Gamma Scale
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Gamma Scale
Neutral third:
just 347.41 cents  ,
ET 350 cents  ,
Gamma scale 351 cents
Comparing the gamma scale's approximations with the just values
Twelve-tone equal temperament vs. just

The ? (gamma) scale is a non-octave repeating musical scale invented by Wendy Carlos while preparing Beauty in the Beast (1986) though it is 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?35.1 cents),[] by splitting the neutral third into two equal parts, or ten equal parts of approximately 35.1 cents each ( ) for 34.188 steps per octave.[1]

The scale step may also precisely be derived from using 20:11 (B?, 1035 cents,  ) to approximate the interval ​,[2] which equals 6:5 (E, 315.64 cents,  ). 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 ${\displaystyle 0.0292487852\times 1200=35.0985422804}$ ( )

"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

## Sources

1. ^ a b Carlos, Wendy (1989-96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.
2. ^ a b Benson, Dave (2006). Music: A Mathematical Offering, p.232-233. ISBN 0-521-85387-7. "Carlos has 34.188 ?-scale degrees to the octave, corresponding to a scale degree of 35.1 cents."
3. ^ Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.