Alpha scale

Minor third (just: 315.64 cents <phonos file="Just minor third on C.mid">Play</phonos>,
12 TET: 300 cents <phonos file="Minor third on C.mid">Play</phonos>,
Alpha scale: 312 cents <phonos file="Alpha scale minor third on C.mid">Play</phonos>

Chromatic circle

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 ${\left(\frac{3}{2}\right)}^{\frac{1}{9}},$ or by dividing the minor third (6:5) into four frequency ratio steps of ${\left(\frac{6}{5}\right)}^{\frac{1}{4}}.$

The size of this scale step may also be precisely derived from using 9:5 (B♭, 1017.60 cents, <phonos file="Greater just minor seventh on C.mid">Play</phonos>) to approximate the interval ⁠ 3:2 / 5:4 ⁠ = 6:5   (E♭, 315.64 cents, <phonos file="Just minor third on C.mid">Play</phonos> ).

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.

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

$\frac{9{\log }_2\left(\frac{3}{2}\right)+5{\log }_2\left(\frac{5}{4}\right)+4{\log }_2\left(\frac{6}{5}\right)}{9^2+5^2+4^2}\approx 0.06497082462$

and $0.06497082462\times 1200=77.964989544$ (<phonos file="Alpha scale step on C.mid">Play</phonos>)

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," (<phonos file="Alpha scale major triad on C.mid">Play major</phonos> and <phonos file="Alpha scale minor triad on C.mid">minor triad</phonos>) and the beta scale has similar properties but the sevenths are more in tune. However, the alpha scale has

"excellent harmonic seventh chords ... using the [octave] inversion of ⁠ 7 / 4 ⁠, i.e., ⁠8/7⁠ [<phonos file="Alpha scale harmonic seventh chord on C.mid">Play</phonos>]."

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