The (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 or by dividing the minor third (6:5) into four frequency ratio steps of
The size of this scale step may also be precisely derived by putting the perfect fifth and the major third in a 9:5 ratio (not to be confused with the interval 9/5).
The formula below finds the minimum by setting the derivative of the mean square deviation with respect to the
and ()
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.
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. However, the alpha scale has