Structure implies multiplicity

In diatonic set theory structure implies multiplicity is quality of a collection or scale for which the interval series formed by the distance around a diatonic circle of fifths between member 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. (Johnson 2003, p.68)

Structure implies multiplicity is true of the diatonic collection and the pentatonic scale, and any subset.

For example, cardinality equals variety dictates that a three member diatonic subset of the C major scale, C-D-E transposed to all scale degrees gives three interval patterns: M2-M2, M2-m2, m2-M2.

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.

Source

• Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1930190808.