Gamma scale

The γ (gamma) scale is a non-octave repeating musical scale invented by Wendy Carlos while preparing Beauty in the Beast (1986) though it does not appear on the album. It is derived from approximating just intervals using multiples of a single interval without, as is standard in equal temperaments, requiring an octave (2:1). It may be approximated by splitting the perfect fifth (3:2) into 20 equal parts (3:2^1⁄20≈35.1 cents),^{[citation needed]} by splitting the neutral third into two equal parts, or ten equal parts of approximately 35.1 cents each (Play^ⓘ) for 34.188 steps per octave.^[1]

Neutral third:
just 347.41 cents Play^ⓘ,
ET 350 cents Play^ⓘ,
Gamma scale 351 cents Play^ⓘ

Chromatic circle

Comparing the gamma scale's approximations with the just values

Twelve-tone equal temperament vs. just

The scale step may also precisely be derived from using 20:11 (B↓, 1035 cents, Play^ⓘ) to approximate the interval 3:2⁄5:4,^[2] which equals 6:5 (E♭, 315.64 cents, Play^ⓘ). Thus the step is approximately 35.099 cents and there are 34.1895 per octave.^[2]

${\displaystyle {\frac {20\log _{2}{(3/2)}+11\log _{2}{(5/4)}+9\log _{2}{(6/5)}}{20^{2}+11^{2}+9^{2}}}=0.0292487852}$ and ${\displaystyle 0.0292487852\times 1200=35.0985422804}$ (Play^ⓘ)

"It produces nearly perfect triads."^[3] "A 'third flavor', sort of intermediate to 'alpha' and 'beta', although a melodic diatonic scale is easily available."^[1]

interval name	size (steps)	size (cents)	just ratio	just (cents)	error
minor third	9	315.89	6:5	315.64	+0.25
major third	11	386.09	5:4	386.31	−0.22
perfect fifth	20	701.98	3:2	701.96	+0.02

See also

• Alpha scale
• Beta scale
• Delta scale
• Bohlen–Pierce scale
• Gamma chord