Beta scale

Perfect fourth (just: 498.04 cents Playⓘ, 12-tet: 500 cents Playⓘ, Beta scale: 512 cents Playⓘ)
Chromatic circle
Comparing the beta scale's approximations with the just values
Twelve-tone equal temperament vs. just

The β (beta) 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 splitting the perfect fifth (3:2) into eleven equal parts [(3:2)1⁄11 ≈ 63.8 cents]. It may be approximated by splitting the perfect fourth (4:3) into two equal parts [(4:3)1⁄2],[1] or eight equal parts [(4:3)1⁄8 = 64 cents],[2] totaling approximately 18.8 steps per octave.

The scale step may also precisely be derived from using 11:6 (B↑♭-, 1049.36 cents, Playⓘ) to approximate the interval 3:2⁄5:4,[3] which equals 6:5 Playⓘ.

In order to make the approximation as good as possible we minimize the mean square deviation. ... We choose a value of the scale degree so that eleven of them approximate a 3:2 perfect fifth, six of them approximate a 5:4 major third, and five of them approximate a 6:5 minor third.[3]

and (Playⓘ)

Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, Playⓘ is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[3][4] Playⓘ though both have nice triads[1] (Play major triadⓘ, minor triadⓘ, and dominant seventhⓘ). "According to Carlos, beta has almost the same properties as the alpha scale, except that the sevenths are slightly more in tune."[1]

The delta scale may be regarded as the beta scale's reciprocal since it is "as far 'down' the (0 3 6 9) circle from α as β is 'up'."[5]

interval name size
(steps) 		size
(cents) 		just ratio just
(cents) 		error
major second 3 191.50 9:8 203.91 −12.41
minor third 5 319.16 6:5 315.64 +3.52
major third 6 383.00 5:4 386.31 −3.32
perfect fifth 11 702.16 3:2 701.96 +0.21
harmonic seventh 15 957.49 7:4 968.83 −11.33
octave 18 1148.99 2:1 1200.00 −51.01
octave 19 1212.83 2:1 1200.00 +12.83
  1. ^ a b c Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
  2. ^ Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
  3. ^ a b c Benson, Dave (2006). Music: A Mathematical Offering, p.232-233. ISBN 0-521-85387-7. "Carlos has 18.809 β-scale degrees to the octave, corresponding to a scale degree of 63.8 cents."
  4. ^ Sethares, William (2004). Tuning, Timbre, Spectrum, Scale, p.60. ISBN 1-85233-797-4. Scale step of 63.8 cents.
  5. ^ Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p.1394. ISBN 0-520-07099-2.