Acoustics & Music Theory
why it sounds like that
Concept

Set class — where it appears

The family of pitch-class sets reachable from one another by transposition and inversion, which is what set theory treats as one object. Two sets in the same class share an interval content, and the census results here are counts over classes rather than sets.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two modes, one set, and every measure of a set that cannot tell them apart. Raga Bhupali and Raga Deshkar drawn as the moves each allows: an arrow from one degree to another means the tradition's ascent or descent goes that way. Raga Deshkar's ascent omits Re, so the two graphs differ by an edge. Below, every standard measure of a scale, evaluated on both — and they are identical in every row, which the drawing checks before it is made. An ascent and a descent that between them use all 5 degrees can be chosen in 63 ways, 62 of them asymmetric. That is how many modes collapse onto one pitch set under the simplest order model there is, and a census over subsets counts the set once.

A degree is where it goes next

The first essay on scales beyond twelve said a mode is not a set and listed four things a set cannot record. This one makes the sharpest of them arithmetic. Specify a mode as an ascent and a descent — which is how a living tradition specifies one — and it becomes a directed graph on its degrees; sixty-three such graphs collapse onto a single pentatonic set, sixty-two of them with an ascent that is not the descent reversed, and every standard measure of a scale returns the same value for all of them.

scales · Beyond twelve How many modes a set has, and why some have fewer than notes. Every non-empty subset of the twelve — 4095 of them — sorted by size, with how many have a transposition that returns the same set. 75 do, which is 1.8 per cent, and they reduce to 16 distinct step patterns. A set of size k with a symmetry of order s has exactly k/s distinct rotations, so the number of modes is arithmetic rather than musical. Sets of five, seven and eleven notes have none at all, because those sizes share no factor with twelve — which is why every seven-note scale has seven modes before any musical question is asked.

The set with fewer modes than notes

Eight earlier essays rotate one collection, and the seven modes were never a result — they are arithmetic. A set has as many modes as it has notes divided by the order of its own transposition symmetry, so the whole-tone scale has six notes and one mode, the octatonic has eight and two, and the question asked just before — how much has to be heard before the mode is settled — is not merely hard for those collections but undefined. Seventy-five of the twelve-note universe's 4,095 subsets are built this way and they reduce to sixteen step patterns. Five, seven and eleven notes cannot be among them, which is why every seven-note scale has seven modes before anybody plays one.

scales · Modes

Named alongside it

The objects these essays reach for when they reach for this one.

EnumerationArohaDiatonic scaleIntervalInterval contentMaximal evennessModeOctatonicRagaRotationScale degreeTransposition

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