Alpha scale

Comparison of the alpha scale's approximations with the just values
Twelve-tone equal temperament vs. just

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]

The size of this scale step may also be precisely derived from using 9:5 (B, 1017.60 cents, ) to approximate the interval = 6:5 (E, 315.64 cents, ).[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 ${\displaystyle \ 0.06497082462\times 1200=77.964989544\ }$ ()

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," ( and ) 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 []."[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