Harmonic scale

Harmonic series on C, partials 1–5 numbered

Play (help·info).

Harmonic series on G, partials 1–5 numbered

Play (help·info).

The Harmonic scale is a "super-just" musical scale allowing extended just intonation, beyond 5-limit to the 19th harmonic (Play (help·info)), 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 (help·info)	0
C♯	C♯	D♭	17 (help·info)	104.96
D	D	D♮	18 (help·info)	203.91
E♭	E♭	E♭	19 (help·info)	297.51
E♮	E♮	E↓	20 (help·info)	386.31
F	F+	F	21 (help·info)	470.78
F♯	F↑	F♯	22 (help·info)	551.32
G	G	G	24 (help·info)	701.96
A♭	A♭	A♭	26 (help·info)	840.53
A♮	A♮+	A♮	27 (help·info)	905.87
B♭	B♭	B♭	28 (help·info)	968.83
B♮	B♮	B♮↓	30 (help·info)	1088.27
C'	C'	C'	32 (help·info)	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 (help·info)

Harmonic-scales chromatic on C and G.

Play chromatic scale on C (help·info)

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[]

^ Jump up to: ^a ^b 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

External links[]

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

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[Milano-1] Jump up to: ^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

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show v t Wendy Carlos
Albums	Switched-On Bach (1968) The Well-Tempered Synthesizer (1969) Sonic Seasonings (1972) Switched-On Bach II (1973) Switched-On Brandenburgs (1980) Digital Moonscapes (1984) Beauty in the Beast (1986) Peter & the Wolf/Carnival of the Animals – Part II (1988)
Soundtrack albums	Stanley Kubrick's A Clockwork Orange (1972) Walter Carlos's Clockwork Orange (1972) The Shining (Original Sound Track) (1980) Tron: Original Motion Picture Soundtrack (1982)
Musical tunings	Alpha scale Beta scale Gamma scale Harmonic scale