Alpha Scale
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Alpha Scale
Minor third (just: 315.64 cents  ,
12-tet: 300 cents  ,
Alpha scale: 312 cents
Comparing the alpha scale's approximations with the just values
Twelve-tone equal temperament vs. just

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 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] or by dividing the minor third (6:5) into four steps (6:5).[1][2][3]

The 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' ? (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 ${\displaystyle 0.06497082462\times 1200=77.965}$ ( )

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," ( 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 ​, i.e., [ ]."[1]

 interval name size (steps) size (cents) just ratio just (cents) error septimal major second 3 233.90 8:7 231.17 +2.72 major third 5 389.83 5:4 386.31 +3.51 perfect fifth 9 701.69 3:2 701.96 −0.27 harmonic seventh 12 935.58 7:4 968.83 −33.25 octave 15 1169.48 2:1 1200.00 −30.52 octave 16 1247.44 2:1 1200.00 +47.44

## Sources

1. ^ a b c Carlos, Wendy (1989-96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com. "9 steps to the perfect (no kidding) fifth." The alpha scale, "splits the minor third exactly in half (also into quarters)."
2. ^ a b Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard. "The idea was to split a minor third into two equal parts. Then that was divided again."
3. ^ Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
4. ^ a b c Benson, Dave (2006). Music: A Mathematical Offering, p.232-233. ISBN 0-521-85387-7. "This actually differs very slightly from Carlos' figure of 15.385 ?-scale degrees to the octave. This is obtained by approximating the scale degree to 78.0 cents."
5. ^ Sethares, William (2004). Tuning, Timbre, Spectrum, Scale, p.60. ISBN 1-85233-797-4. Scale step of 78 cents.