In diatonic set theory

Diatonic set theory

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

, a generated collection is a collection

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects...

 or scale formed by repeatedly adding a constant interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 in integer notation, the generator, also known as an interval cycle

Interval cycle

In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached In music, an interval cycle is a collection...

, around the chromatic circle

Chromatic circle

The chromatic circle is a geometrical space that shows relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale...

 until a complete collection or scale is formed. All scales which have the deep scale property

Deep scale property

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. Examples include the diatonic scale...

 may be generated by any interval coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 with (in twelve-tone equal temperament) twelve. (Johnson 2003, p.83)

The C major diatonic collection may be generated by adding a cycle of perfect fifth

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

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

Perfect fourth

In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

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

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...

 corresponds to the single generator or interval cycle used is a MOS 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 will have only one specific interval

Specific interval

In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between notes. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal...

 between scale members (in this case 6) which corresponds to the generic interval (4, a fifth) but to not the generator (7) and this one differing interval 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

Pentatonic scale

A pentatonic scale is a musical scale with five notes per octave in contrast to a heptatonic scale such as the major scale and minor scale...

 is also well formed. (ibid)

The properties of generated and well-formedness were described by Norman Carey and David Clampitt in "Aspects of Well-Formed Scales" (1989), (ibid, p. 151.) In earlier (1975) work, non-academic theoretician Erv Wilson

Erv Wilson

Ervin Wilson is a Mexican/American music theorist. Despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design...

 considered the idea, and called such a scale a MOS, an acronym for "Moment of Symmetry". While unpublished, this terminology became widely known and used in the microtonal music

Microtonal music

Microtonal music is music using microtones—intervals of less than an equally spaced semitone. Microtonal music can also refer to music which uses intervals not found in the Western system of 12 equal intervals to the octave.-Terminology:...

 community.

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)

A bisector

Bisector (music)

In diatonic set theory, a bisector divides the octave approximately in half and may be used in place of a generator to derive collections for which structure implies multiplicity is not true such as the ascending melodic minor, harmonic minor, and octatonic scales. Well formed generated collections...

is a weaker substitute used to create collections which may not be generated.