13edt: Difference between revisions
Contribution (talk | contribs) No edit summary |
changed notation system |
||
| Line 28: | Line 28: | ||
! Comments | ! Comments | ||
! Generator for... | ! Generator for... | ||
! [[ | ! [[4L 5s (3/1-equivalent)#Notation|BPS enneatonic notation]] (J = 1/1) | ||
|- | |||
| 0 | |||
| 0 | |||
| 0 | |||
| P1 | |||
| 1/1 | |||
| | |||
| | |||
| J | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 34: | Line 43: | ||
| 100 | | 100 | ||
| A1/m2 | | A1/m2 | ||
| 27/25 | | 27/25, 49/45 | ||
| | | | ||
| | | | ||
| | | K | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 46: | Line 55: | ||
| | | | ||
| [[Sirius]] | | [[Sirius]] | ||
| | | K#, Lb | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 55: | Line 64: | ||
| | | | ||
| [[Bohlen-Pierce|Linear BP]] | | [[Bohlen-Pierce|Linear BP]] | ||
| | | L | ||
|- | |- | ||
| 4 | | 4 | ||
| Line 64: | Line 73: | ||
| | | | ||
| [[Canopus]] | | [[Canopus]] | ||
| | | M | ||
|- | |- | ||
| 5 | | 5 | ||
| Line 73: | Line 82: | ||
| false 3/2 | | false 3/2 | ||
| false Father | | false Father | ||
| | | M#, Nb | ||
|- | |- | ||
| 6 | | 6 | ||
| Line 82: | Line 91: | ||
| | | | ||
| [[Arcturus]] | | [[Arcturus]] | ||
| | | N | ||
|- | |- | ||
| 7 | | 7 | ||
| Line 91: | Line 100: | ||
| | | | ||
| Arcturus | | Arcturus | ||
| | | O | ||
|- | |- | ||
| 8 | | 8 | ||
| Line 100: | Line 109: | ||
| false 2/1 | | false 2/1 | ||
| false Father | | false Father | ||
| | | O#, Pb | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 109: | Line 118: | ||
| | | | ||
| Canopus | | Canopus | ||
| | | P | ||
|- | |- | ||
| 10 | | 10 | ||
| Line 118: | Line 127: | ||
| | | | ||
| Linear BP | | Linear BP | ||
| | | Q | ||
|- | |- | ||
| 11 | | 11 | ||
| Line 127: | Line 136: | ||
| | | | ||
| Sirius | | Sirius | ||
| Q | | Q#, Rb | ||
|- | |- | ||
| 12 | | 12 | ||
| Line 133: | Line 142: | ||
| 1200 | | 1200 | ||
| M9/d10 | | M9/d10 | ||
| 25/9 | | 25/9, 135/49 | ||
| | | | ||
| | | | ||
Revision as of 18:08, 2 October 2024
| ← 12edt | 13edt | 14edt → |
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, and therefore has received by far the most attention among equal divisions of the tritave.
It provides an excellent approximation to the 3.5.7 subgroup, especially for its size, being comparable to 34edo's accuracy in the 5-limit. In this subgroup, 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.
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 no-2 3/1-equave-7-limit, 13edt maintains the smallest relative error of any EDT until 258edt and 271edt, and the smallest absolute error until 56edt.
Theory
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -29.57 | +0.00 | -59.13 | -6.53 | -29.57 | -3.83 | +57.61 | +0.00 | -36.10 | -54.80 | -59.13 |
| Relative (%) | -20.2 | +0.0 | -40.4 | -4.5 | -20.2 | -2.6 | +39.4 | +0.0 | -24.7 | -37.5 | -40.4 | |
| Steps (reduced) |
8 (8) |
13 (0) |
16 (3) |
19 (6) |
21 (8) |
23 (10) |
25 (12) |
26 (0) |
27 (1) |
28 (2) |
29 (3) | |
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | 33 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -6.53 | -3.83 | +0.00 | -54.80 | -51.40 | -6.53 | +69.39 | +23.14 | -3.83 | -15.02 | -13.07 | +0.00 | +22.59 | +53.44 | -54.80 |
| Relative (%) | +0.0 | -4.5 | -2.6 | +0.0 | -37.5 | -35.1 | -4.5 | +47.4 | +15.8 | -2.6 | -10.3 | -8.9 | +0.0 | +15.4 | +36.5 | -37.5 | |
| Steps (reduced) |
13 (0) |
19 (6) |
23 (10) |
26 (0) |
28 (2) |
30 (4) |
32 (6) |
34 (8) |
35 (9) |
36 (10) |
37 (11) |
38 (12) |
39 (0) |
40 (1) |
41 (2) |
41 (2) | |
Intervals
| Steps | Cents | Hekts | BP nonatonic degree | Corresponding JI intervals | Comments | Generator for... | BPS enneatonic notation (J = 1/1) |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | P1 | 1/1 | J | ||
| 1 | 146.3 | 100 | A1/m2 | 27/25, 49/45 | K | ||
| 2 | 292.6 | 200 | M2/d3 | 25/21 | Sirius | K#, Lb | |
| 3 | 438.9 | 300 | A2/P3/d4 | 9/7 | Linear BP | L | |
| 4 | 585.2 | 400 | A3/m4/d5 | 7/5 | Canopus | M | |
| 5 | 731.5 | 500 | M4/m5 | 75/49 | false 3/2 | false Father | M#, Nb |
| 6 | 877.8 | 600 | A4/M5 | 5/3 | Arcturus | N | |
| 7 | 1024.1 | 700 | A5/m6/d7 | 9/5 | Arcturus | O | |
| 8 | 1170.4 | 800 | M6/m7 | 49/25 | false 2/1 | false Father | O#, Pb |
| 9 | 1316.7 | 900 | A6/M7/d8 | 15/7 | Canopus | P | |
| 10 | 1463.0 | 1000 | P8/d9 | 7/3 | Linear BP | Q | |
| 11 | 1609.3 | 1100 | A8/m9 | 63/25 | Sirius | Q#, Rb | |
| 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
| Subgroup | Comma list | Mapping | Optimal Equave stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 3.5.7 | 245/243, 3125/3087 | [⟨13 19 23]] (b13) | +1.393 | 1.150 | 0.79 |
Rank-2 temperaments
| Periods per tritave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperament |
|---|---|---|---|---|
| 1 | 1\13 | 146.30 | 49/45 | Procyon |
| 1 | 2\13 | 292.61 | 25/21 | Sirius |
| 1 | 3\13 | 438.91 | 9/7 | BPS |
| 1 | 4\13 | 585.22 | 7/5 | Canopus |
| 1 | 5\13 | 731.63 | 75/49 | |
| 1 | 6\13 | 877.83 | 5/3 | Arcturus |
See also
- Bohlen-p_et
- Catalog of 3.5.7 subgroup rank two temperaments
- No-twos subgroup temperaments#3.5.7 subgroup temperaments
- 19ED5: relative ED5
- 23ED7: relative ED7