Myhill's property

From Wikipedia, the free encyclopedia
Jump to: navigation, search
The major scale is maximally even. For example, for every generic interval of a second there are only two possible specific intervals: 1 semitone (a minor second) or 2 semitones (a major second).

In diatonic set theory Myhill's property is the quality of musical scales or collections with exactly two specific intervals for every generic interval, and thus also have the properties of maximal evenness, cardinality equals variety, structure implies multiplicity, and be a well formed generated collection. In other words, each generic interval can be made from one of two possible different specific intervals. For example, there are major or minor and perfect or augmented/diminished variants of all the diatonic intervals:

Diatonic interval Generic interval Diatonic intervals Specific intervals
2nd 1 m2 and M2 1 and 2
3rd 2 m3 and M3 3 and 4
4th 3 P4 and A4 5 and 6
5th 4 d5 and P5 6 and 7
6th 5 m6 and M6 8 and 9
7th 6 m7 and M7 10 and 11

The diatonic and pentatonic collections possess Myhill's property. The concept appears to have been first described by John Clough and Gerald Myerson and named after their associate the mathematician John Myhill. (Johnson 2003, p.106, 158)

Further reading[edit]

  • Clough, Engebretsen, and Kochavi. "Scales, Sets, and Interval Cycles": 78-84.

Source[edit]

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