scale (music) 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 ...

theory, a maximally even set (scale) is one in which every

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. (Johnson 2003, p. 26) A specific interval is the cl ...

has either one or two consecutive integers

specific 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. (Johnson 2003, p. 26) A specific interval is the clo ...

s-in other words a scale whose notes (pcs) are "spread out as much as possible." This property was first described by John Clough and Jack Douthett. Clough and Douthett also introduced the maximally even algorithm. For a chromatic cardinality ''c'' and pc-set cardinality ''d'' a maximally even set is

D =

where ''k'' ranges from 0 to ''d'' − 1 and ''m'', 0 ≤ ''m'' ≤ ''c'' − 1 is fixed and the bracket pair is the

floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...

. A discussion on these concepts can be found in Timothy Johnson's book on the mathematical foundations of diatonic scale theory. Jack Douthett and Richard Krantz introduced maximally even sets to the mathematics literature. A scale is said to have

Myhill's property In diatonic set theory a generic interval is the number of scale Step (music), steps between note (music), notes of a Set (music), collection or scale (music), scale. The largest generic interval (music), interval is one less than the number of sc ...

if every

comes in two

sizes, and a scale with Myhill's property is said to be a

well-formed scale 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 ...

. 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, ...

is both a well-formed scale and is maximally even. The

whole-tone scale In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or ''hexatonic'' sc ...

is also maximally even, but it is not well-formed since each generic interval comes in only one size. Second-order maximal evenness is maximal evenness of a subcollection of a larger collection that is maximally even. Diatonic triads and seventh chords possess second-order maximal evenness, being maximally even in regard to the maximally even diatonic scale—but are not maximally even with regard to the chromatic scale. (ibid, p.115) This nested quality resembles

Fred Lerdahl Alfred Whitford (Fred) Lerdahl (born March 10, 1943, in Madison, Wisconsin) is the Fritz Reiner Professor Emeritus of Musical Composition at Columbia University, and a composer and music theorist best known for his work on musical grammar and cogn ...

's "reductional format" for

pitch space In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apa ...

from the bottom up: ::(Lerdahl, 1992) In a

dynamical In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...

approach, spinning

concentric circles In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...

and iterated maximally even sets have been constructed. This approach has implications in

Neo-Riemannian theory Neo-Riemannian theory is a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harmonies directly t ...

, and leads to some interesting connections between

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

chromatic 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 pair, ...

theory. Emmanuel Amiot has discovered yet another way to define maximally even sets by employing

discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...

s. Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", Music Theory Spectrum 11: 187–206.

References

{{Set theory (music) Musical terminology