Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of set theory to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. Musical set theory is a atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ... Musical analysis can be defined as a process attempting to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst. ... In music theory, a diatonic scale is a scale whose notes are built on the natural staff positions of lines and spaces, with no accidentals, with or without a key signature. ... In diatonic set theory maximal evenness is the quality of a collection or scale which for every generic interval there are is either one or two consecutive (adjacent) specific intervals, in other words a scale which is spread out as much as possible. ... In diatonic set theory Myhills property is the quality of 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 diatonic set theory a generated collection is a collection or scale formed by repeatedly adding a constant interval in integer notation, the generator, also known as an interval cycle, around the chromatic circle until a complete collection or scale is formed. ... In diatonic set theory the deep scale property is the quality of pitch class collections or scales containing each interval class a unique number of times. ... In diatonic set theory cardinality equals variety is quality of a collection or scale for which the number of notes in a series indicates the number of unique interval patterns formed by diatonic transpositions. ... In diatonic set theory structure implies multiplicity is quality of a collection or scale for which 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 (adjacently, rather than around the circle of fifths...

Music theorists working in diatonic set theory include Eytan Agmon, Gerald J. Balzano, Norman Carey, David Clampitt, John Clough, Jay Rahn, and mathematician Jack Douthett.

See also: Bisector, generic interval, and specific interval. For the numerical analysis algorithm, see bisection method. ... In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. ... In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle (interval class), in other words the number of half steps between notes. ...

Further reading

• Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1930190808.
• Carey, Norman and Clampitt, David (1996). "Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues", Perspectives of New Music 34, no. 2: 62-87.

Precursors

• Rahn, Jay (1977). "Some Recurrent Features of Scales", In Theory Only 2, no. 11-12: 43-52.