1547edo: Difference between revisions
Jump to navigation
Jump to search
→Regular temperament properties: completed rtp table |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| (23 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | |||
1547edo is [[consistent]] to the [[15-odd-limit]] and is excellent in the 7-limit. As an equal temperament, it [[tempering out|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 | 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 sixty [[15/8]]'s with [[4/3]], and sixty-one of them make [[5/4]]. | ||
In the | In the 7-limit, it provides the [[optimal patent val]] for 7-limit [[brahmagupta]], the {{nowrap|441 & 1106}} temperament, and supports an alternative 11-limit extension to it. It also supports [[semidimi]], the {{nowrap|171 & 1376}} temperament. | ||
In higher limits, it supports 91th-octave temperament [[protactinium]]. | 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]]. | ||
1547 | === Prime harmonics === | ||
==Regular temperament properties== | {{Harmonics in equal|1547}} | ||
=== Subsets and supersets === | |||
Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos {{EDOs| 7, 13, 17, 91, 119, and 221 }}. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
![[ | ! rowspan="2" | [[Subgroup]] | ||
![[ | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |- | ||
| | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|- | | {{monzo| 2452 -1547 }} | ||
| | | {{mapping| 1547 2542 }} | ||
| | | −0.015 | ||
|0. | | 0.015 | ||
| | | 1.99 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
| | | {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }} | ||
| | | {{mapping| 1547 2542 3592 }} | ||
| | | −0.008 | ||
|0. | | 0.017 | ||
| | | 2.14 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|4375/4374, | | 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3 -6 }} | ||
| | | {{mapping| 1547 2542 3592 4343 }} | ||
| | | −0.007 | ||
|0. | | 0.014 | ||
| | | 1.86 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|4375/4374, 4096/4095, | | 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 | ||
| | | {{mapping| 1547 2542 3592 4343 5352 }} | ||
| | | −0.017 | ||
|0.034 | | 0.024 | ||
|4.42 | | 3.10 | ||
|- | |||
| 2.3.5.7.11.13 | |||
| 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 | |||
| {{mapping| 1547 2542 3592 4343 5352 5725 }} | |||
| −0.029 | |||
| 0.034 | |||
| 4.42 | |||
|} | |} | ||
===Rank-2 temperaments=== | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|118\1547 | | 118\1547 | ||
|91.532 | | 91.532 | ||
| | | {{monzo| 9 -32 18 }} | ||
|[[Gross]] | | [[Gross]] | ||
|- | |||
| 1 | |||
| 579\1547 | |||
| 449.127 | |||
| 35/27 | |||
| [[Semidimi]] | |||
|- | |- | ||
| | | 7 | ||
| | | 670\1547<br />(7\1547) | ||
| | | 519.715<br />(5.429) | ||
| | | 27/20<br />(325/324) | ||
|[[ | | [[Brahmagupta]] (7-limit) | ||
|- | |- | ||
|7 | | 7 | ||
| | | 11\1547 | ||
| | | 8.533 | ||
| | | 1029/1024 | ||
|[[ | | [[Nitrogen]] | ||
|- | |||
| 13 | |||
| 642\1547<br />(47\1547) | |||
| 497.996<br />(36.458) | |||
| 4/3<br />(?) | |||
| [[Aluminium]] | |||
|- | |||
| 17 | |||
| 321\1547<br />(48\1547) | |||
| 248.998<br />(37.233) | |||
| {{monzo| -23 5 9 -2 }}<br />(100352/98415) | |||
| [[Chlorine]] | |||
|- | |- | ||
| 91 | | 91 | ||
| | | 642\1547<br />(13\1547) | ||
| | | 497.996<br />(10.084) | ||
| 3 | | 4/3<br />(176/175) | ||
| [[Protactinium]] | | [[Protactinium]] | ||
|} | |} | ||
[[ | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Music == | |||
; [[Eliora]] | |||
* [https://www.youtube.com/watch?v=oac5JZ9FtB8 ''Prelude and Fugue in Two Elements''] | |||
[[Category:Listen]] | |||
Latest revision as of 23:08, 20 February 2025
| ← 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. As an equal temperament, 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 distinct