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 which have the deep scale property may be generated by any interval coprime with (in twelve-tone equal temperament) twelve. (Johnson 2003, p.83) Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of set theory to properties of the diatonic collection such as maximal evenness, Myhills property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. ... In common usage, a collection is any group of items that has one or more properties in common. ... In music, a scale is an unordered collection of notes or pitches, as opposed to a series of intervals, which is a musical mode. ... In music theory, an interval is the distance in pitch between two notes, the lower and higher members of the interval. ... Music notation is a system of writing for music. ... In music, interval cycles, unfold a single recurrent interval in a series that closes with a return to the initial pitch class, and are notated by George Perle using the letter C, for cycle, with an interval class integer to distinguish the interval. ... In music, the total chromatic is the saturation of the diatonic scale and is also commonly used in place of aggregate. ... In diatonic set theory the deep scale property is the quality of pitch class collections or scales containing each interval class a unique number of times. ... In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...

The C major diatonic collection may 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 musical interval of a perfect fifth is the relationship between the first note (the root or tonic) and the fifth note in a major scale. ...

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. The musical interval of a perfect fourth, often P4, is the relationship between the first note (the root or tonic) and the fourth note (subdominant) in a major scale. ...

A generated collection for which a single generic interval corresponds to the single generator or interval cycle used is a 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 will have only one specific interval between scale members (in this case 6) which corresponds to the generic interval (4, a fifth) but not the generator (7) and it will always be the generator (7) plus or minus one (7-1 = 6) if the total number of specific intervals (12) and the generic interval's corresponding specific interval (7) are coprime (12 and 7 are). The pentatonic scale is also well formed. (ibid) In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. ... In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle (interval class), in other words the number of half steps between notes. ... In music, a pentatonic scale is a scale with five notes per octave. ...

The properties of generated and well-formedness were first described by Norman Carey and David Clampitt in "Aspects of Well-Formed Scales" (1989). (ibid, p.151)

A degenerate well-formed collection are scales 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. (ibid, p.158n14) In music, a whole tone scale is a scale in which each note is separated from its neighbors by the interval of a whole step. ...

A bisector is a weaker substitute used to create collections which may not be generated. For the numerical analysis algorithm, see bisection method. ...

Further reading

• 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.

Source

• Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1930190808.