.
• Non-chord tone treatment. Every non-chord tone has a clear function (passing, neighbor, suspension, anticipation, escape, or shadow) and is treated accordingly.
• Wing-color awareness. The piece either commits to a wing or makes wing crossings deliberate.
• Cadence landing. The final note of each phrase is on a chord tone of the cadential chord.
• Voice ranges respected. No unintended voice crossing.

If all ten boxes are checked, the piece passes the basic OTH counterpoint test. The next level — fifth species and beyond — concerns expressive judgment, motivic invention, formal architecture, and the use of OTH-native devices in service of musical narrative. That level is not assessable by checklist; it is assessable by listening.

Tools for Counterpoint

A.1 The 14 orbits

A.2 The five species — at a glance

A.3 The cadential formula

The voice motion across this two-edge cadence is two single-semitone descents (or ascents), each widening a tritone (d5→P5) or contracting an augmented fifth (A5→P5). When both motions occur simultaneously across the two-edge cadence, the listener perceives a unified resolution event — the OTH analog of the V→I tonal cadence.

A.4 The eight positive rules (from the working memo)

1. Path Continuity. Every transition between successive chords is a legal chord-graph motion type.
2. Cadential Direction. Cadences are approached through tension chords (Saddle, Slope, or related).
3. Tension Preparation. Saddle chords should be preceded by a chord at chord-graph distance 1 from them.
4. Resolution Pacing. From a Saddle, both tritones should resolve to perfect fifths within two chord-graph edges.
5. Avoid Orbit-Persistence. Three or more consecutive chord-graph edges within the same orbit produce voice fusion.
6. Wing Awareness. The piece should have a coherent wing-color identity, with wing crossings made deliberately.
7. Degree-Gradient Direction. Phrase-level motion should follow a meaningful trajectory through the degree landscape.
8. Fiber as Color, Not Function. Fiber inversions vary registral character without altering harmonic identity.

The Window Slide

Part IV — The Full Grammar

The Saddle

The Quintal Cadence

Color and Function

Modulation and the T₆ Axis

Extended Forms

Part V — The Deep Structure

The Mathematical Turn

The Stack

The Terrain

The Spiral

The Deep Structure

Conclusion

Art for Dessert

[Add note]

Appendix A — Complete Orbit Reference

Appendix B — Mathematical Foundations

Appendix C — Glossary

Foundational Terms

I. Foundational Terms

Open Tone Harmony (OTH)

A non-tertian harmonic system built from stacked fifths and fourths, governed by the [6,8] constraint: consecutive stacking intervals in four-voice chords must be drawn from {d5, P5, A5} = {6, 7, 8 semitones}. The name derives from the characteristic acoustic quality of quintal/quartal sonorities — their open, resonant, unresolved-yet-stable sound.

The [6,8] Constraint

The defining rule of the system. A four-note chord belongs to OTH if and only if it can be stacked (bottom to top) with consecutive intervals each equal to a diminished fifth (6), perfect fifth (7), or augmented fifth (8). This single constraint generates the entire 228-chord space.

Base Space (B)

The metric space of 228 four-note pitch-class sets satisfying the [6,8] constraint. B has diameter 8 (the maximum geodesic distance between any two chords) and eccentricity range 7–8 (54 central chords with eccentricity 7; 174 peripheral chords with eccentricity 8). B is the “ground” on which all of OTH is built.

Extended Space (E)

The full space of voiced (registral) chords obtained by applying Tymoczko’s interscalar transposition to each chord in B. E is a ℤ₄-fiber bundle over B: each chord in B has four inversional positions, of which typically 1 (Class A orbits) or 2 (Class B orbits) land back in [6,8].

Chord-Graph

The graph whose vertices are the 228 chords of B and whose edges connect chords differing by a single-semitone single-voice move (three voices held, one moved ±1). This is the lattice of legal atomic voice motion in OTH.

Edge

One step in the chord-graph: one voice moves by one semitone while three voices hold. The atomic unit of harmonic motion.

Geodesic Distance

The length of the shortest path between two chords in B, measured in edges. Ranges from 1 (neighbors) to 8 (antipodal chords, e.g., C–G–D–A to A♭–E♭–B♭–F).

Orbit

An equivalence class of chords under the combined action of transposition (T_n) and pitch-class inversion (I_n). There are 14 T/I orbits in B, each with a characteristic interval recipe. Orbits are the “chord types” of OTH — analogous to major triads, minor triads, etc. in tertian harmony.

Isometry Group

The group of distance-preserving transformations of B: ℤ₁₂ ⋊ ℤ₂ (the T/I group, 24 isometries). This is the same symmetry group that governs the triadic Tonnetz and standard pitch-class set theory.

Interval Recipe

The ordered triple of stacking intervals (up to reversal) that characterizes an orbit. For example, [7,7,7] for the Summit, [6,8,6] for the Saddle, [7,6,7] for a Valley. Non-palindromic recipes like [6,7,7] are equivalent to their reversal [7,7,6] under inversion.

Degree

The number of valid single-semitone neighbors a chord has in B. Ranges from 4 (minimum — Valleys, Narrows) to 8 (maximum — Summit, upper Plateau, Saddle). Degree is the primary measure of harmonic connectivity and stability.

The Topographic Landscape — Functional Regions

II. The Topographic Landscape — Functional Regions

All function names are terrain features — natural landforms, not human constructions.

Summit

Orbit: Q777 — [7,7,7] | Forte: 4-23 | Degree: 8 | Size: 12 chords Fiber class: A | Step sequence: [2,5,2,3] | Parent scale: Major Pentatonic

The all-perfect-fifth orbit. Maximum connectivity, maximum stability. The OTH tonic — the point of arrival and departure. The word evokes the apex of the degree landscape: a vantage from which one can move in many directions (unlike “tonic,” which implies gravitational pull toward). The 12 Summit chords define the 12 quintal keys of OTH.

Example: C–G–D–A (the “C stack”)

Plateau

Two orbits sharing the name, distinguished by degree:

Upper Plateau — Q787 — [7,8,7] | Forte: 4-26 | Degree: 8 | Size: 12 chords Palindromic. Shares degree 8 with the Summit. The minor seventh chord type (Cm7). The “other peak” — same elevation, different color. Parent scale: minor pentatonic rotation.

Plateau — Q877 — [7,7,8]/[8,7,7] | Forte: 4-22 | Degree: 6 | Size: 24 chords Non-palindromic. One step from pure quintal. Parent scale: major pentatonic.

Plateaus are stable enough to linger on but not the final destination. Moderate tension, multiple available paths. The OTH analogue of the subdominant region.

Slope

Four orbits at two degree levels, functioning as transitional terrain:

Upper Slope — Q786 — [7,8,6]/[6,8,7] | Forte: 4-27 | Degree: 6 | Size: 24 chords The dominant/half-diminished seventh chord type. Bridges the quintal core to the d5-containing chords. Parent scale: chromatic-pentatonic (E♭ variant).

Upper Slope — Q776 — [7,7,6]/[6,7,7] | Forte: 4-16 | Degree: 5 | Size: 24 chords The first orbit with a semitone step. Parent scale: chromatic-pentatonic (D♭ variant).

Lower Slope — Q876 — [8,7,6]/[6,7,8] | Forte: 4-18 | Degree: 5 | Size: 24 chords Maximally heterogeneous — one each of d5, P5, A5. Parent scale: chromatic-pentatonic (D♭ variant).

Lower Slope — Q867 — [8,6,7]/[7,6,8] | Forte: 4-Z15 | Degree: 4 | Size: 24 chords The all-interval tetrachord. Maximum interval diversity. Parent scale: All-Interval Heptatonic (7 notes, 7 modes — the richest scale in the system).

Slopes are the connector tissue of OTH — transitional chords mediating between the stable upper regions and the tense lower regions. The OTH analogue of the pre-dominant.

Valley

Three orbits sharing the name, all at degree 4:

Valley — Q767 — [7,6,7] | Forte: 4-8 | Degree: 4 | Size: 12 chords The “pinched” quintal chord — two d5-adjacent pairs. The most dissonant playable scale in the system (heptatonic chromatic-pentatonic with twin semitone clusters). d5-wing.

Valley — Q868 — [8,6,8] | Forte: 4-21 | Degree: 4 | Size: 12 chords The whole-tone tetrachord. Symmetric but constrained. Parent scale: whole-tone (shared with Saddle). Cross-wing bridge.

Valley — Q878 — [8,7,8] | Forte: 4-17 | Degree: 4 | Size: 12 chords Two A5 intervals flanking P5. Parent scale: chromatic-pentatonic (D♭ variant). A5-wing.

Valleys are low points in the degree landscape — maximum tension, minimum connectivity. Chords that demand resolution.

Saddle

Orbit: Q686 — [6,8,6] | Forte: 4-25 | Degree: 8 | Size: 6 chords (T₆ symmetric) Fiber class: B (2 inversions in [6,8]) | Step sequence: [2,4,2,4] (period-2) Parent scale: Whole-tone (6 notes, 1 mode)

Formerly “Crossroads.” Renamed to maintain the terrain-only naming principle — a saddle is a natural landform; a crossroads is human infrastructure.

The structural dominant of OTH. A saddle point in the mathematical sense: a critical point of the degree landscape that is simultaneously the low point along the ridge connecting two Summits and the high point of the pass between two lower regions. The saddle has maximum betweenness centrality (~9.1% of all shortest paths per chord, ~54.8% total for the 6-chord orbit) — more paths through the space flow through the Saddle than through any other orbit.

The terrain metaphor is precise: in mountain geography, a saddle (also called a col or pass) is the low point on a ridge between two peaks, and simultaneously the high point of the route between two valleys. It is the natural transit point — the place where paths converge, the pass through which travelers must go. Every geodesic between two Summit chords in different keys passes through a Saddle chord. This makes the Saddle the structural bottleneck of the space: it creates directed tension because, once you arrive at the Saddle, the paths forward are constrained (degree 8, but limited orbit-level exits), and the most natural exit is upward to a Summit — the OTH cadence.

The Saddle’s T₆ symmetry means each Saddle chord belongs to exactly two quintal keys simultaneously (it equals its own tritone transposition). This dual allegiance is the OTH mechanism for modulation: the Saddle is the pivot through which key changes flow.

The 6 Saddle chords are: {C,D,F♯,A♭}, {C,E,F♯,B♭}, {D♭,F,G,B}, {E♭,F,A,B}, {D♭,E♭,G,A}, {D,E,A♭,B♭}.

Tertian identity: French augmented sixth chord.

Precipice

Two orbits at the extreme edge of the degree landscape:

Precipice — Q688 — [6,8,8]/[8,8,6] | Forte: 4-24 | Degree: 6 | Size: 12 chords Fiber class B. Parent scale: whole-tone. The “edge of stability” — low-degree, A5-dominant.

Precipice — Q788 — [7,8,8]/[8,8,7] | Forte: 4-19 | Degree: 4 | Size: 24 chords The augmented-major seventh chord type (Cm(maj7)). Extreme chromatic tension. Parent scale: heptatonic chromatic-pentatonic.

Precipices are chords teetering at the edge — very few exits, strong directional tendency. The word conveys the sense of standing at a cliff face before a dramatic resolution.

Narrows

Orbit: Q676 — [6,7,6] | Forte: 4-9 | Degree: 4 | Size: 6 chords (T₆ symmetric) Fiber class: B (2 inversions in [6,8]) | Step sequence: [1,5,1,5] (period-2) Parent scale: Octatonic only

The most constrained chord in B — a geographic bottleneck with minimal options. Only two exit orbits at the orbit level (both to the Upper Slope Q776). The OTH analogue of the fully diminished seventh chord’s extreme restlessness, but even more constrained. The most extreme step distribution: two semitone clusters separated by perfect fourths.

The Narrows is the most isolated orbit scalarly — it has no pentatonic, diatonic, whole-tone, or blues parent. Only octatonic containment.

The Ridgeline

III. The Ridgeline

Ridge

Not a single orbit but a structural feature of the degree landscape: the connected path of degree-8 orbits that forms the “spine” of the terrain.

Three orbits share the maximum degree of 8: Summit (Q777), upper Plateau (Q787), and Saddle (Q686). These three orbit types form the high ground of B — the continuous elevated terrain from which all descents begin and to which all ascents return.

The ridge is the path along this high ground. In mountain geography, a ridgeline is a continuous line of high elevation connecting peaks, with the terrain falling away on both sides. In OTH:

• The Summit is the true peak (maximum stability, pentatonic parent, all-P5).
• The upper Plateau Q787 is the adjacent peak at the same elevation (degree 8) but with different character (one A5, minor-seventh flavor, minor-pentatonic parent).
• The Saddle is the low point on the ridge — still at degree 8, but functionally a pass rather than a peak (maximum betweenness, whole-tone parent, structural dominant).

The ridge separates the two wings of the space:

• To one side: the d5-wing (Q767, Q686, Q676 — orbits rich in tritones, descending from the ridge through darker, more dissonant territory)
• To the other side: the A5-wing (Q878, Q788, Q688, Q868 — orbits rich in augmented fifths, descending through brighter, more expansive territory)

Walking along the ridge (Summit ↔ Plateau Q787 ↔ Saddle) maintains maximum connectivity while shifting harmonic color. Walking off the ridge in either direction descends into one of the two wings. This is the fundamental topographic metaphor of OTH: the ridge is the high road; the wings are the slopes falling away on either side; the Saddle is the pass where the ridge dips and the two sides come closest together.

Note: The ridge is an emergent structural feature, not a formally defined term in the same sense as the orbit function names. It names a relationship between orbits rather than a single orbit. Whether “ridge” or “ridgeline” should be promoted to a primary structural term alongside the orbit function names is a compositional and pedagogical question — does calling attention to the connected high ground help composers navigate the space?

Wing Structure

IV. Wing Structure

Wing

A subset of orbits sharing a characteristic interval coloring. The OTH analogue of major/minor modality.

d5-Wing

Orbits rich in tritones (diminished fifths): Q676 (Narrows), Q686 (Saddle), Q767 (Valley). Connected through the hub orbit Q776 (Upper Slope) to the core. Produces a darker, more dissonant harmonic color. The tritone is the least stable interval in Hindemith’s hierarchy.

A5-Wing

Orbits rich in augmented fifths: Q878 (Valley), Q788 (Precipice), Q688 (Precipice), Q868 (Valley). Connected through the hub orbit Q877 (Plateau) to the core. Produces a brighter, more expansive harmonic color. The augmented fifth (= minor sixth) is a relatively consonant interval.

Wing Color

The distinction between d5-wing and A5-wing harmonic character within a progression or passage. The OTH analogue of mode. A passage committed to one wing has a consistent harmonic color; crossing between wings creates mode mixture.

Hub Orbit

An orbit with high orbit-level connectivity that bridges a wing to the core. Q776 (Upper Slope, 5 orbit-level connections) bridges the d5-wing; Q877 (Plateau, 6 orbit-level connections) bridges the A5-wing.

Fiber Bundle Structure

V. Fiber Bundle Structure

Fiber

The four inversional positions (voicings) of a single chord, obtained via Tymoczko’s interscalar transposition. Motion within the fiber changes registral distribution without changing pitch-class content — “color, not function.”

Coloration

A specific fiber position (inversion) of a chord in B. Following Persichetti: colorations vary the registral distribution of a chord without altering its harmonic identity.

Fiber Class

Class A: 1 of 4 inversions lands back in [6,8]. (11 orbits) Class B: 2 of 4 inversions land back in [6,8]. (3 orbits: Saddle Q686, Narrows Q676, Precipice Q688)

Class B membership correlates with period-2 step-sequence symmetry (Step Symmetry Conjecture — computationally verified, not yet analytically proved).

Universal L1 Law

Every inversion step costs exactly 12 semitones (L1 displacement) across all 14 orbits. The full four-step inversion cycle costs 72 semitones in the pattern [12, 12, 12, 36]. This universality justifies treating the fiber as metrically orthogonal to the base space.

Quartal/Quintal Duality Theorem

The quartal traversal (t₋₁) visits the same four voicings as the quintal traversal (t₁) in reverse order. Calling a chord “quartal” or “quintal” names the direction of fiber traversal — a ℤ₂ orientation symmetry on the ℤ₄ fiber.

Double Stability Theorem

The Saddle orbit [6,8,6] is the unique orbit satisfying both: (a) maximum degree 8 and maximum betweenness centrality in B, and (b) two inversions landing in [6,8] (maximum among all degree-8 orbits). No other orbit achieves both criteria simultaneously.

Octave-Transfer Incompatibility

The naïve cyclic inversion operator ρ (moving the bottom note up an octave) is compatible with [6,8] only for the all-tritone chord [6,6,6] — which is not in B. For all chords in B, ρ exits the [6,8] space. This is why Tymoczko’s interscalar transposition is the correct inversion mechanism for OTH.

Progression Motion

VI. Progression & Motion Terminology

Ascent

A degree-ascending progression (from lower to higher degree). The “strong” direction in OTH grammar. An ascent from Saddle to Summit is the OTH cadence.

Descent

A degree-descending progression (from higher to lower degree). The “weak” direction — departure from stability toward tension. A descent from Summit to Saddle sets up a cadence.

Traverse

A progression that stays within the same degree level, moving laterally through B. Traverses provide harmonic variety without changing tension level.

Window Slide

Transposition of an entire chord by semitone (or by any interval), moving between different key regions of B. The OTH analogue of modulation. Geodesic distance 4 in B. The 12 Summit chords connected by T₁ transposition form the window-slide cycle — the 12 keys of OTH.

OTH Cadence

The progression Saddle → Slope → Summit ([6,8,6] → [6,7,7] or [6,8,7] → [7,7,7]). The fundamental resolution formula. The voice motion involves two single-semitone steps, each widening a tritone (d5→P5) or contracting an augmented fifth (A5→P5).

OTH Departure

The progression Summit → Plateau → Slope → Saddle. The fundamental tension-building formula.

Leading-Interval-Pair

The two coordinated semitone resolutions at the cadential moment — both voices moving simultaneously to convert the Saddle’s d5 intervals into the Summit’s P5 intervals. The OTH analogue of the leading tone’s resolution.

Motion Categories

VII. Motion Categories

Fiber Inversion

Motion along the ℤ₄ fiber. Preserves the pitch-class set. Atomic in E. Perfect pitch-direction correlation: ±12 semitones per step.

Single-Voice Perturbation

One voice moves by one semitone in B; three voices hold. Atomic in B. The fundamental edge of the chord-graph.

Compound Perturbation

Multiple voices move by one semitone each in a single compositional gesture. Distance-2 (2 voices, 2 common tones) or distance-3 (3 voices, 1 common tone). Not atomic — decomposes into sequential single-voice perturbations.

Window Slide

All four voices move, transposing the entire chord. Distance 4 in B. Perfect pitch-direction correlation: ±28 semitones per step. The OTH analogue of key change.

Geodesic Traversal

Motion along a shortest path between two chords in B. Composite motion type — the path determines the sequence of passing chords. Up to 298 distinct geodesics may exist between antipodal chords.

Scale and Mode

VIII. Scale & Mode Terminology

Parent Scale

A traditional scale (pentatonic, diatonic, whole-tone, etc.) that contains an orbit’s pitch-class set as a subset. Parent-scale containment is a set-theoretic fact, not a claim about derivation.

Step Sequence

The ordered tuple of semitone gaps between consecutive pitch classes of a chord (within one octave, including wrap-around). Always sums to 12.

Mode

A cyclic rotation of the step sequence — starting on a different note of the chord. Each mode is a different “entry point” into the same pitch collection.

Step-Size Multiset

The step sequence sorted in ascending order. Rotation-invariant. Characterizes the orbit’s melodic vocabulary regardless of mode. Two orbits may share a multiset (multiset collision) while having different cyclic orderings.

Multiset Collision

Two orbits sharing the same step-size multiset but different cyclic orderings. Collisions: {2,2,3,5} = Summit Q777 and Plateau Q877; {2,3,3,4} = Plateau Q787 and Upper Slope Q786; {2,2,4,4} = Saddle Q686 and Precipice Q688; {1,3,3,5} = Valley Q878 and Lower Slope Q876. Used compositionally for “shadow” effects.

Scale Families

The eight parent scales of OTH, organized into four families:

Total: 8 distinct parent scales, 43 distinct modes.

Cadential Scalar Arc

The characteristic progression of scale families through a full OTH cadence: Pentatonic → Chromatic-Pentatonic → Whole-Tone → Chromatic-Pentatonic → Pentatonic

The whole-tone region at the center is the “eye of the storm” — structurally the highest tension (Saddle, structural dominant), but melodically the smoothest (no semitones, no direction). Character lives at home; the dominant is the void you pass through.

Structural Terms

IX. Structural Terminology

Stack

A chord in the Summit orbit — the quintal reference sonority. “The C stack” = the [7,7,7] chord rooted on C = {C, G, D, A}.

Axis Pair

Two Summit chords separated by a tritone (T₆), sharing the same Saddle chord(s). The OTH analogue of tritone-related keys. Example: the C stack {C,G,D,A} and the F♯ stack {F♯,C♯,G♯,D♯}.

Key Region

The set of chords in B gravitationally organized around a particular Summit chord. The 12 Summit chords define 12 key regions, analogous to the 12 major keys.

Degree Landscape

The assignment of degree values (4, 5, 6, or 8) to every chord in B, visualized as a terrain map. The landscape has peaks (degree 8), slopes (degree 5–6), and valleys (degree 4). The overall structure is a ridge of degree-8 chords with two wings descending on either side.

Maths

X. Advanced Mathematical Framework

ℤ₄-Voltage Graph

The Gross-Tucker voltage graph construction applied to OTH: the base graph is the chord-graph of B, with ℤ₄ voltage labels on edges encoding the fiber (inversion) shift at each step. The derived graph is the chord-graph of E. This is the exact discrete analogue of the continuous fiber bundle.

Metrized Holonomy Groupoid

The proposed unifying framework for all OTH motion types: a groupoid over the voltage graph whose morphisms encode all possible paths through E, equipped with the metric inherited from the L1 voice-leading norm. The holonomy of a path (the net fiber shift accumulated around a closed loop) captures the “geometric phase” phenomenon — progressions that return to the same base chord but in a different inversion.

Discrete Gauge Theory

The interpretation of the voltage graph as a discrete gauge field, where the ℤ₄ fiber plays the role of the gauge group and the voltage assignment is the discrete connection. Parallel transport around loops produces holonomy (fiber phase shift). This situates OTH within the framework of discrete differential geometry.

Ollivier-Ricci Curvature

A discrete analogue of Ricci curvature applicable to graphs. An open computational question for OTH: does the Ollivier-Ricci curvature of B correlate with the degree landscape and harmonic stability? Positive curvature at the Summit would indicate that geodesics converge there (a geometric formalization of tonic stability).

Counterpoint Terms

XI. Counterpoint Terminology

Species (OTH)

Five species of OTH counterpoint, adapted from the Fuxian tradition:

Shadow Note

A non-chord tone that belongs to the step-size multiset of a sister orbit (an orbit sharing the same multiset). For example, a pitch from Q877’s set sounded over a Q777 chord. Shadow notes are not errors — they are deliberate coloristic devices.

Orbit Persistence

Three or more consecutive chord-graph edges within the same orbit. The OTH analogue of parallel fifths — a voice-fusion risk to be managed deliberately.

Appendix D — Notation Conventions

References

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