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. Cardinality being the number of notes in the series, variety being the number of different interval patterns. 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, an interval is the distance in pitch between two notes, the lower and higher members of the interval. ...

Cardinality equals variety is true of the diatonic collection and the pentatonic scale, and any subset, but this property is absent from scales, such as whole tone scale, which consist of only one adjacent interval, since transposition of any number of notes by any interval of the whole tone scale produces the same interval pattern. In music, a pentatonic scale is a scale with five notes per octave. ... In music, a whole tone scale is a scale in which each note is separated from its neighbors by the interval of a whole step. ...

For example, 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 music or music theory a scale degree is an individual note of a scale, both its pitch and its diatonic function. ...

Cardinality equals variety and structure implies multiplicity are true of all collections with Myhill's property or maximal evenness. 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. ...

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.