just 347.41 cents Playⓘ,
ET 350 cents Playⓘ,
Gamma scale 351 cents Playⓘ
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 (3:21⁄20≈35.1 cents),[citation needed] of approximately 35.1 cents each (Playⓘ) for 34.188 steps per octave.[1]
As 20 is even, this scale contains true neutral thirds, which are not found in alpha or beta.
The size of this scale step may also be precisely derived by putting the perfect fifth and the major third in a 20:11 ratio (not to be confused with the interval 20/11). 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
[edit]References
[edit]- ^ a b Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.
- ^ Benson, Dave (2006). Music: A Mathematical Offering, p.232-233. ISBN 0-521-85387-7. "Carlos has 34.188 γ-scale degrees to the octave, corresponding to a scale degree of 35.1 cents."
- ^ Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
