Harmonic scale

For other uses, see Harmonic scale (disambiguation).

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, 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

1. 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

• "Wendy Carlos Harmonic scale", Microtonal-Synthesis.com.