Beta scale
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 i ) to approximate the interval 3:2⁄5:4 ,[3] which equals 6:5 Play i .
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]
{\displaystyle {\frac {11\log _{2}{(3/2)}+6\log _{2}{(5/4)}+5\log _{2}{(6/5)}}{11^{2}+6^{2}+5^{2}}}=0.05319411048} and {\displaystyle 0.05319411048\times 1200=63.832932576} (Play i )
Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, Play i is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[3] [4] Play i though both have nice triads [1] (Play major triad i , minor triad i , and dominant seventh i ). "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]
(steps) size
(cents) just ratio just
(cents) error
See also
[edit ]References
[edit ]- ^ a b c Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
- ^ Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
- ^ 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."
- ^ 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.
External links
[edit ]- Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.