Harmonic scale

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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.

For example, if the harmonic scale is tuned to a fundamental of C, then harmonics 16–32 are as follows:

Notation	Harmonics	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: 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.

Though described by Carlos as containing "144 [= 122] distinct pitches to the octave", the twelve scales include 78 (= .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠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.

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

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