125px|thumb|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.
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.
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 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"[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 scales are also well formed.
The properties of generated and well-formedness were described by . 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. Although unpublished until 1999,
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Sources
- Scarecrow Press 2008 ed. .