Structure implies multiplicity

In

diatonic transpositions of that series. Structure refers to the intervals in relation to the circle of fifths; multiplicity refers to the number of times each different (adjacent) interval pattern occurs. The property was first described by John Clough and Gerald Myerson in "Variety and Multiplicity in Diatonic Systems" (1985). (Johnson 2003

, pp. 68, 151)

Structure implies multiplicity is true of the

diatonic collection and the pentatonic scale

, and any subset.

For example,

scale degrees

gives three interval patterns: M2-M2, M2-m2, m2-M2.

C-D-E on the circle of fifths

On the circle of fifths:

C G D A E B F (C)
  1 2 1 2 1 2  3

E and C are three notes apart, C and D are two notes apart, D and E two notes apart. Just as the distance around the circle of fifths between forms the interval pattern 3-2-2, M2-M2 occurs three times, M2-m2 occurs twice, and m2-M2 occurs twice.

Myhill's property or maximal evenness

.

References

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

v t e Musical set theory
All-interval tetrachord All-trichord hexachord Complement Forte number Identity Interval class Interval vector Multiplication Permutation Pitch class Pitch interval Pitch-interval class Set List Similarity relation Transformation Z-relation
Diatonic set theory	Bisector Cardinality equals variety Common tone (Deep scale property) Diatonic scale Generated collection Generic and specific intervals (Myhill's property) Maximal evenness Rothenberg propriety Structure implies multiplicity

References

Further reading