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 (3:21⁄11≈63.8 cents). It may be approximated by splitting the perfect fourth (4:3) into two equal parts (4:31⁄2), or eight equal parts ((4:31⁄8=64 cents)), totaling approximately 18.8 steps per octave.
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.
Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents,
- Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
- Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
- 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."
- Sethares, William (2004). Tuning, Timbre, Spectrum, Scale, p.60. ISBN 1-85233-797-4. Scale step of 63.8 cents.
- Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p.1394. ISBN 0-520-07099-2.
- Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.