Alpha scale explained

The (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, to approximate the interval E, 315.64 cents, .

Carlos' (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.

The formula below finds the minimum by setting the derivative of the mean square deviation with respect to the

(9 log _{2 \left(} (3)/(2) \right) + 5log _{2\left(} (5)/(4) \right) + 4 log _{2\left(} (6)/(5) \right))/(9 ^{2} +5 ^{2} +4 ^{2}) ≈ 0 . 06497082462

and

0 . 06497082462 x 1200 = 77 . 964989544

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," (and) 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, i.e., [{{audio|Alpha scale harmonic seventh chord on C.mid|Play}}]."^[1]

align=center bgcolor="#ffffb4"	interval name	align=center bgcolor="#ffffb4"	size (steps)	align=center bgcolor="#ffffb4"	size (cents)	align=center bgcolor="#ffffb4"	just ratio	align=center bgcolor="#ffffb4"	just (cents)	align=center bgcolor="#ffffb4"	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

See also

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

Notes and References

1. Carlos . Wendy . Wendy Carlos . 1989–1996 . Three asymmetric divisions of the octave . WendyCarlos.com . 9 steps to the perfect (no kidding) fifth." The alpha scale "splits the minor third exactly in half (also into quarters). . 2010-06-13 . 2017-07-12 . https://web.archive.org/web/20170712065803/http://www.wendycarlos.com/resources/pitch.html . live .
2. Milano . Dominic . November 1986 . A many-colored jungle of exotic tunings . wendycarlos.com . Keyboard . The idea was to split a minor third into two equal parts. Then that was divided again. . 2010-06-13 . 2010-12-02 . https://web.archive.org/web/20101202134950/http://wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf . live .
3. Carlos . Wendy . Wendy Carlos . 2000 . 1986 . record liner notes . Beauty in the Beast . ESD 81552.
4. Book: Benson, Dave . 2006 . Music: A mathematical offering . 232–233 . Cambridge University Press . 0-521-85387-7 . This actually differs very slightly from Carlos' figure of 15.385 -scale degrees to the octave. This is obtained by approximating the scale degree to 78.0 cents..
5. Book: Sethares, W. . William Sethares . 2004 . Tuning, Timbre, Spectrum, Scale . 60 . Springer . 1-85233-797-4 . ... scale step of 78 cents..