Alpha scale

The α (alpha) scale is a non-octave-repeating musical scale invented by Wendy Carlos and first used on her album Beauty in the Beast (1986). It is derived from approximating just intervals using multiples of a single interval, but without requiring (as temperaments normally do) an octave (2:1). It may be approximated by dividing the perfect fifth (3:2) into nine equal steps, with frequency ratio ${\displaystyle \ \left({\tfrac {\ 3\ }{2}}\right)^{\tfrac {1}{9}}\ ,}$^[1] or by dividing the minor third (6:5) into four frequency ratio steps of ${\displaystyle \ \left({\tfrac {\ 6\ }{5}}\right)^{\tfrac {1}{4}}~.}$^[1][2][3]

Minor third (just: 315.64 cents Play^ⓘ,
12 TET: 300 cents Play^ⓘ,
Alpha scale: 312 cents Play^ⓘ

Chromatic circle

Comparison of the alpha scale's approximations with the just values

Twelve-tone equal temperament vs. just

The size of this scale step may also be precisely derived from using 9:5 (B♭, 1017.60 cents, Play^ⓘ) to approximate the interval ⁠ 3:2 / 5:4 ⁠ = 6:5   (E♭, 315.64 cents, Play^ⓘ ).^[4]

Carlos' α (alpha) scale arises from ... taking a value for the scale degree so that nine of them approximate a 3:2 perfect fifth, five of them approximate a 5:4 major third, and four of them approximate a 6:5 minor third. In order to make the approximation as good as possible we minimize the mean square deviation.^[4]

The formula below finds the minimum by setting the derivative of the mean square deviation with respect to the scale step size to 0 .

${\displaystyle \ {\frac {\ 9\ \log _{2}\left({\frac {\ 3\ }{2}}\right)+5\log _{2}\left({\frac {\ 5\ }{4}}\right)+4\ \log _{2}\left({\frac {\ 6\ }{5}}\right)\ }{\ 9^{2}+5^{2}+4^{2}\ }}\approx 0.06497082462\ }$

and ${\displaystyle \ 0.06497082462\times 1200=77.964989544\ }$ (Play^ⓘ)

At 78 cents per step, this totals approximately 15.385 steps per octave, however, more accurately, the alpha scale step is 77.965 cents and there are 15.3915 steps per octave.^[4][5]

Though it does not have a perfect octave, the alpha scale produces "wonderful triads," (Play major^ⓘ and minor triad^ⓘ) and the beta scale has similar properties but the sevenths are more in tune.^[2] However, the alpha scale has

"excellent harmonic seventh chords ... using the [octave] inversion of ⁠ 7 / 4 ⁠, i.e., ⁠8/7⁠ [Play^ⓘ]."^[1]

interval name	size (steps)	size (cents)	just ratio	just (cents)	error
septimal major second	3	233.89	8:7	231.17	+2.72
minor third	4	311.86	6:5	315.64	−3.78
major third	5	389.82	5:4	386.31	+3.51
perfect fifth	9	701.68	3:2	701.96	−0.27
harmonic seventh	octave−3	966.11	7:4	968.83	−2.72
octave	15	1169.47	2:1	1200.00	−30.53
octave	16	1247.44	2:1	1200.00	+47.44

See also

• Bohlen–Pierce scale
• Beta scale
• Gamma scale
• Delta scale
• Harmonic scale