Английская Википедия:Alpha scale
12 TET: 300 cents Шаблон:Audio,
Alpha scale: 312 cents Шаблон:Audio
The Шаблон:Mvar (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 <math>\ \left( \tfrac{\ 3\ }{ 2 } \right)^{\tfrac{1}{9} }\ ,</math>[1] or by dividing the minor third (6:5) into four frequency ratio steps of <math>\ \left( \tfrac{\ 6\ }{ 5 } \right)^{\tfrac{1}{4} } ~.</math>[1][2][3]
The size of this scale step may also be precisely derived from using 9:5 Шаблон:BigBШаблон:Music, 1017.60 cents, Шаблон:AudioШаблон:Big to approximate the interval Шаблон:Math Шаблон:BigEШаблон:Music, 315.64 cents, Шаблон:Audio Шаблон:Big.[4]
- Carlos' Шаблон:Big (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 .
- <math>\ \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\ </math>
and <math>\ 0.06497082462 \times 1200 = 77.964989544\ </math> (Шаблон:Audio)
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 cents per octave.[4][5]
Though it does not have a perfect octave, the alpha scale produces "wonderful triads," (Шаблон:Audio and Шаблон:Audio) 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 Шаблон:Sfrac, i.e., [[septimal whole tone|Шаблон:Sfrac]] [[[:Шаблон:Audio]]]."[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
References
Шаблон:Microtonal music Шаблон:Musical tuning Шаблон:Scales Шаблон:Wendy Carlos