Harmonic scale

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

The harmonic scale is a "super-just" musical scale allowing extended just intonation, beyond 5-limit to the 19th harmonic (Play), 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 16 0
C C17 D 17 104.96
D D D 18 203.91
E E19 E 19 297.51
E E E 20 386.31
F F7+ F 21 470.78
F F F 22 551.32
G G G 24 701.96
A A13 A 26 840.53
A A+ A 27 905.87
B B7 B 28 968.83
B B B 30 1088.27
C' C' C' 32 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. Play diatonic scale

Harmonic-scales chromatic on C and G. Play chromatic scale on C

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, Franz Richter Herf and Gosheven.[3]

Number of notes[edit]

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.


  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, doi:10.2307/3680228, JSTOR 3680228
  4. ^ Carlos, Wendy (1987), "Tuning: At the Crossroads", Computer Music Journal, 11 (1): 29–43, doi:10.2307/3680176, JSTOR 3680176

External links[edit]