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 6:5Playⓘ.
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]
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]
^Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
^ abcBenson, Dave (2006). Music: A Mathematical Offering, p.232-233. ISBN0-521-85387-7. "Carlos has 18.809 β-scale degrees to the octave, corresponding to a scale degree of 63.8 cents."
