Harmonic Scale

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 C D  17  104.96
D D D  18  203.91
E E E  19  297.51
E E E  20  386.31
F F+ F  21  470.78
F F F  22  551.32
G G G  24  701.96
A A A  26  840.53
A A+ A  27  905.87
B B 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 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.


  1. 1 2 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

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