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

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

.

See also

• Structure implies multiplicity

Sources

• Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing.
  ISBN 1-930190-80-8
  .

Further reading

• Clough, John and Myerson, Gerald (1985). "Variety and Multiplicity in Diatonic Systems", Journal of Music Theory 29: 249-70.
• Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", Music Theory Spectrum 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.