Editing Cardinality equals variety (section)

[[File:Cardinality equals variety diatonic scale thirds in the chromatic circle.png|thumb|Three note sets from the diatonic scale in the [[chromatic circle]]: M2M2=red, M2m2=yellow, and m2M2=blue]]

The musical operation of  [[Transposition (music)|scalar transposition]] shifts every note in a melody by the same number of scale steps.  The musical operation of [[Transposition (music)|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 theory (music)|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 degree]]s in the C major scale, we obtain three interval patterns: M2-M2, M2-m2, m2-M2.

[[Image:Cardinality equals variety CDE.PNG|400px|three member diatonic subset of the C major scale, C-D-E transposed to all scale degrees]]

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" (1985) (Johnson 2003, p.&nbsp;68, 151).  Cardinality equals variety in the [[diatonic collection]] and the [[pentatonic scale]], and, more generally, what Carey and Clampitt (1989) call "nondegenerate well-formed scales."  "Nondegenerate well-formed scales" are those that possess [[Myhill's property]].