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.