1547edo: Difference between revisions
→Rank-2 temperaments: missed a monzo there |
→Rank-2 temperaments: based on what 1\364 and 13\1547 map to |
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1547edo is excellent in the 7-limit. It tempers out [[4375/4374]] and it is a member of the [[optimal GPV sequence]] for the rank-3 temperament associated with this comma. | 1547edo is excellent in the 7-limit. It tempers out [[4375/4374]] and it is a member of the [[optimal GPV 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 {{Monzo|236 -61 -60}}, thus associating a stack of 60 [[15/8]]<nowiki/>s with [[8/5]], and 61 of them make a [[5/4]]. | 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 {{Monzo|236 -61 -60}}, thus associating a stack of 60 [[15/8]]<nowiki/>s with [[8/5]], and 61 of them make a [[5/4]]. | ||
In the 7-limit, it supports [[semidimi]]. Another edo which is quite strong in the 7-limit is like 1547edo is 441edo, and 1547edo thus supports the [[brahmagupta]] temperament produced by merging 441 & 1547. | In the 7-limit, it supports [[semidimi]]. Another edo which is quite strong in the 7-limit is like 1547edo is 441edo, and 1547edo thus supports the [[brahmagupta]] temperament produced by merging 441 & 1547. | ||
| Line 97: | Line 97: | ||
|642\1547<br>(13\1547) | |642\1547<br>(13\1547) | ||
|497.996<br>(10.084) | |497.996<br>(10.084) | ||
|4/3<br>( | |4/3<br>(3025/3024) | ||
|[[Protactinium]] | |[[Protactinium]] | ||
|} | |} | ||
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number --> | [[Category:Equal divisions of the octave|####]] <!-- 4-digit number --> | ||
Revision as of 13:37, 5 January 2023
| ← 1546edo | 1547edo | 1548edo → |
Theory
| 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) | |
1547edo is excellent in the 7-limit. It tempers out 4375/4374 and it is a member of the optimal GPV 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 60 15/8s with 8/5, and 61 of them make a 5/4.
In the 7-limit, it supports semidimi. Another edo which is quite strong in the 7-limit is like 1547edo is 441edo, and 1547edo thus supports the brahmagupta temperament produced by merging 441 & 1547.
In the 11-limit, it is a tuning for the rank-3 temperament heimdall.
In higher limits, it supports 91th-octave temperament protactinium.
1547's divisors are 1, 7, 13, 17, 91, 119, 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, 151263/151250, 820125/819896, 2097152/2096325 | [⟨1547 2542 3592 4343 5352]] | -0.017 | 0.024 | 3.10 |
| 2.3.5.7.11.13 | 4375/4374, 4096/4095, 10648/10647, 91125/91091, 105644/105625 | [⟨1547 2542 3592 4343 5352 5725]] | -0.029 | 0.034 | 4.42 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
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 |
| 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 (3025/3024) |
Protactinium |