1911edo: Difference between revisions
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== Theory == | |||
1911edo is [[consistent]] in the [[11-odd-limit]]. The equal temperament [[Tempering out|tempers out]] the [[aluminium comma]] in the 5-limit, and it provides the [[optimal patent val]] for the [[protactinium]] temperament in the 17-limit. However as may stem from consistency only in the 11-limit, the 13th harmonic has a large relative error. As such, 1911edo is best considered as a 2.3.5.7.11.17.19 [[subgroup]] tuning. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|1911}} | {{Harmonics in equal|1911}} | ||
=== Subsets and supersets === | |||
Since 1911 factors into {{factorization|1911}}, 1911edo has subset edos {{EDOs| 3, 7, 13, 21, 39, 49, 91, 147, 273, 637 }}. | |||
== Regular temperament properties == | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 13 | |||
| 793\1911<br />(58\1911) | |||
| 497.959<br />(36.421) | |||
| 4/3<br />(?) | |||
| [[Aluminium]] | |||
|- | |||
| 91 | |||
| 793\1911<br />(16\1911) | |||
| 497.959<br />(10.047) | |||
| 4/3<br />(176/175) | |||
| [[Protactinium]] | |||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
Latest revision as of 06:04, 21 February 2025
| ← 1910edo | 1911edo | 1912edo → |
1911 equal divisions of the octave (abbreviated 1911edo or 1911ed2), also called 1911-tone equal temperament (1911tet) or 1911 equal temperament (1911et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1911 equal parts of about 0.628 ¢ each. Each step represents a frequency ratio of 21/1911, or the 1911th root of 2.
Theory
1911edo is consistent in the 11-odd-limit. The equal temperament tempers out the aluminium comma in the 5-limit, and it provides the optimal patent val for the protactinium temperament in the 17-limit. However as may stem from consistency only in the 11-limit, the 13th harmonic has a large relative error. As such, 1911edo is best considered as a 2.3.5.7.11.17.19 subgroup tuning.
Odd harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.086 | -0.128 | +0.091 | +0.016 | +0.289 | -0.089 | +0.132 | +0.297 | +0.250 | -0.295 |
| Relative (%) | +0.0 | +13.7 | -20.5 | +14.5 | +2.6 | +46.0 | -14.1 | +21.1 | +47.3 | +39.8 | -46.9 | |
| Steps (reduced) |
1911 (0) |
3029 (1118) |
4437 (615) |
5365 (1543) |
6611 (878) |
7072 (1339) |
7811 (167) |
8118 (474) |
8645 (1001) |
9284 (1640) |
9467 (1823) | |
Subsets and supersets
Since 1911 factors into 3 × 72 × 13, 1911edo has subset edos 3, 7, 13, 21, 39, 49, 91, 147, 273, 637.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 13 | 793\1911 (58\1911) |
497.959 (36.421) |
4/3 (?) |
Aluminium |
| 91 | 793\1911 (16\1911) |
497.959 (10.047) |
4/3 (176/175) |
Protactinium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct