13edt
13 equal divisions of the tritave (13edt) is the nonoctave tuning system derived by dividing the tritave (3/1) into 13 equal steps of 146.3 cents each, or the thirteenth root of 3. It is best known as the equal-tempered version of the Bohlen-Pierce scale.
| ← 12edt | 13edt | 14edt → |
13edt can be described as approximately 8.202edo. This implies that each step of 13edt can be approximated by 5 steps of 41edo.
In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as bohpier temperament. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 (26edt, 39edt and 52edt) come to the fore.
Theory
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -29.57 | +0.00 | -59.13 | -6.53 | -29.57 | -3.83 | +57.61 | +0.00 | -36.10 | -54.80 | -59.13 | -51.40 | -33.39 | -6.53 | +28.04 |
| Relative (%) | -20.2 | +0.0 | -40.4 | -4.5 | -20.2 | -2.6 | +39.4 | +0.0 | -24.7 | -37.5 | -40.4 | -35.1 | -22.8 | -4.5 | +19.2 | |
| Steps (reduced) |
8 (8) |
13 (0) |
16 (3) |
19 (6) |
21 (8) |
23 (10) |
25 (12) |
26 (0) |
27 (1) |
28 (2) |
29 (3) |
30 (4) |
31 (5) |
32 (6) |
33 (7) | |
Intervals
| Steps | Cents | Hekts | BP nonatonic degree | Corresponding JI intervals | Comments | Generator for... | Arcturus nonatonic notation (J = 1/1) |
|---|---|---|---|---|---|---|---|
| 1 | 146.3 | 100 | A1/m2 | 27/25~49/45 | J# | ||
| 2 | 292.6 | 200 | M2/d3 | 25/21 | Sirius | Kb | |
| 3 | 438.9 | 300 | A2/P3/d4 | 9/7 | Linear BP | K | |
| 4 | 585.2 | 400 | A3/m4/d5 | 7/5 | Canopus | K#, Lb | |
| 5 | 731.5 | 500 | M4/m5 | 75/49 | false 3/2 | false Father | L |
| 6 | 877.8 | 600 | A4/M5 | 5/3 | Arcturus | M | |
| 7 | 1024.1 | 700 | A5/m6/d7 | 9/5 | Arcturus | N | |
| 8 | 1170.4 | 800 | M6/m7 | 49/25 | false 2/1 | false Father | N#, Ob |
| 9 | 1316.7 | 900 | A6/M7/d8 | 15/7 | Canopus | O | |
| 10 | 1463.0 | 1000 | P8/d9 | 7/3 | Linear BP | P | |
| 11 | 1609.3 | 1100 | A8/m9 | 63/25 | Sirius | Q | |
| 12 | 1755.7 | 1200 | M9/d10 | 25/9~135/49 | R | ||
| 13 | 1902.0 | 1300 | A9/P10 | 3/1 | Tritave | J |
JI approximation
Regular temperament properties
See also
- Bohlen-p_et
- Catalog of 3.5.7 subgroup rank two temperaments
- No-twos subgroup temperaments#3.5.7 subgroup
- 19ED5: relative ED5
- 23ED7: relative ED7