Alpha scale
Play (help·info),12 TET: 300 cents
Play (help·info),Alpha scale: 312 cents
Play (help·info)
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 [1] or by dividing the minor third (6:5) into four frequency ratio steps of [1][2][3]
The size of this scale step may also be precisely derived by putting the perfect fifth and the major third in a 9:5 ratio (not to be confused with the interval 9/5).[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 .
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 (help·info) and minor triad (help·info)) 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 (help·info)]."[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 |
See also
- Bohlen–Pierce scale
- Beta scale
- Gamma scale
- Delta scale
- Harmonic scale
References
- ↑ 1.0 1.1 1.2 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).
Unknown parameter|url-status=ignored (help) - ↑ 2.0 2.1 Milano, Dominic (November 1986). "A many-colored jungle of exotic tunings" (PDF). Keyboard. Archived from the original (PDF) 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.
Unknown parameter|url-status=ignored (help) - ↑ Carlos, Wendy (2000) [1986]. Beauty in the Beast (record liner notes). ESD 81552.
- ↑ 4.0 4.1 4.2 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.
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- ↑ Sethares, W. (2004). Tuning, Timbre, Spectrum, Scale. Springer. p. 60. ISBN 1-85233-797-4.
... scale step of 78 cents.
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