Английская Википедия:Delta scale
The δ (delta) scale is a non-octave repeating musical scale. It may be regarded as the beta scale's reciprocal, since it is "as far 'down' the (0 3 6 9) circle from α as β is 'up'".[1] As such it would split the minor second (presumably 16:15) into eight equal parts of approximately 14 cents each Шаблон:Audio. This would total approximately 85.7 steps per octave.
The scale step may also precisely be derived from using 50:28 (25:14, 1003.8 cents, AШаблон:MusicШаблон:Music, Шаблон:Audio) to approximate the interval Шаблон:Frac, which equals 6:5 (EШаблон:Music, 315.64 cents, Шаблон:Audio). Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave.
- <math>\begin{align}
\frac{50\log_2{\left(\frac32\right)} + 28\log_2{\left(\frac54\right)} + 23\log_2{\left(\frac65\right)}}{50^2+28^2+23^2} = 0.011\,621\,2701 \\
0.011\,621\,2701 \times 1200 = 13.945\,524\,1627
\end{align}</math>
The Bohlen–Pierce delta scale is based on the tritave and the 7:5:3 "wide" triad (Шаблон:Audio) and the 9:7:5 "narrow" triad (Шаблон:Audio) (rather than the conventional 4:5:6 triad). Notes include:[2]
- 1:1 Шаблон:Audio
- 25:21 Шаблон:Audio
- 9:7 Шаблон:Audio
- 75:49 Шаблон:Audio
- 5:3 Шаблон:Audio
- 9:5 Шаблон:Audio
- 15:7 Шаблон:Audio
- 7:3 Шаблон:Audio
- 25:9 Шаблон:Audio
- 3:1 Шаблон:Audio
| interval name | size (steps) |
size (cents) |
just ratio | just (cents) |
error |
| minor third | 23 | 321.23 | 6:5 | 315.64 | +5.59 |
| major third | 28 | 391.06 | 5:4 | 386.31 | +4.75 |
| perfect fifth | 50 | 698.32 | 3:2 | 701.96 | −3.63 |
See also
References
Further reading
- Bohlen, Heinz: "13 Tonstufen in der Duodezime", Acustica, vol. 39 no. 2, S. Hirzel Verlag, Stuttgart, 1978, pp. 76–86. Шаблон:In lang
Шаблон:Microtonal music Шаблон:Musical tuning Шаблон:Scales
- ↑ Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p. 1394. Шаблон:ISBN.
- ↑ "What about BP tonality?", The Bohlen-Pierce Site.
