In
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 t ...
theory, a
maximally even set (scale) is one in which every
generic interval 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 cl ...
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
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 ...
. 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 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 cloc ...
if every
generic interval comes in two
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 cl ...
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