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)Template:Sup ≈ 63.8 cents]. It may be approximated by splitting the perfect fourth (4:3) into two equal parts [(4:3)Template:Sup],[1] or eight equal parts [(4:3)Template:Sup = 64 cents],[2] totaling approximately 18.8 steps per octave.
The scale step may also precisely be derived from using 11:6 (BTemplate:MusicTemplate:MusicTemplate:Music, 1049.36 cents, Template:Audio) to approximate the interval Template:Frac,[3] which equals 6:5 Template:Audio. Template:Blockquote
and (Template:Audio)
Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, Template:Audio is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[3][4] Template:Audio though both have nice triads[1] (Template:Audio, Template:Audio, and Template:Audio). "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
References
External links
- Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.
Template:Microtonal music Template:Musical tuning Template:Scales Template:Wendy Carlos
- ↑ 1.0 1.1 1.2 Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
- ↑ Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
- ↑ 3.0 3.1 Benson, Dave (2006). Music: A Mathematical Offering, p.232-233. Template:ISBN. "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. Template:ISBN. Scale step of 63.8 cents.
- ↑ Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p.1394. Template:ISBN.