Delta scale
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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 . 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, A♯, ) to approximate the interval 3:2⁄5:4, which equals 6:5 (E♭, 315.64 cents, ). Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave.
(
)The Bohlen–Pierce delta scale is based on the tritave and the 7:5:3 "wide" triad ( ) and the 9:7:5 "narrow" triad ( ) (rather than the conventional 4:5:6 triad). Notes include:[2]
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
[edit]References
[edit]- ^ Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p. 1394. ISBN 978-0-520-07099-8.
- ^ "What about BP tonality?", The Bohlen-Pierce Site.
Further reading
[edit]- Bohlen, Heinz: "13 Tonstufen in der Duodezime", Acustica, vol. 39 no. 2, S. Hirzel Verlag, Stuttgart, 1978, pp. 76–86. (in German)