Bohlen-Pierce
The Bohlen-Pierce system is a non-octave tuning system of (exact or well-tempered) 13 equal divisions of the perfect twelfth (or tritave). In the Bohlen-Pierce system, the tritave is generally seen as the interval of equivalence, and harmony emphasizes the odd harmonics (such as the chord 3:5:7:9). It is the smallest EDT that has a tuning of lambda, the 9-note scale of the sensamagic temperament (which serves an analogous role to meantone in tritave-based harmony).
General theory
JI approximation
13edt's 2/1 is flat about 30 cents, making it most prominently a 3.5.7 subgroup temperament, although it also contains approximations of 19, 23, and 29. 7/3 (the tritave-reduced 7th harmonic) is flattened by a small amount, so that a sharpened 9/7 stacks twice to a flattened 5/3, as in sensamagic temperament. As a full 7-limit temperament, it supports sensi, albeit with a severely flattened octave.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -29.6 | +0.0 | -6.5 | -3.8 | -54.8 | -51.4 | +69.4 | +23.2 | -15.0 | +22.7 | +53.5 |
| Relative (%) | -20.2 | +0.0 | -4.4 | -2.6 | -37.4 | -35.1 | +47.5 | +15.8 | -10.2 | +15.5 | +36.6 | |
| Steps (reduced) |
8 (-0.202) |
13 (4.798) |
19 (2.596) |
23 (6.596) |
28 (3.394) |
30 (5.394) |
34 (1.192) |
35 (2.192) |
37 (4.192) |
40 (7.192) |
41 (8.192) | |
Note: Due to a bug with the template, the step counts are octave-reduced instead of tritave-reduced.
Edtstep interpretations
The step of 13edt can be interpreted as the following ratios in the 3.5.7 subgroup.
- 49/45
- 27/25
Multiples
65edt
See 41edo.
