Delta Scale
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Delta Scale
The delta scale's approximations compared with the just values
Twelve-tone equal temperament vs. just

The ? (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 ? as ? is 'up.'"[1] As such it would split the minor second (presumably 16:15) into eight equal parts of approximately 14 cents each  . 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,  ) to approximate the interval ​, which equals 6:5 (E, 315.64 cents,  ). 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}}}

( )

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

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

## Sources

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.