Editing Generated collection

[[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.&nbsp;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&nbsp;+&nbsp;7&nbsp;=&nbsp;0, 0&nbsp;+&nbsp;7&nbsp;=&nbsp;7, 7&nbsp;+&nbsp;7&nbsp;=&nbsp;2, 2&nbsp;+&nbsp;7&nbsp;=&nbsp;9, 9&nbsp;+&nbsp;7&nbsp;=&nbsp;4, 4&nbsp;+&nbsp;7&nbsp;=&nbsp;11.

[[Image:7-note segment of C5.svg|384px|center|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 fourth]]s (C5), as 12 minus any coprime of twelve is also coprime with twelve: 12&nbsp;&minus;&nbsp;7&nbsp;=&nbsp;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.&nbsp;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).

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 [[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==
*[[833 cents scale]]
*[[Cyclic group]]
*[[Distance model]]
*[[Pythagorean tuning]]

==References==
{{Reflist}}

==Sources==
*{{citation
 | last = Carey | first = Norman
 | date = July 2007
 | doi = 10.1080/17459730701376743
 | issue = 2
 | journal = Journal of Mathematics and Music
 | pages = 79–98
 | title = Coherence and sameness in well-formed and pairwise well-formed scales
 | volume = 1| s2cid = 120586231
 }}
*Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", ''Music Theory Spectrum'' 11: 187–206.
*Clough, Engebretsen, and Kochavi. "Scales, Sets, and Interval Cycles", 79.
*Johnson, Timothy (2003). ''Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals''. Key College Publishing. {{ISBN|1-930190-80-8}}.

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

{{Set theory (music)}}

[[Category:Diatonic set theory]]
[[Category:Music theory]]