In music theory, agenerated collection is acollection orscale formed by repeatedly adding a constantinterval ininteger notation, the generator, also known as aninterval cycle, around thechromatic circle until a complete collection or scale is formed. All scales with thedeep scale property can be generated by any intervalcoprime with the number of notes per octave. (Johnson, 2003, p. 83)

The C major diatonic collection can be generated by adding a cycle ofperfect fifths (C7) starting at F: F-C-G-D-A-E-B = C-D-E-F-G-A-B. Using integer notation and12-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 scale could also be generated using cycle ofperfect 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 singlegeneric interval corresponds to the single generator or interval cycle used is aMOS (for "moment of symmetry"[1]) orwell 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 onespecific 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 minorpentatonic scales are also well formed. (Johnson, 2003, p. 83)

The properties of generated and well-formedness were described byNorman Carey andDavid Clampitt in "Aspects of Well-Formed Scales" (1989), (Johnson, 2003, p. 151.) In earlier (1975) work, theoreticianErv Wilson defined the properties of the idea, and called such a scale aMOS, an acronym for "Moment of Symmetry".^[1] While unpublished until it appeared online in 1999, this paper was widely distributed and well known throughout themicrotonal music community, which adopted the term. The paper also remains more inclusive of further developments of the concept. For instance, thethree-gap theorem implies that every generated collection has at most three different steps, the intervals between adjacent tones in the collection (Carey 2007).

Adegenerate 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 thewhole-tone scale. (Johnson, 2003, p. 158, n. 14)

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

• 833 cents scale
• Cyclic group
• Distance model
• Pythagorean tuning

• Carey, Norman (July 2007), "Coherence and sameness in well-formed and pairwise well-formed scales",Journal of Mathematics and Music,1 (2):79–98,doi:10.1080/17459730701376743,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.

• Original concept of MOS as presented in a 1975 letter by Erv Wilson