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)Шаблон:Sup ≈ 63.8 cents]. It may be approximated by splitting the perfect fourth (4:3) into two equal parts [(4:3)Шаблон:Sup],[1] or eight equal parts [(4:3)Шаблон:Sup = 64 cents],[2] totaling approximately 18.8 steps per octave.
<math>\frac{11\log_2{(3/2)}+6\log_2{(5/4)}+5\log_2{(6/5)}}{11^2+6^2+5^2}=0.05319411048</math> and <math>0.05319411048\times1200=63.832932576</math> (Шаблон:Audio)
Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, Шаблон:Audio is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[3][4]Шаблон:Audio though both have nice triads[1] (Шаблон:Audio, Шаблон:Audio, and Шаблон:Audio). "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.
↑ 3,03,1Benson, Dave (2006). Music: A Mathematical Offering, p.232-233. Шаблон:ISBN. "Carlos has 18.809 β-scale degrees to the octave, corresponding to a scale degree of 63.8 cents."
