diatonic set theory Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the ...

structure implies multiplicity is a quality of a collection or scale. This is that for the interval series formed by the shortest distance around a diatonic

circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...

between members of a series indicates the number of unique interval patterns (adjacently, rather than around the circle of fifths) formed by diatonic transpositions of that series. Structure being the intervals in relation to the circle of fifths, multiplicity being the number of times each different (adjacent) interval pattern occurs. The property was first described by John Clough and

Gerald Myerson Gerald is a male Germanic given name meaning "rule of the spear" from the prefix ''ger-'' ("spear") and suffix ''-wald'' ("rule"). Variants include the English given name Jerrold, the feminine nickname Jeri and the Welsh language Gerallt and Irish ...

in "Variety and Multiplicity in Diatonic Systems" (1985). () Structure implies multiplicity is true of the

diatonic collection In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...

and the

pentatonic scale A pentatonic scale is a musical scale with five notes per octave, in contrast to the heptatonic scale, which has seven notes per octave (such as the major scale and minor scale). Pentatonic scales were developed independently by many ancien ...

, and any subset. For example,

cardinality equals variety The musical operation of scalar transposition shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition shifts every note in a melody by the same distance in pitch class space. In genera ...

dictates that a three member diatonic subset of the C major scale, C-D-E transposed to all

scale degree In music theory, the scale degree is the position of a particular note on a scale relative to the tonic, the first and main note of the scale from which each octave is assumed to begin. Degrees are useful for indicating the size of intervals and ...

s gives three interval patterns: M2-M2, M2-m2, m2-M2. Structure implies multiplicity circle of fifths

On the circle of fifths: C G D A E B F (C) 1 2 1 2 1 2 3 E and C are three notes apart, C and D are two notes apart, D and E two notes apart. Just as the distance around the circle of fifths between forms the interval pattern 3-2-2, M2-M2 occurs three times, M2-m2 occurs twice, and m2-M2 occurs twice.

Cardinality equals variety The musical operation of scalar transposition shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition shifts every note in a melody by the same distance in pitch class space. In genera ...

and structure implies multiplicity are true of all collections with

Myhill's property In diatonic set theory a generic interval is the number of scale Step (music), steps between note (music), notes of a Set (music), collection or scale (music), scale. The largest generic interval (music), interval is one less than the number of sc ...

maximal evenness In scale (music) theory, a maximally even set (scale) is one in which every generic interval has either one or two consecutive integers specific intervals-in other words a scale whose notes (pcs) are "spread out as much as possible." This proper ...

References

Further reading