diatonic set theory Diatonic set theory is a subdivision or application of set theory (music), musical set theory which applies the techniques and musical analysis, analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myh ...

a generic interval is the number of scale steps between

notes Note, notes, or NOTE may refer to: Music and entertainment * Musical note, a pitched sound (or a symbol for a sound) in music * ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian * ''Notes'', a common (yet unofficial) shortened versi ...

of a collection or scale. The largest generic interval is one less than the number of scale members. (Johnson 2003, p. 26) A specific interval is the clockwise distance between

pitch class In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...

es on the chromatic circle ( interval class), in other words the number of half steps between

. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal temperament the largest specific interval is 11. (Johnson 2003, p. 26) In the diatonic collection the generic interval is one less than the corresponding diatonic interval: * Adjacent intervals,

second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...

s, are 1 * Thirds = 2 * Fourths = 3 * Fifths = 4 * Sixths = 5 * Sevenths = 6 The largest generic interval in the diatonic scale being 7 − 1 = 6.

Myhill's property

Myhill's property is the quality of

musical scale In music theory, a scale is "any consecutive series of notes that form a progression between one note and its octave", typically by order of pitch or fundamental frequency. The word "scale" originates from the Latin ''scala'', which literal ...

s or collections with exactly two specific intervals for every generic interval, and thus also have the properties of cardinality equals variety, structure implies multiplicity, and being 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: The

diatonic Diatonic and chromatic are terms in music theory that are used to characterize scales. The terms are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair ...

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)

Sources

* Johnson, Timothy (2003). ''Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals''. Key College Publishing. .

Myhill's property

Sources

Further reading