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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.