Red line indicates the major scale on C within the outer circle of fifths
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.
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"[1]) 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 scales 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Wilson defined the properties of the idea, and called such a scale a MOS, an acronym for "Moment of Symmetry".[1] While unpublished until it appeared online in 1999, this paper was widely distributed and well known throughout the microtonal music which adopted the term.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 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.
^"Introduction to Erv Wilson's Moments of Symmetry".
Sourcesedit
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. ISBN1-930190-80-8.
External linksedit
Original concept of MOS as presented in a 1975 letter by Erv Wilson
January 01, 1970
generated, collection, diatonic, theory, generated, collection, collection, scale, formed, repeatedly, adding, constant, interval, integer, notation, generator, also, known, interval, cycle, around, chromatic, circle, until, complete, collection, scale, formed. In diatonic set 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 in twelve tone equal temperament twelve Johnson 2003 p 83 Red line indicates the major scale on C within the outer circle of fifths 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 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 1 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 scales 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 defined the properties of the idea and called such a scale a MOS an acronym for Moment of Symmetry 1 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 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 Contents 1 See also 2 References 3 Sources 4 External linksSee also edit833 cents scale Cyclic group Distance model Pythagorean tuningReferences edit Introduction to Erv Wilson s Moments of Symmetry Sources editCarey 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 External links editOriginal concept of MOS as presented in a 1975 letter by Erv Wilson Retrieved from https en wikipedia org w index php title Generated collection amp oldid 1089282675, wikipedia, wiki, book, books, library,
