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.

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

External links

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