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, th ...

, a bisector divides the

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

approximately in half (the equal tempered

tritone In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three a ...

is exactly half the octave) and may be used in place of a

generator Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...

to derive

collections 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 * Collection ...

for which structure implies multiplicity is not true such as the ascending melodic minor,

harmonic minor In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which also ...

, and

octatonic scale An octatonic scale is any eight-note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), this symmetrical ...

s. Well formed generated collections generators and bisectors coincide, such as the

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 ...

(

circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...

) in the

diatonic collection In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...

. The term was introduced by Jay Rahn (1977), who considers any division between one and two thirds as approximately half (

major third In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...

minor sixth In Western classical music, a minor sixth is a musical interval encompassing six staff positions (see Interval number for more details), and is one of two commonly occurring sixths (the other one being the major sixth). It is qualified as ''mi ...

or 400 to 800 cents) and who applied the term only the equally spaced collections. Clough and Johnson both adapt the term to apply to generic scale steps. Rahn also uses ''aliquant bisector'' for bisectors which may be used to generate every note in a collection, in which case the bisector and the number of notes must be

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 equivale ...

. Bisectors may be used to produce the

diatonic Diatonic and chromatic are terms in music theory that are most often used to characterize Scale (music), scales, and are also applied to musical instruments, Interval (music), intervals, Chord (music), chords, Musical note, notes, musical sty ...

, and ascending melodic minor collections. (Johnson 2003, p.97, 101, 158n10-12) The diatonic scale may be derived from a chain of perfect fifths: P5 P5 P5 P5 P5 P5 F C G D A E B = C D E F G A B C. 5, 0, 7, 2, 9, 4, e = 0, 2, 4, 5, 7, 9, e, 0. +7 +7 +7 +7 +7 +7 (mod 12) For example, the octatonic scale may be derived similarly to derivations of the diatonic scale by a chain of perfect fifths (a generator), by using a bisector of 5 scale steps (3 may also be used). However, five steps in the octatonic scale alternates between 7 and 8 semitones, so it is a bisector and not a generator: A5 P5 A5 P5 A5 P5 A5 P5 C A♭ E♭ B G♭ D A F C = C D E♭ F G♭ A♭ A B C. 0, 8, 3, e, 6, 2, 9, 5, 0 = 0, 2, 3, 5, 6, 8, 9, e, 0. +8 +7 +8 +7 +8 +7 +8 +7 Octatonic scale bisector

References

*Johnson, Timothy (2003). ''Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals''. Key College Publishing. . *Rahn, Jay (1977). "Some Recurrent Features of Scales", ''In Theory Only'' 2, no. 11-12: 43-52 Diatonic set theory Musicology {{music-theory-stub