Editing Generic and specific intervals

[[Image:Maximal evenness seconds.png|thumb|The [[major scale]] is [[maximal evenness|maximally even]]. For example, for every generic interval of a second there are only two possible specific intervals: 1 semitone (a minor second) or 2 semitones (a major second).]]

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 scale members. (Johnson 2003, p.&nbsp;26)

A '''specific interval''' is the clockwise distance between [[pitch class]]es on the [[chromatic circle]] ([[interval class]]), in other words the number of [[half step]]s between [[note (music)|notes]]. The largest specific [[interval (music)|interval]] is one less than the number of "chromatic" pitches. In twelve tone equal temperament the largest specific interval is 11. (Johnson 2003, p.&nbsp;26)

In the [[diatonic collection]] the generic interval is one less than the corresponding diatonic interval:
* Adjacent intervals, [[Major second|second]]s, are 1
* [[Major third|Third]]s = 2
* [[Perfect fourth|Fourth]]s = 3
* [[Perfect fifth|Fifth]]s = 4
* [[Major sixth|Sixth]]s = 5
* [[Major seventh|Seventh]]s = 6
The largest generic interval in the diatonic scale being 7 − 1 = 6.

==Myhill's property<!--'Myhill's property' redirects here-->==
'''Myhill's property'''<!--boldface per WP:R#PLA--> is the quality of [[musical scale]]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]], 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:

{| class="wikitable"
! Diatonic<br/> interval
! Generic<br/> interval
! Diatonic<br/> intervals
! Specific<br/> intervals
|-
|2nd
|1
|m2 and M2
|1 and 2
|-
|3rd
|2
|m3 and M3
|3 and 4
|-
|4th
|3
|P4 and A4
|5 and 6
|-
|5th
|4
|d5 and P5
|6 and 7
|-
|6th
|5
|m6 and M6
|8 and 9
|-
|7th
|6
|m7 and M7
|10 and 11
|}

The [[diatonic scale|diatonic]] and [[pentatonic collection]]s 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.&nbsp;106, 158)

==Sources==
* Johnson, Timothy (2003). ''Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals''. Key College Publishing. {{ISBN|1-930190-80-8}}.

==Further reading==
*Clough, Engebretsen, and Kochavi. "Scales, Sets, and Interval Cycles": 78&ndash;84.

{{Set theory (music)}}

[[Category:Diatonic set theory]]
[[Category:Intervals (music)]]