Alpha scale

The α (alpha) 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, but without requiring (as temperaments normally do) an octave (2:1). It may be approximated by dividing the perfect fifth (3:2) into nine equal steps, with frequency ratio ${\displaystyle \ \left({\tfrac {\ 3\ }{2}}\right)^{\tfrac {1}{9}}\ ,}${\displaystyle \ \left({\tfrac {\ 3\ }{2}}\right)^{\tfrac {1}{9}}\ ,}^[1] or by dividing the minor third (6:5) into four frequency ratio steps of ${\displaystyle \ \left({\tfrac {\ 6\ }{5}}\right)^{\tfrac {1}{4}}~.}${\displaystyle \ \left({\tfrac {\ 6\ }{5}}\right)^{\tfrac {1}{4}}~.}^{[1] [2] [3]}

The size of this scale step may also be precisely derived from using 9:5 (B♭, 1017.60 cents, Play ⁱ ) to approximate the interval ⁠ 3:2 / 5:4 ⁠ = 6:5 (E♭, 315.64 cents, Play ⁱ ).^[4]

Carlos' α (alpha) scale arises from ... taking a value for the scale degree so that nine of them approximate a 3:2 perfect fifth, five of them approximate a 5:4 major third, and four of them approximate a 6:5 minor third. In order to make the approximation as good as possible we minimize the mean square deviation.^[4]

The formula below finds the minimum by setting the derivative of the mean square deviation with respect to the scale step size to 0 .

${\displaystyle \ {\frac {\ 9\ \log _{2}\left({\frac {\ 3\ }{2}}\right)+5\log _{2}\left({\frac {\ 5\ }{4}}\right)+4\ \log _{2}\left({\frac {\ 6\ }{5}}\right)\ }{\ 9^{2}+5^{2}+4^{2}\ }}\approx 0.06497082462\ }${\displaystyle \ {\frac {\ 9\ \log _{2}\left({\frac {\ 3\ }{2}}\right)+5\log _{2}\left({\frac {\ 5\ }{4}}\right)+4\ \log _{2}\left({\frac {\ 6\ }{5}}\right)\ }{\ 9^{2}+5^{2}+4^{2}\ }}\approx 0.06497082462\ }

and ${\displaystyle \ 0.06497082462\times 1200=77.964989544\ }${\displaystyle \ 0.06497082462\times 1200=77.964989544\ } (Play ⁱ )

At 78 cents per step, this totals approximately 15.385 steps per octave, however, more accurately, the alpha scale step is 77.965 cents and there are 15.3915 steps per octave.^{[4] [5]}

Though it does not have a perfect octave, the alpha scale produces "wonderful triads," (Play major ⁱ and minor triad ⁱ ) and the beta scale has similar properties but the sevenths are more in tune.^[2] However, the alpha scale has

"excellent harmonic seventh chords ... using the [octave] inversion of ⁠ 7 / 4 ⁠, i.e., ⁠8/7⁠ [Play ⁱ ]."^[1]

• Bohlen–Pierce scale
• Beta scale
• Gamma scale
• Delta scale
• Harmonic scale

• Equal temperaments
• Non–octave-repeating scales
• Wendy Carlos
• Music theory stubs

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