Theβ (beta)scale is a non-octave-repeatingmusical scale invented byWendy Carlos and first used on her albumBeauty in the Beast (1986). It is derived from approximatingjust intervals using multiples of a single interval without, as is standard inequal temperaments, requiring anoctave (2:1). It may be approximated by splitting theperfect fifth (3:2) into eleven equal parts [(3:2)1⁄11 ≈ 63.8 cents]. It may be approximated by splitting theperfect 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 peroctave.
Thescale step may also precisely be derived from using11:6 (B↑♭-, 1049.36 cents,Playⓘ) to approximate the interval3:2⁄5:4,[3] which equals6:5Playⓘ.
In order to make the approximation as good as possible we minimize themean 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]
Although neither has an octave, one advantage to the beta scale over thealpha scale is that 15 steps, 957.494 cents,Playⓘ is a reasonable approximation to theseventh harmonic (7:4, 968.826 cents)[3][4]Playⓘ though both have nicetriads[1] (Play major triadⓘ,minor triadⓘ, anddominant seventhⓘ). "According to Carlos, beta has almost the same properties as the alpha scale, except that the sevenths are slightly more in tune."[1]
Thedelta 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 |
