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fixed error in math notation
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clarified the formula. Music: A Mathematical Offering also explains this process so not original research
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The scale step may also be precisely derived from using [[minor seventh|9:5]] (B{{music|b}}, 1017.60 cents, {{audio|Greater just minor seventh on C.mid|Play}}) to approximate the interval {{frac|3:2|[[major third|5:4]]}} (=6:5, E{{music|b}}, 315.64 cents, {{audio|Just minor third on C.mid|Play}}).<ref name="Benson"/>
The scale step may also be precisely derived from using [[minor seventh|9:5]] (B{{music|b}}, 1017.60 cents, {{audio|Greater just minor seventh on C.mid|Play}}) to approximate the interval {{frac|3:2|[[major third|5:4]]}} (=6:5, E{{music|b}}, 315.64 cents, {{audio|Just minor third on C.mid|Play}}).<ref name="Benson"/>
{{Quote|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]].<ref name="Benson"/>}}
{{Quote|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]].<ref name="Benson"/>}}

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


<math>\frac{9\log_2(3/2)+5\log_2(5/4)+4\log_2(6/5)}{9^2+5^2+4^2}\approx0.06497082462</math> and <math>0.06497082462\times1200=77.965</math> ({{audio|Alpha scale step on C.mid|Play}})
<math>\frac{9\log_2(3/2)+5\log_2(5/4)+4\log_2(6/5)}{9^2+5^2+4^2}\approx0.06497082462</math> and <math>0.06497082462\times1200=77.965</math> ({{audio|Alpha scale step on C.mid|Play}})

Revision as of 01:36, 10 February 2021

Minor third (just: 315.64 cents Playⓘ,
12-tet: 300 cents Playⓘ,
Alpha scale: 312 cents Playⓘ
Chromatic circle
Comparing 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 without, as is standard in equal temperaments, requiring an octave (2:1). It may be approximated by dividing the perfect fifth (3:2) into nine equal steps (3:2)1⁄9,[1] or by dividing the minor third (6:5) into four steps (6:5)1⁄4.[1][2][3]

The scale step may also be precisely derived from using 9:5 (B♭, 1017.60 cents, Playⓘ) to approximate the interval 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.

and (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 per octave.[4][5]

Though it does not have an 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., 8⁄7 [Playⓘ]."[1]

interval name size
(steps) 		size
(cents) 		just ratio just
(cents) 		error
septimal major second 3 233.90 8:7 231.17 +2.72
major third 5 389.83 5:4 386.31 +3.51
perfect fifth 9 701.69 3:2 701.96 −0.27
harmonic seventh 12 935.58 7:4 968.83 −33.25
octave 15 1169.48 2:1 1200.00 −30.52
octave 16 1247.44 2:1 1200.00 +47.44

See also

Sources

  1. ^ a b c Carlos, Wendy (1989–96). "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)."
  2. ^ a b Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard. "The idea was to split a minor third into two equal parts. Then that was divided again."
  3. ^ Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
  4. ^ a b c Benson, Dave (2006). Music: A Mathematical Offering, p.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, William (2004). Tuning, Timbre, Spectrum, Scale, p.60. ISBN 1-85233-797-4. Scale step of 78 cents.
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