diatonic set theory Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the ...

, a generated collection is a

collection Collection or Collections may refer to: * Cash collection, the function of an accounts receivable department * Collection (church), money donated by the congregation during a church service * Collection agency, agency to collect cash * Collectio ...

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 positio ...

, 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 (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 Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...

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 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 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 clo ...

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 clo ...

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. (Johnson, 2003, p. 83) The properties of generated and well-formedness were described by

Norman Carey Norman or Normans may refer to: Ethnic and cultural identity * The Normans, a people partly descended from Norse Vikings who settled in the territory of Normandy in France in the 10th and 11th centuries ** People or things connected with the Norm ...

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 Ervin Wilson (June 11, 1928 – December 8, 2016) was a Mexico, Mexican/United States, American (dual citizen) music theory, music theorist. Early life Ervin Wilson was born in a remote area of northwest Chihuahua (state), Chihuahua, Mexico, wher ...

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 tw ...

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