Harmonic scale
The Harmonic scale is a "super-just" musical scale allowing extended just intonation, beyond 5-limit to the 19th harmonic (
![](../I/m/Harmonic_series_klang.png)
![](../I/m/Loudspeaker.svg.png)
![](../I/m/Harmonic_series_klang_G.png)
![](../I/m/Loudspeaker.svg.png)
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.
![](../I/m/Harmonic_scale_chromatic_on_C_and_G.png)
![](../I/m/Loudspeaker.svg.png)
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
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
- Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
- Benson, Dave (2007). Music: A Mathematical Offering, p. 212. ISBN 9780521853873.
- Sims, Ezra (1987), "Observations on Microtonality Issue: Letters", Computer Music Journal, 11 (4): 8–9, JSTOR 3680228
- Carlos, Wendy (1987), "Tuning: At the Crossroads", Computer Music Journal, 11 (1): 29–43, JSTOR 3680176