gamma scale

File:Neutral third on C.png

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| header = Chromatic circle

| image1 = Gamma scale chromatic circle.png

| caption1 = Comparing the gamma scale's approximations with the just values

| image2 = 12-tet scale chromatic circle for comparison with alpha scale et al.png

| caption2 = Twelve-tone equal temperament vs. just

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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{{sup|{{frac|20}}}}≈35.1 cents),{{Citation needed|date=January 2012}} by splitting the neutral third into two equal parts, or ten equal parts of approximately 35.1 cents each ({{audio|Gamma scale step on C.mid|Play}}) for 34.188 steps per octave.Carlos, Wendy (1989–96). [http://www.wendycarlos.com/resources/pitch.html "Three Asymmetric Divisions of the Octave"], WendyCarlos.com.

The scale step may also precisely be derived from using 20:11 (B{{music|U11}}, 1035 cents, {{audio|Lesser undecimal neutral seventh on C.mid|Play}}) to approximate the interval {{Frac|3:2|5:4}}, which equals 6:5 (E{{music|b}}, 315.64 cents, {{audio|Just minor third on C.mid|Play}}). Thus the step is approximately 35.099 cents and there are 34.1895 per octave.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."

$\backslash frac\{20\backslash log\_2\{(3/2)\}+11\backslash log\_2\{(5/4)\}+9\backslash log\_2\{(6/5)\}\}\{20^2+11^2+9^2\}=0.0292487852$ and $0.0292487852\backslash times1200=35.0985422804$

({{audio|Gamma scale step on C.mid|Play}})

"It produces nearly perfect triads."Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], Keyboard. "A 'third flavor', sort of intermediate to 'alpha' and 'beta', although a melodic diatonic scale is easily available."

class="wikitable" |align=center bgcolor="#ffffb4"|interval name |align=center bgcolor="#ffffb4"|size (steps) |align=center bgcolor="#ffffb4"|size (cents) |align=center bgcolor="#ffffb4"|just ratio |align=center bgcolor="#ffffb4"|just (cents) |align=center bgcolor="#ffffb4"|error

align=center|minor third |align=center|9 |align=center|315.89 |align=center|6:5 |align=center|315.64 |align=center|+0.25

align=center|major third |align=center|11 |align=center|386.09 |align=center|5:4 |align=center|386.31 |align=center|−0.22

align=center|perfect fifth |align=center|20 |align=center|701.98 |align=center|3:2 |align=center|701.96 |align=center|+0.02