Delta scale

Minor second Play (help·info)

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 (help·info). 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♯, Play (help·info)) to approximate the interval 3:2⁄5:4, which equals 6:5 (E♭, 315.64 cents, Play (help·info)). Thus the step is approximately 13.946 cents and there are 86.049 per octave.

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

(Play (help·info))

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

1:1 Play (help·info)

25:21 Play (help·info)

9:7 Play (help·info)

75:49 Play (help·info)

5:3 Play (help·info)

9:5 Play (help·info)

15:7 Play (help·info)

7:3 Play (help·info)

25:9 Play (help·info)

3:1 Play (help·info)

See also[]

• Alpha scale
• Beta scale
• Gamma scale

Further reading[]

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

Sources[]

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