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

a generic interval is the number of scale

steps Step(s) or STEP may refer to: Common meanings * Steps, making a staircase * Walking * Dance move * Military step, or march ** Marching Arts Films and television * ''Steps'' (TV series), Hong Kong * ''Step'' (film), US, 2017 Literature * ...

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 version ...

of a collection or

scale Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number ...

. 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 positi ...

es on the

chromatic circle The chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle. Explanation If one starts on any equal-tempered pitch and repeatedly ascends by ...

(

interval class In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'" (; ), is the shortest distance in pitc ...

), in other words the number of

half step A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

s 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 In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps ...

the generic interval is one less than the corresponding diatonic interval: * Adjacent intervals, seconds, are 1 *

Third Third or 3rd may refer to: Numbers * 3rd, the ordinal form of the cardinal number 3 * , a fraction of one third * 1⁄60 of a ''second'', or 1⁄3600 of a ''minute'' Places * 3rd Street (disambiguation) * Third Avenue (disambiguation) * Hi ...

s = 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 set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in t ...

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 most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a p ...

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