Delta scale

Minor second Playⓘ
Chromatic circle
The delta scale's approximations compared with the just values
Twelve-tone equal temperament vs. just

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 Playⓘ. 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, A7 upside-down♯, Playⓘ) to approximate the interval 3:2⁄5:4, which equals 6:5 (E♭, 315.64 cents, Playⓘ). Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave.

50 log 2 ⁡ ( 3 2 ) + 28 log 2 ⁡ ( 5 4 ) + 23 log 2 ⁡ ( 6 5 ) 50 2 + 28 2 + 23 2 = 0.011 621 2701 0.011 621 2701 × 1200 = 13.945 524 1627 {\displaystyle {\begin{aligned}{\frac {50\log _{2}{\left({\frac {3}{2}}\right)}+28\log _{2}{\left({\frac {5}{4}}\right)}+23\log _{2}{\left({\frac {6}{5}}\right)}}{50^{2}+28^{2}+23^{2}}}=0.011\,621\,2701\\0.011\,621\,2701\times 1200=13.945\,524\,1627\end{aligned}}}

(Playⓘ)

The Bohlen–Pierce delta scale is based on the tritave and the 7:5:3 "wide" triad (Playⓘ) and the 9:7:5 "narrow" triad (Playⓘ) (rather than the conventional 4:5:6 triad). Notes include:[2]

1:1 Playⓘ
25:21 Playⓘ
9:7 Playⓘ
75:49 Playⓘ
5:3 Playⓘ
9:5 Playⓘ
15:7 Playⓘ
7:3 Playⓘ
25:9 Playⓘ
3:1 Playⓘ
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

  1. ^ Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p. 1394. ISBN 978-0-520-07099-8.
  2. ^ "What about BP tonality?", The Bohlen-Pierce Site.

Further reading

  • Bohlen, Heinz: "13 Tonstufen in der Duodezime", Acustica, vol. 39 no. 2, S. Hirzel Verlag, Stuttgart, 1978, pp. 76–86. (in German)
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