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 may be approximated by dividing the perfect fifth (3:2) into nine equal steps, with frequency ratio

[1] or by dividing the minor third (6:5) into four frequency ratio steps of

[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 .

[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.