Harmonic scale

Harmonic series on C, partials 1–5 numbered  .
Harmonic series on G, partials 1–5 numbered  .

The Harmonic scale is a "super-just" musical scale allowing extended just intonation, beyond 5-limit to the 19th harmonic ( ), and free modulation through the use of synthesizers. Transpositions and tuning tables are controlled by the left hand on the appropriate note on a one-octave keyboard.[1]

For example, if the harmonic scale is tuned to a fundamental of C, then harmonics 16–32 are as follows:

Notation Harmonics[2] Cents
C C C   0
C C D   104.96
D D D   203.91
E E E   297.51
E E E   386.31
F F+ F   470.78
F F F   551.32
G G G   701.96
A A A   840.53
A A+ A   905.87
B B B   968.83
B B B   1088.27
C' C' C'   1200

Some harmonics are not included:[1] 23, 25, 29, & 31. The 21st is a natural seventh above G, but not a great interval above C, and the 27th is a just fifth above D.

Harmonic-scales chromatic on C and G.

It was invented by Wendy Carlos and used on three pieces on her album Beauty in the Beast (1986): Just Imaginings, That's Just It, and Yusae-Aisae. Versions of the scale have also been used by Ezra Sims and Frans Richter Herf.[3]

Number of notes

Though described by Carlos as containing "144 [= 122] distinct pitches to the octave",[4] the twelve scales include 78 (= 12(12+1)/2) notes per octave.

Technically there should then be duplicates and thus 57 (= 78 − 21) pitches (21 = 6(6+1)/2). For example, a perfect fifth above G (D) is the major tone above C.

References

1. ^ a b Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
2. ^ Benson, Dave (2007). Music: A Mathematical Offering, p. 212. ISBN 9780521853873.
3. ^ Sims, Ezra (1987), "Observations on Microtonality Issue: Letters", Computer Music Journal, 11 (4): 8–9, JSTOR 3680228
4. ^ Carlos, Wendy (1987), "Tuning: At the Crossroads", Computer Music Journal, 11 (1): 29–43, JSTOR 3680176