Beta Scale
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Beta Scale
Perfect fourth (just: 498.04 cents About this soundPlay , 12-tet: 500 cents About this soundPlay , Beta scale: 512 cents About this soundPlay )
Comparing the beta scale's approximations with the just values
Twelve-tone equal temperament vs. just

The ? (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?63.8 cents). It may be approximated by splitting the perfect fourth (4:3) into two equal parts (4:3),[1] or eight equal parts ((4:3=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, About this soundPlay ) to approximate the interval ​,[3] which equals 6:5 About this soundPlay .

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 (About this soundPlay )

Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, About this soundPlay  is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[3][4]About this soundPlay  though both have nice triads[1] (About this soundPlay major triad , About this soundminor triad , and About this sounddominant 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 ? as ? is 'up'."[5]

interval name size
(steps)
size
(cents)
just ratio just
(cents)
error
minor third 5 319.00 6:5 315.64 +3.35
major third 6 382.80 5:4 386.31 −3.52
perfect fifth 11 701.79 3:2 701.96 −0.16
harmonic seventh 15 956.99 7:4 968.83 −11.84

See also

Sources

  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 ?-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.

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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