In diatonic set theory

Diatonic set theory

Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and...

 Myhill's property is the quality of musical scale

Musical scale

In music, a scale is a sequence of musical notes in ascending and descending order. Most commonly, especially in the context of the common practice period, the notes of a scale will belong to a single key, thus providing material for or being used to conveniently represent part or all of a musical...

s or collections with exactly two specific interval

Specific interval

In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between notes. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal...

s for every generic interval

Generic interval

In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members...

, and thus also have the properties of maximal evenness

Maximal evenness

In diatonic set theory maximal evenness is the quality of a collection or scale which for every generic interval there are either one or two consecutive specific intervals, in other words a scale which is "spread out as much as possible." This property was first described by music theorist John...

, cardinality equals variety

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...

, structure implies multiplicity

Structure implies multiplicity

In diatonic set theory structure implies multiplicity is a quality of a collection or scale. This is that for the interval series formed by the shortest distance around a diatonic circle of fifths between member of a series indicates the number of unique interval patterns formed by diatonic...

, 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

Diatonic scale

In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...

 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

John Myhill

John R. Myhill was a mathematician, born in 1923. He received his Ph.D. from Harvard University under Willard Van Orman Quine in 1949. He was professor at SUNY Buffalo from 1966 until his death in 1987...

. (Johnson 2003, p.106, 158)

Source

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