Chromatic circle

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)^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^ⓘ) to approximate the interval 3:2⁄5:4,^[3] which equals 6:5 Play^ⓘ.

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^ⓘ)

Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, Play^ⓘ is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)^[3][4] Play^ⓘ though both have nice triads^[1] (Play major triad^ⓘ, minor triad^ⓘ, and dominant 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 α as β is 'up'."^[5]