In diatonic set theory, a generated collection is a collection or

scale Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number ...

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 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 equival ...

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 fifth In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five ...

s (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 th ...

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 In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members. (Johnson 2003, p. 26) A specific interval is the cl ...

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 ancien ...

s are also well formed. (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. 151.) In earlier (1975) work, theoretician

Erv Wilson Ervin Wilson (June 11, 1928 – December 8, 2016) was a Mexican/American (dual citizen) music theorist. Early life Ervin Wilson was born in a remote area of northwest Chihuahua, Mexico, where he lived until the age of fifteen. His mother taught ...

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 Microtonal music or microtonality is the use in music of microtones— intervals smaller than a semitone, also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of ...

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. 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