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 interval]]s 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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