Delta scale

Minor second Template:Audio

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'". As such it would split the minor second (presumably 16:15) into eight equal parts of approximately 14 cents each Template: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, ATemplate:MusicTemplate:Music, Template:Audio) to approximate the interval Template:Frac, which equals 6:5 (ETemplate:Music, 315.64 cents, Template:Audio). Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave.

$\begin{array}{c}\frac{50{\log }_2\left(\frac{3}{2}\right)+28{\log }_2\left(\frac{5}{4}\right)+23{\log }_2\left(\frac{6}{5}\right)}{50^2+28^2+23^2}=0.0116212701\\ 0.0116212701\times 1200=13.9455241627\end{array}$

(Template:Audio)

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

1:1 Template:Audio

25:21 Template:Audio

9:7 Template:Audio

75:49 Template:Audio

5:3 Template:Audio

9:5 Template:Audio

15:7 Template:Audio

7:3 Template:Audio

25:9 Template:Audio

3:1 Template: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

• 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. Template:In lang

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