Delta scale

Theδ(delta)scaleis a non-octave repeatingmusical scale.It may be regarded as thebeta scale's reciprocal, since it is "as far 'down' the (0 3 6 9) circle fromαas β is 'up' ".As such it would split theminor second(presumably 16:15) into eight equal parts of approximately 14 cents each.This would total approximately 85.7 steps per octave.

Minor second

Chromatic circle

The delta scale's approximations compared with the just values

Twelve-tone equal temperament vs. just

Thescale stepmay also precisely be derived from using 50:28 (25:14, 1003.8 cents, A♯,) to approximate the interval3:2⁄5:4,which equals6:5(E♭,315.64 cents,). Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave.

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

()

TheBohlen–Piercedelta scale is based on thetritaveand the 7:5:3 "wide"triad() and the 9:7:5 "narrow" triad () (rather than the conventional 4:5:6 triad). Notes include:

1:1

25:21

9:7

75:49

5:3

9:5

15:7

7:3

25:9

3:1

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.(in German)