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 without, as is standard in equal temperaments, requiring an octave (2:1). It may be approximated by dividing the perfect fifth (3:2) into nine equal steps (3:2)^​1⁄₉,[1] or by dividing the minor third (6:5) into four steps (6:5)^​1⁄₄.[1][2][3]

Minor third (just: 315.64 cents Play ,
12-tet: 300 cents Play ,
Alpha scale: 312 cents Play 

Chromatic circle

Comparing the alpha scale's approximations with the just values

Twelve-tone equal temperament vs. just

The 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]

${\displaystyle {\frac {9\log _{2}(3/2)+5\log _{2}(5/4)+4\log _{2}(6/5)}{9^{2}+5^{2}+4^{2}}}\approx 0.06497082462}$ and ${\displaystyle 0.06497082462\times 1200=77.965}$ (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 per octave.[4][5]

Though it does not have an 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]