Alpha scale

Comparison of the alpha scale's approximations with the just values

Twelve-tone equal temperament vs. just

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 $\ \left({\tfrac {\ 3\ }{2}}\right)^{\tfrac {1}{9}}\ ,$ ^[1] or by dividing the minor third (6:5) into four frequency ratio steps of $\ \left({\tfrac {\ 6\ }{5}}\right)^{\tfrac {1}{4}}~.$ ^[1]^[2]^[3]

The size of this scale step may also be precisely derived from using 9:5 (B♭, 1017.60 cents, Play^ⓘ) to approximate the interval $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠ 3:2 / 5:4 ⁠ = 6:5$   (E♭, 315.64 cents, Play^ⓘ ).^[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 .

\ {\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 $\ 0.06497082462\times 1200=77.964989544\$ (Play^ⓘ)

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^ⓘ and minor triad^ⓘ) 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., ⁠87⁠ [Play]."^[1]

interval name	size (steps)	size (cents)	just ratio	just (cents)	error
septimal major second	3	233.89	8:7	231.17	+2.72
minor third	4	311.86	6:5	315.64	−3.78
major third	5	389.82	5:4	386.31	+3.51
perfect fifth	9	701.68	3:2	701.96	−0.27
harmonic seventh	octave−3	966.11	7:4	968.83	−2.72
octave	15	1169.47	2:1	1200.00	−30.53
octave	16	1247.44	2:1	1200.00	+47.44

^ ^a ^b ^c Carlos, Wendy (1989–1996). Three asymmetric divisions of the octave (Report). Archived from the original on 2017-07-12. Retrieved 2010-06-13 – via WendyCarlos.com. 9 steps to the perfect (no kidding) fifth." The alpha scale "splits the minor third exactly in half (also into quarters).
^ ^a ^b Milano, Dominic (November 1986). "A many-colored jungle of exotic tunings" (PDF). Keyboard. Archived (PDF) from the original on 2010-12-02. Retrieved 2010-06-13 – via wendycarlos.com. The idea was to split a minor third into two equal parts. Then that was divided again.
^ Carlos, Wendy (2000) [1986]. Beauty in the Beast (record liner notes). ESD 81552.
^ ^a ^b ^c Benson, Dave (2006). Music: A mathematical offering. Cambridge University Press. pp. 232–233. ISBN 0-521-85387-7. This actually differs very slightly from Carlos' figure of 15.385 $α$ -scale degrees to the octave. This is obtained by approximating the scale degree to 78.0 cents.
^ Sethares, W. (2004). Tuning, Timbre, Spectrum, Scale. Springer. p. 60. ISBN 1-85233-797-4. ... scale step of 78 cents.

[Three-1] Carlos, Wendy (1989–1996). Three asymmetric divisions of the octave (Report). Archived from the original on 2017-07-12. Retrieved 2010-06-13 – via WendyCarlos.com. 9 steps to the perfect (no kidding) fifth." The alpha scale "splits the minor third exactly in half (also into quarters).

[Milano-2] Milano, Dominic (November 1986). "A many-colored jungle of exotic tunings" (PDF). Keyboard. Archived (PDF) from the original on 2010-12-02. Retrieved 2010-06-13 – via wendycarlos.com. The idea was to split a minor third into two equal parts. Then that was divided again.

[Liner-3] Carlos, Wendy (2000) [1986]. Beauty in the Beast (record liner notes). ESD 81552.

[Benson-4] Benson, Dave (2006). Music: A mathematical offering. Cambridge University Press. pp. 232–233. ISBN 0-521-85387-7. This actually differs very slightly from Carlos' figure of 15.385 $α$ -scale degrees to the octave. This is obtained by approximating the scale degree to 78.0 cents.

[5] Sethares, W. (2004). Tuning, Timbre, Spectrum, Scale. Springer. p. 60. ISBN 1-85233-797-4. ... scale step of 78 cents.

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