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 general, for a given scale S, the scalar transpositions of a line L can be grouped into categories, or transpositional set classes, whose members are related by chromatic transposition. In diatonic set theory cardinality equals variety when, for any melodic line L in a particular scale S, the number of these classes is equal to the number of distinct pitch classes in the line L.
For example, the melodic line C-D-E has three distinct pitch classes. When transposed diatonically to all scale degrees in the C major scale, we obtain three interval patterns: M2-M2, M2-m2, m2-M2.
Melodic lines in the C major scale with n distinct pitch classes always generate n distinct patterns.
The property was first described by John Clough and Gerald Myerson in "Variety and Multiplicity in Diatonic Systems" . Cardinality equals variety in the diatonic collection and the pentatonic scale, and, more generally, what Carey and Clampitt call "nondegenerate well-formed scales." "Nondegenerate well-formed scales" are those that possess Myhill's property.
