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 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, A

♯, 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.

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

(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

• Alpha scale
• Beta scale
• Gamma scale

References

Further reading

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