In
diatonic set theory 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 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 classes on the
chromatic circle (
interval class), in other words the number of
half steps between
notes. 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,
seconds, 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 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 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 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. .
Further reading
*Clough, Engebretsen, and Kochavi. "Scales, Sets, and Interval Cycles": 78–84.
{{Set theory (music)
Diatonic set theory
Intervals (music)