Generic and specific intervals: Difference between revisions

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[[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. 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. 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

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* [[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]] is the quality of musical scales or collections with exactly two specific intervals for every generic interval. In other words, each generic interval can be made from one of two possible different specific intervals.

'''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|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)}}

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[[Category:Diatonic set theory]]

[[Category:Intervals (music)]]

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