1547edo
| ← 1546edo | 1547edo | 1548edo → |
1547 equal divisions of the octave (abbreviated 1547edo or 1547ed2), also called 1547-tone equal temperament (1547tet) or 1547 equal temperament (1547et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1547 equal parts of about 0.776 ¢ each. Each step represents a frequency ratio of 21/1547, or the 1547th root of 2.
Theory
1547edo is consistent to the 15-odd-limit and is excellent in the 7-limit. It tempers out 4375/4374 and it is a member of the optimal ET sequence for the rank-3 temperament associated with this comma.
In the 5-limit, it supports gross, which is a very high-accuracy temperament. The 118-tone maximal evenness scale produced by gross is concoctic, since it uses 118\1547 as the generator. In addition, 1547edo tempers out the septendecima and thus supports the chlorine temperament in 5-limit and also in the 7-limit. 1547edo tempers out the 5-limit comma [236 -61 -60⟩, thus associating a stack of sixty 15/8's with 4/3, and sixty-one of them make 5/4.
In the 7-limit, it provides the optimal patent val for 7-limit brahmagupta, the 441 & 1106 temperament, and supports an alternative 11-limit extension to it. It also supports semidimi, the 171 & 1376 temperament.
In the 11-limit, 1547edo provides the optimal patent val for the aluminium temperament, which maps 135/128 to 1/13th of the occtave. It also tempers out 117649/117612, and is a tuning for the rank-3 temperament heimdall. In higher limits, it supports 91th-octave temperament protactinium.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.049 | -0.018 | +0.017 | +0.201 | +0.326 | -0.237 | +0.354 | +0.039 | -0.230 | -0.110 |
| Relative (%) | +0.0 | +6.3 | -2.3 | +2.2 | +25.9 | +42.0 | -30.5 | +45.6 | +5.0 | -29.7 | -14.2 | |
| Steps (reduced) |
1547 (0) |
2452 (905) |
3592 (498) |
4343 (1249) |
5352 (711) |
5725 (1084) |
6323 (135) |
6572 (384) |
6998 (810) |
7515 (1327) |
7664 (1476) | |
Subsets and supersets
Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos 7, 13, 17, 91, 119, and 221.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2452 -1547⟩ | [⟨1547 2542]] | −0.015 | 0.015 | 1.99 |
| 2.3.5 | [-52 -17 34⟩, [40 -56 21⟩ | [⟨1547 2542 3592]] | −0.008 | 0.017 | 2.14 |
| 2.3.5.7 | 4375/4374, [-1 4 11 -11⟩, [46 -14 -3 -6⟩ | [⟨1547 2542 3592 4343]] | −0.007 | 0.014 | 1.86 |
| 2.3.5.7.11 | 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 | [⟨1547 2542 3592 4343 5352]] | −0.017 | 0.024 | 3.10 |
| 2.3.5.7.11.13 | 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 | [⟨1547 2542 3592 4343 5352 5725]] | −0.029 | 0.034 | 4.42 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 118\1547 | 91.532 | [9 -32 18⟩ | Gross |
| 1 | 579\1547 | 449.127 | 35/27 | Semidimi |
| 7 | 670\1547 (7\1547) |
519.715 (5.429) |
27/20 (325/324) |
Brahmagupta (7-limit) |
| 7 | 11\1547 | 8.533 | 1029/1024 | Nitrogen |
| 13 | 642\1547 (47\1547) |
497.996 (36.458) |
4/3 (?) |
Aluminium |
| 17 | 321\1547 (48\1547) |
248.998 (37.233) |
[-23 5 9 -2⟩ (100352/98415) |
Chlorine |
| 91 | 642\1547 (13\1547) |
497.996 (10.084) |
4/3 (176/175) |
Protactinium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct