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Beta scale

## Summary

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

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)111 ≈ 63.8 cents]. It may be approximated by splitting the perfect fourth (4:3) into two equal parts [(4:3)12],[1] or eight equal parts [(4:3)18 = 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,  ) to approximate the interval 3:25:4,[3] which equals 6:5  .

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 ${\displaystyle 0.05319411048\times 1200=63.832932576}$ ( )

Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents,   is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[3][4]   though both have nice triads[1] ( ,  , and  ). "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 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