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