Alpha scale

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Minor third (just: 315.64 cents Play ,
12-tet: 300 cents Play ,
Alpha scale: 312 cents Play 
Comparing 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 equal temperaments normally do) an octave (2:1). It may be approximated by dividing the perfect fifth (3:2) into nine equal steps (3:2)19,[1] or by dividing the minor third (6:5) into four steps (6:5)14.[1][2][3]

The size of this scale step may also be precisely derived from using 9:5 (B, 1017.60 cents, Play ) to approximate the interval 3:25:4 (=6:5, E, 315.64 cents, Play ).[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.

and (Play )

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," (Play major  and minor triad ) 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 74, i.e., 87 [Play ]."[1]

interval name size
just ratio just
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

See also[edit]


  1. ^ a b c Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", "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 α-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.