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
Intervals and notation
The "Intervals represented" column reflects the just (well-tempered) tuning of Bohlen-Pierce.
| Steps | Cents | Intervals represented | Note name (Lambda) | Name (Lambda) | Interval category (ADIN*) |
|---|---|---|---|---|---|
| 0 | 0 | 1/1 | A | Unison | Unison |
| 1 | 146.3 | 27/25 | A#, Bb | Minor second | Neutral second |
| 2 | 292.6 | 25/21 | B | Major second | Farminor third |
| 3 | 438.9 | 9/7 | C | Perfect third | Supermajor third |
| 4 | 585.2 | 7/5 | C#, Db | Minor fourth | Nearaugmented fourth |
| 5 | 731.5 | 75/49 | D | Major fourth, minor fifth | Superfifth |
| 6 | 877.8 | 5/3 | E | Major fifth | Nearmajor sixth |
| 7 | 1024.1 | 9/5 | F | Minor sixth | Nearminor seventh |
| 8 | 1170.4 | 49/25 | F#, Gb | Major sixth, minor seventh | Suboctave |
| 9 | 1316.7 | 15/7 | G | Major seventh | Nearminor ninth |
| 10 | 1463.0 | 7/3 | H | Perfect eighth | Subminor tenth |
| 11 | 1609.3 | 63/25 | H#, Jb | Minor ninth | Farmajor tenth |
| 12 | 1755.7 | 25/9 | J | Major ninth | Neutral eleventh |
| 13 | 1902.0 | 3/1 | A | Tritave | Perfect twelfth |
*As a subset of 41edo
Multiples
39edt
39edt, sometimes known as Triple Bohlen-Pierce, additionally adds approximations of the 11th and 13th harmonics to 13edt.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +19.2 | +0.0 | -6.5 | -3.8 | -6.0 | -2.6 | +20.7 | +23.2 | -15.0 | +22.7 | +4.7 |
| Relative (%) | +39.4 | +0.0 | -13.3 | -7.8 | -12.3 | -5.3 | +42.4 | +47.5 | -30.7 | +46.5 | +9.7 | |
| Steps (reduced) |
25 (0.394) |
39 (14.394) |
57 (7.788) |
69 (19.788) |
85 (11.182) |
91 (17.182) |
101 (2.576) |
105 (6.576) |
111 (12.576) |
120 (21.576) |
122 (23.576) | |
65edt
13edt is approximately 8.2edo. Dividing each step into fifths to obtain a more accurate octave results in a slightly detuned version of 41edo.
Bohpier temperament
Bohpier temperament is somewhat analogous to blackwood or compton temperament, but based on 13edt, adding the octave as a separate generator (in this case, the period). It is supported by 41edo.
