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, Audio file "Neutral seventh on C.mid" not found) to approximate the interval 3:2⁄5:4,[3] which equals 6:5 Audio file "Just minor third on C.mid" not found.
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 (Audio file "Beta scale step on C.mid" not found)
Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, Audio file "Beta scale 15 steps on C.mid" not found is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[3][4] Audio file "Harmonic seventh on C.mid" not found though both have nice triads[1] (Audio file "Beta scale major triad on C.mid" not found, Audio file "Beta scale minor triad on C.mid" not found, and Audio file "Beta scale dominant seventh on C.mid" not found). "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 |
See also
[edit | edit source]References
[edit | edit source]- ^ 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. Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).. "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. Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).. Scale step of 63.8 cents.
- ^ Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p.1394. Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)..
External links
[edit | edit source]- Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.
