Pentatonic family: the four ancestral pentatonic shapes recognised throughout the legacy theory (major, Okinawan, Iwato, Kumoi).

Chroma (pitch-class set) for a pentatonic family, rooted at 0.

A pentatonic scale is a family + a root pitch class.

Concrete chroma of a rooted pentatonic.

Reverse lookup: given a 5-PC set, identify its family+root if any.

28-mode taxonomy ported from the legacy toMode function.

A mode is a quality rooted at a pitch class.

Concrete chroma of a rooted mode.

Identify a 7-PC set as a mode by trying each element as root and pattern-matching against the 28 templates.

Pinned-root classifier: identify a 7-PC set as a mode rooted on the given pitch class, exhaustively checking the 28 mode quality patterns. Mirrors legacy toMode semantics (root explicit, not inferred). Returns Nothing when the set isn't 7 unique PCs or when no quality matches the shifted interval pattern.

Each of the 28 ModeQuality constructors belongs to exactly one of four parent scale families.

Map a ModeQuality to its parent scale family.

Semitone offset of each mode's tonic from its parent scale's root. E.g. Aeolian sits on the 6th degree of major → 9 semitones above the parent root → modeDegree Aeolian = 9.

Parent (root pitch class, family) for any mode. E.g. parentKey (Mode Aeolian (P 1)) == (P 4, Major) — C# Aeolian is a mode of E Major.

Render a scale family as a human-readable string.

Legacy toMode mode-quality strings from theHarmonicAlgorithmLegacy/src/MusicData.hs. Preferred over the derived Show (which yields compact constructor names like IonS5) for musician-facing diagnostic output.

Result of a triad-anchored mode classification. ModeOk carries a normally classified mode; ModeInvalid carries the offending pair-union pitch classes when the union doesn't have exactly 7 unique PCs (only reachable under absStrata overrides that violate tristrata adjacency).

The eleven canonical strata labels (Roman I–XI).

Chroma (E-minor base) of each strata.

Prime-form dissonance of each strata (from the octatripentatonic spec).

All strata labels in order.

A tristrata is three strata whose pair-unions are diatonic 7-note sets and whose three-way union is the canonical 8-note set [1,2,4,6,7,9,10,11].

Twelve canonical tristrata (records 1..12 of the octatripentatonic corpus).

Index into validTristrata using a 1-based ordinal.

Project a tristrata to its strata at position 1, 2, or 3.

Sum of prime-form dissonances of the tristrata's three strata.

All (tristrata, position) pairs that contain a given strata.

Classify the three pair-unions of a tristrata as diatonic modes. Returns (mode(ts1∪ts2), mode(ts1∪ts3), mode(ts2∪ts3)).

Parse a tristrata allow-list string to a list of 1-based indices. "" → empty (meaning "all allowed"); "5" → [5]; "1 2 5" → [1,2,5].

Parse a per-bar position sequence (elements ∈ {1,2,3}).

Parse a per-bar absolute strata sequence. Accepts "I V X" or "[I V X]".