Gamma scale

Summary

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

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 (3:2120≈35.1 cents),[citation needed] by splitting the neutral third into two equal parts, or ten equal parts of approximately 35.1 cents each (audio speaker iconPlay ) for 34.188 steps per octave.[1]

The scale step may also precisely be derived from using 20:11 (B, 1035 cents, audio speaker iconPlay ) to approximate the interval 3:25:4,[2] which equals 6:5 (E, 315.64 cents, audio speaker iconPlay ). Thus the step is approximately 35.099 cents and there are 34.1895 per octave.[2]

and (audio speaker iconPlay )

"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

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 γ-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.