Harmonic scale

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

The Harmonic scale is a "super-just" musical scale allowing extended just intonation, beyond 5-limit to the 19th harmonic (About this soundPlay ), 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 About this sound16  0
C C17 D About this sound17  104.96
D D D About this sound18  203.91
E E19 E About this sound19  297.51
E E E About this sound20  386.31
F F7+ FHalf down arrow.png About this sound21  470.78
F F FSims flagged arrow down.svg About this sound22  551.32
G G G About this sound24  701.96
A A13 AHalf up arrow.png About this sound26  840.53
A A+ A About this sound27  905.87
B B7 BHalf down arrow.png About this sound28  968.83
B B B About this sound30  1088.27
C' C' C' About this sound32  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. About this soundPlay diatonic scale 

Harmonic-scales chromatic on C and G. About this soundPlay 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, JSTOR 3680228
  4. ^ Carlos, Wendy (1987), "Tuning: At the Crossroads", Computer Music Journal, 11 (1): 29–43, JSTOR 3680176

External links[edit]