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, the ...

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), ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian * ''Notes'', a common (yet unofficial) sho ...

of a

collection Collection or Collections may refer to: * Cash collection, the function of an accounts receivable department * Collection (church), money donated by the congregation during a church service * Collection agency, agency to collect cash * Collectio ...

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 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 t ...

(

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 pitch ...

), 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,

second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...

s, are 1 *

Third Third or 3rd may refer to: Numbers * 3rd, the ordinal form of the cardinal number 3 * , a fraction of one third * Second#Sexagesimal divisions of calendar time and day, 1⁄60 of a ''second'', or 1⁄3600 of a ''minute'' Places * 3rd Street (d ...

s = 2 * Fourths = 3 * Fifths = 4 * Sixths = 5 *

Seventh Seventh is the ordinal form of the number seven. Seventh may refer to: * Seventh Amendment to the United States Constitution * A fraction (mathematics), , equal to one of seven equal parts Film and television *"The Seventh", a second-season e ...

s = 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 the ...

s or collections with exactly two specific intervals for every generic interval, and thus also have the properties of

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. In genera ...

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 members of a series indicates the number of ...

, and being 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 ...

. 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 Scale (music), scales, and are also applied to musical instruments, Interval (music), intervals, Chord (music), chords, Musical note, notes, musical sty ...

and

pentatonic collection A pentatonic scale is a musical scale with five notes per octave, in contrast to the heptatonic scale, which has seven notes per octave (such as the major scale and minor scale). Pentatonic scales were developed independently by many ancien ...

s possess Myhill's property. The concept appears to have been first described by John Clough and

Gerald Myerson Gerald is a male Germanic given name meaning "rule of the spear" from the prefix ''ger-'' ("spear") and suffix ''-wald'' ("rule"). Variants include the English given name Jerrold, the feminine nickname Jeri and the Welsh language Gerallt and Irish ...

and named after their associate the mathematician

John Myhill John R. Myhill Sr. (11 August 1923 – 15 February 1987) was a British mathematician. Education Myhill 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 ...

. (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