page_title: Cardinality equals variety
hierarchical_section_title: Cardinality equals variety
caption_attribution_description: English: Three note groups of the diatonic scale in the chromatic circle. Because the diatonic scale possesses a quality known as "cardinality equals variety" given a consecutive three note group, such as CDE, there are three classes of three note groups in the diatonic scale&#160;: MM/22 (red), Mm/21 (yellow), and mM/12 (blue).
context_page_description: 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".
context_section_description: 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" (1985) (Johnson 2003, p. 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.
caption: Three note sets from the diatonic scale in the chromatic circle: M2M2=red, M2m2=yellow, and m2M2=blue
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