In diatonic set theory structure implies multiplicity is quality of a collection or scale for which the interval series formed by the shortest 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. The property was first described by John Clough and Gerald Myerson in "Variety and Multiplicity in Diatonic Systems" (1985). (Johnson 2003, p.68, 151) Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of set theory to properties of the diatonic collection such as maximal evenness, Myhills property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. ... In music, a scale is an unordered collection of notes or pitches, as opposed to a series of intervals, which is a musical mode. ... In music theory, the circle of fifths is a model of pitch space and is the series encompassing all of the notes in the equally tempered chromatic scale. ... In music theory, an interval is the distance in pitch between two notes, the lower and higher members of the interval. ...

Structure implies multiplicity is true of the diatonic collection and the pentatonic scale, and any subset. In music, a pentatonic scale is a scale with five notes per octave. ...

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. In diatonic set theory cardinality equals variety is quality of a collection or scale for which the number of notes in a series indicates the number of unique interval patterns formed by diatonic transpositions. ... In music or music theory a scale degree is an individual note of a scale, both its pitch and its diatonic function. ...

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 and structure implies multiplicity are true of all collections with Myhill's property or maximal evenness. In diatonic set theory cardinality equals variety is quality of a collection or scale for which the number of notes in a series indicates the number of unique interval patterns formed by diatonic transpositions. ... In diatonic set theory Myhills property is the quality of scales or collections with exactly two specific intervals for every generic interval, and thus also have the properties of maximal evenness, cardinality equals variety, structure implies multiplicity, and be a well formed generated collection. ... In diatonic set theory maximal evenness is the quality of a collection or scale which for every generic interval there are is either one or two consecutive (adjacent) specific intervals, in other words a scale which is spread out as much as possible. ...

Further reading

• Clough, John and Myerson, Gerald (1985). "Variety and Multiplicity in Diatonic Systems", Journal of Music Theory 29: 249-70.
• Agmon, Eytan (1989). "A Mathematical Model of the Diatonic System", Journal of Music Theory 33: 1-25.
• Agmon, Eytan (1996). "Coherent Tone-Systems: A Study in the Theory of Diatonicism", Journal of Music Theory 40: 39-59.

Source

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