In diatonic set theory, a generated collection is a collection or scale formed by repeatedly adding a constant interval in

integer notation In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave posit ...

, the generator, also known as an

interval cycle In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class.Whittall, Arnold. 2008. ''The Cambridge Introduction to Serialism'', p. 273-74. New York: Cambridge University Press. (pbk). In other ...

, around the

chromatic circle The chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle. Explanation If one starts on any equal-tempered pitch and repeatedly ascends by ...

until a complete collection or scale is formed. All scales with the

deep scale property In music, a common tone is a pitch class that is a member of, or common to (shared by) two or more scales or sets. Common tone theorem A common tone is a pitch class that is a member of, or common to, a musical scale and a transposition of ...

can be generated by any interval

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

with (in twelve-tone equal temperament) twelve. (Johnson, 2003, p. 83) 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 modulo 12: 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 of

perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...

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 symmetr

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, tritone, or 6), 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 A pentatonic scale is a musical scale with five notes per octave, in contrast to the heptatonic scale, which has seven notes per octave (such as the major scale and minor scale). Pentatonic scales were developed independently by many an ...

s are also well formed. (Johnson, 2003, p. 83) The properties of generated and well-formedness were described by Norman Carey and

David Clampitt David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...

in "Aspects of Well-Formed Scales" (1989), (Johnson, 2003, p. 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". While unpublished until it appeared online in 1999, this paper was widely distributed and well known throughout the microtonal music which adopted the term. the paper also remains more inclusive of further developments of the concept. For instance, the

three-gap theorem In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places points on a circle, at angles of , , , ... from the starting point, then there will be at most three distinct distances between pairs ...

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 In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or ''hexatonic'' s ...

. (Johnson, 2003, p. 158, n. 14) A bisector is a more general concept used to create collections that cannot be generated but includes all collections which can be generated.

References

Sources

* *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. .

External links

Original concept of MOS as presented in a 1975 letter by Erv Wilson
{{Set theory (music) Diatonic set theory

See also

References

Sources

External links