Bisector (music)

In

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

References

• Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing.
  ISBN 1-930190-80-8
  .
• Rahn, Jay (1977). "Some Recurrent Features of Scales", In Theory Only 2, no. 11-12: 43-52