Cardinality equals variety
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In diatonic set theory cardinality equals variety is a quality of a collection or scale relating the number of notes in a series to the number of unique interval patterns formed by diatonic transpositions. Cardinality means the number of notes in the series, variety is 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)
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.
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.
Cardinality equals variety and structure implies multiplicity are true of all collections with Myhill's property or maximal evenness.
[edit] 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.
[edit] Source
- Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.
