Delta scale
# 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'". 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 steps per octave.

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

(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:

    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)

| 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 |
