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[[File:Circle of Fifths.svg|125px|thumb|Red line indicates the major scale on C within the outer [[circle of fifths]]]]
[[File:Circle of Fifths.svg|125px|thumb|Red line indicates the major scale on C within the outer [[circle of fifths]]]]


In music theory, a '''generated collection''' is a [[set (music)|collection]] or [[scale (music)|scale]] formed by repeatedly adding a constant [[interval (music)|interval]] in [[integer notation]], the generator, also known as an [[interval cycle]], around the [[chromatic circle]] until a complete collection or scale is formed. All scales with the [[Common tone (scale)#Deep scale property|deep scale property]] can be generated by any interval [[coprime]] with the number of notes per octave. (Johnson, 2003, p. 83)
In music theory, a '''generated collection''' is a [[set (music)|collection]] or [[scale (music)|scale]] formed by repeatedly adding a constant [[interval (music)|interval]] in [[integer notation]], the generator, also known as an [[interval cycle]], around the [[chromatic circle]] until a complete collection or scale is formed. All scales with the [[Common tone (scale)#Deep scale property|deep scale property]] can be generated by any interval [[coprime]] with the number of notes per octave.{{sfnp|Johnson|2003|p=83}}


The C major diatonic collection can be generated by adding a cycle of [[perfect fifth]]s (C7) starting at F: F-C-G-D-A-E-B = C-D-E-F-G-A-B. Using integer notation and [[12-tone equal temperament]], the standard tuning of Western music: 5 + 7 = 0, 0 + 7 = 7, 7 + 7 = 2, 2 + 7 = 9, 9 + 7 = 4, 4 + 7 = 11.
The C major diatonic collection can be generated by adding a cycle of [[perfect fifth]]s (C7) starting at F: F-C-G-D-A-E-B = C-D-E-F-G-A-B. Using integer notation and [[12-tone equal temperament]], the standard tuning of Western music: 5 + 7 = 0, 0 + 7 = 7, 7 + 7 = 2, 2 + 7 = 9, 9 + 7 = 4, 4 + 7 = 11.
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The C major scale could also be generated using cycle of [[perfect fourth]]s (C5), as 12 minus any coprime of twelve is also coprime with twelve: 12 − 7 = 5. B-E-A-D-G-C-F.
The C major scale could also be generated using cycle of [[perfect fourth]]s (C5), as 12 minus any coprime of twelve is also coprime with twelve: 12 − 7 = 5. B-E-A-D-G-C-F.


A generated collection for which a single [[generic interval]] corresponds to the single generator or interval cycle used is a '''MOS''' (for "moment of symmetry"[http://www.tonalsoft.com/enc/m/mos.aspx]) or '''well formed generated collection'''. For example, the diatonic collection is well formed, for the perfect fifth (the generic interval 4) corresponds to the generator 7. Though not all fifths in the diatonic collection are perfect (B-F is a diminished fifth), a well formed generated collection has only one [[specific interval]] between scale members (in this case 6)—which corresponds to the generic interval (4, a fifth) but to not the generator (7). The major and minor [[pentatonic scale]]s are also well formed. (Johnson, 2003, p. 83)
A generated collection for which a single [[generic interval]] corresponds to the single generator or interval cycle used is a '''MOS''' (for "moment of symmetry"[http://www.tonalsoft.com/enc/m/mos.aspx]) or '''well formed generated collection'''. For example, the diatonic collection is well formed, for the perfect fifth (the generic interval 4) corresponds to the generator 7. Though not all fifths in the diatonic collection are perfect (B-F is a diminished fifth), a well formed generated collection has only one [[specific interval]] between scale members (in this case 6)—which corresponds to the generic interval (4, a fifth) but to not the generator (7). The major and minor [[pentatonic scale]]s are also well formed.{{sfnp|Johnson|2003|p=83}}


The properties of generated and well-formedness were described by [[Norman Carey]] and [[David Clampitt]] in "Aspects of Well-Formed Scales" (1989), (Johnson, 2003, p.&nbsp;151.) In earlier (1975) work, theoretician [[Erv Wilson]] defined the properties of the idea, and called such a scale a ''MOS'', an acronym for "Moment of Symmetry".<ref>{{Cite web|url=http://anaphoria.com/wilsonintroMOS.html|title = Introduction to Erv Wilson's Moments of Symmetry}}</ref> While unpublished until it appeared online in 1999, this paper was widely distributed and well known throughout the [[microtonal music]] community, which adopted the term. The paper also remains more inclusive of further developments of the concept. For instance, the [[three-gap theorem]] implies that every generated collection has at most three different steps, the intervals between adjacent tones in the collection (Carey 2007).
The properties of generated and well-formedness were described by {{harvp|Carey|Clampitt|1989}}.{{sfnp|Johnson|2003|p=151}} Earlier, in 1975, theoretician [[Erv Wilson]] defined the properties of the idea, and called such a scale a ''MOS'', an acronym for ''moment of symmetry''.{{sfnp|Wilson|1975}} Although unpublished until 1999,<ref name=Wilson-1999-intro/> when it was posted online, this paper was widely distributed and well known throughout the community of [[microtonal music]]ians, who adopted the term. The paper also remains more inclusive of further developments of the concept. For instance, the [[three-gap theorem]] implies that every generated collection has at most three different steps, the intervals between adjacent tones in the collection,{{sfnp|Carey|2007}}


A '''degenerate well-formed collection''' is a scale in which the generator and the interval required to complete the circle or return to the initial note are equivalent and include all scales with equal notes, such as the [[whole-tone scale]]. (Johnson, 2003, p.&nbsp;158, n. 14)
A '''degenerate well-formed collection''' is a scale in which the generator and the interval required to complete the circle or return to the initial note are equivalent and include all scales with equal notes, such as the [[whole-tone scale]].{{sfnp|Johnson|2003|at=p.&nbsp;158, n.&nbsp;14}}


A [[bisector (music)|bisector]] is a more general concept used to create collections that cannot be generated but includes all collections which can be generated.
A [[bisector (music)|bisector]] is a more general concept used to create collections that cannot be generated but includes all collections which can be generated.


==See also==
==See also==
*[[833 cents scale]]
{{div col begin|colwidth=15em}}
*[[Cyclic group]]
* [[833 cents scale]]
*[[Distance model]]
* [[Cyclic group]]
*[[Pythagorean tuning]]
* [[Dissonance and consonance]]
* [[Distance model]]
* [[Just intonation]]
* [[Meantone temperament]]
* [[Pythagorean tuning]]
{{div col end}}


==References==
==References==
{{Reflist}}
{{reflist|18em|refs=
 
<ref name=Wilson-1999-intro>
{{cite web
|title = Introduction to Erv Wilson's moments of symmetry
|year = 1999 |orig-year = 1975
|url=http://anaphoria.com/wilsonintroMOS.html
|via=anaphoria.com
}}
</ref>
 
}} <!-- end "refs=" -->


==Sources==
==Sources==
*{{citation
{{refbegin|25em|small=yes}}
  | last = Carey | first = Norman
* {{cite journal
  | last = Carey | first = Norman | author-link = Norman Carey
  | date = July 2007
  | date = July 2007
| title = Coherence and sameness in well-formed and pairwise well-formed scales
| journal = [[Journal of Mathematics and Music]]
| volume = 1  | issue = 2  | pages = 79–98
| s2cid = 120586231
  | doi = 10.1080/17459730701376743
  | doi = 10.1080/17459730701376743
  | issue = 2
}}
  | journal = Journal of Mathematics and Music
* {{cite journal
  | pages = 79–98
  |last1 = Carey    |first1 = Norman  |author1-link = Norman Carey
  | title = Coherence and sameness in well-formed and pairwise well-formed scales
  |last2 = Clampitt |first2 = David  <!-- |author2-link = David Clampitt -->
  | volume = 1| s2cid = 120586231
  |year = 1989
}}
  |title = Aspects of well-formed scales
*Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", ''Music Theory Spectrum'' 11: 187–206.
|journal = [[Music Theory Spectrum]]
*Clough, Engebretsen, and Kochavi. "Scales, Sets, and Interval Cycles", 79.
  |volume = 11  |pages = 187–206
*Johnson, Timothy (2003). ''Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals''. Key College Publishing. {{ISBN|1-930190-80-8}}.
}}
* {{cite journal
|last1 = Clough     |first1 = John
|last2 = Engebretsen |first2 = Nora
|last3 = Kochavi     |first3 = Jonathan
|date = March 1999
|title = Scales, sets, and interval cycles: A taxonomy
|journal = [[Music Theory Spectrum]]
|volume = 21  |issue = 1  |pages = 74–104, esp p&nbsp;79
|jstor = 745921
|doi = 10.2307/745921
}}
* {{cite book
|last = Johnson |first = Timothy A.
|year = 2003
|title = Foundations of Diatonic Theory: A mathematically based approach to music fundamentals
|place = Emeryville, CA
|publisher = Key College Publishing
|ISBN = 978-1-9301-9080-1  |lccn = 2002075736
|url = https://archive.org/details/foundationsofdia0000john |via = [[Internet Archive]]
|url-access = registration
}} Scarecrow Press 2008&nbsp;ed. {{isbn|978-0-8108-6213-5}}.
* {{cite letter
|first=Erv |last=Wilson
|date=26 April 1975
|recipient = John
|subject=The basic structure of "moments of symmetry"
|location = Poinsettia Drive, L.A., California
|url = http://www.anaphoria.com/mos.PDF  |url-status = dead  |access-date = 16 Oct 2025
|archive-url = https://web.archive.org/web/20221006112108/https://anaphoria.com/mos.pdf
|archive-date = 2022-10-06  |df = dmy-all
}}


==External links==
{{refend}}
* [http://www.anaphoria.com/mos.PDF Original concept of MOS as presented in a 1975 letter by Erv Wilson]


{{Set theory (music)}}
{{Set theory (music)}}

Latest revision as of 09:09, 16 October 2025

File:Circle of Fifths.svg
Red line indicates the major scale on C within the outer circle of fifths

In music theory, a generated collection is a collection or scale formed by repeatedly adding a constant interval in integer notation, the generator, also known as an interval cycle, around the chromatic circle until a complete collection or scale is formed. All scales with the deep scale property can be generated by any interval coprime with the number of notes per octave.Template:Sfnp

The C major diatonic collection can be generated by adding a cycle of perfect fifths (C7) starting at F: F-C-G-D-A-E-B = C-D-E-F-G-A-B. Using integer notation and 12-tone equal temperament, the standard tuning of Western music: 5 + 7 = 0, 0 + 7 = 7, 7 + 7 = 2, 2 + 7 = 9, 9 + 7 = 4, 4 + 7 = 11.

7-note segment of C5: the C major scale as a generated collection
7-note segment of C5: the C major scale as a generated collection

The C major scale could also be generated using cycle of perfect fourths (C5), as 12 minus any coprime of twelve is also coprime with twelve: 12 − 7 = 5. B-E-A-D-G-C-F.

A generated collection for which a single generic interval corresponds to the single generator or interval cycle used is a MOS (for "moment of symmetry"[1]) or well formed generated collection. For example, the diatonic collection is well formed, for the perfect fifth (the generic interval 4) corresponds to the generator 7. Though not all fifths in the diatonic collection are perfect (B-F is a diminished fifth), a well formed generated collection has only one specific interval between scale members (in this case 6)—which corresponds to the generic interval (4, a fifth) but to not the generator (7). The major and minor pentatonic scales are also well formed.Template:Sfnp

The properties of generated and well-formedness were described by Template:Harvp.Template:Sfnp Earlier, in 1975, theoretician Erv Wilson defined the properties of the idea, and called such a scale a MOS, an acronym for moment of symmetry.Template:Sfnp Although unpublished until 1999,[1] when it was posted online, this paper was widely distributed and well known throughout the community of microtonal musicians, who adopted the term. The paper also remains more inclusive of further developments of the concept. For instance, the three-gap theorem implies that every generated collection has at most three different steps, the intervals between adjacent tones in the collection,Template:Sfnp

A degenerate well-formed collection is a scale in which the generator and the interval required to complete the circle or return to the initial note are equivalent and include all scales with equal notes, such as the whole-tone scale.Template:Sfnp

A bisector is a more general concept used to create collections that cannot be generated but includes all collections which can be generated.

See also

Template:Div col begin

References

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