2814edo: Difference between revisions
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{{Infobox ET}} | |||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
2814edo has all the | 2814edo has all the [[harmonic]]s from 3 to 17 approximated below 1/3 relative error and it is as a corollary [[consistent]] in the [[17-odd-limit]]. | ||
In the 7-limit, it is | In the 7-limit, it is [[Enfactoring|enfactored]], with the same [[comma]]s [[Tempering out|tempered out]] as [[1407edo]]. In the 11-limit, it supports rank-3 [[odin]] temperament. In the 13-limit it tempers out [[6656/6655]] and supports the 2.5.7.11.13 [[subgroup]] [[double bastille]] temperament. | ||
In the 11-limit, it supports rank-3 [[odin]] temperament. | |||
In the 13-limit it tempers out [[6656/6655]] and supports the 2.5.7.11.13 subgroup [[ | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|2814}} | {{Harmonics in equal|2814}} | ||
=== Subsets and supersets === | |||
Since 2814 factors into {{factorization|2814}}, 2814edo has subset edos {{EDOs| 2, 3, 6, 7, 14, 21, 42, 67, 134, 201, 402, 469, 938 and 1407 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
Note: 7-limit temperaments represented by [[1407edo]] are not included. | Note: 7-limit temperaments represented by [[1407edo]] are not included. | ||
{| class="wikitable center-all left- | {| class="wikitable center-all left-5" | ||
! Periods<br>per 8ve | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Generator | |- | ||
! Cents | ! Periods<br />per 8ve | ||
! Associated<br> | ! Generator* | ||
! | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 28: | Line 29: | ||
| 252.878 | | 252.878 | ||
| 53094899/45875200 | | 53094899/45875200 | ||
| [[Double | | [[Double bastille]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[Category:Jacobin]] | [[Category:Jacobin]] | ||
Latest revision as of 23:08, 20 February 2025
| ← 2813edo | 2814edo | 2815edo → |
2814 equal divisions of the octave (abbreviated 2814edo or 2814ed2), also called 2814-tone equal temperament (2814tet) or 2814 equal temperament (2814et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2814 equal parts of about 0.426 ¢ each. Each step represents a frequency ratio of 21/2814, or the 2814th root of 2.
Theory
2814edo has all the harmonics from 3 to 17 approximated below 1/3 relative error and it is as a corollary consistent in the 17-odd-limit.
In the 7-limit, it is enfactored, with the same commas tempered out as 1407edo. In the 11-limit, it supports rank-3 odin temperament. In the 13-limit it tempers out 6656/6655 and supports the 2.5.7.11.13 subgroup double bastille temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.036 | +0.040 | +0.044 | +0.068 | -0.016 | -0.051 | +0.142 | -0.129 | -0.153 | -0.046 |
| Relative (%) | +0.0 | -8.4 | +9.4 | +10.3 | +15.9 | -3.7 | -12.0 | +33.2 | -30.3 | -35.9 | -10.8 | |
| Steps (reduced) |
2814 (0) |
4460 (1646) |
6534 (906) |
7900 (2272) |
9735 (1293) |
10413 (1971) |
11502 (246) |
11954 (698) |
12729 (1473) |
13670 (2414) |
13941 (2685) | |
Subsets and supersets
Since 2814 factors into 2 × 3 × 7 × 67, 2814edo has subset edos 2, 3, 6, 7, 14, 21, 42, 67, 134, 201, 402, 469, 938 and 1407.
Regular temperament properties
Rank-2 temperaments
Note: 7-limit temperaments represented by 1407edo are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 593\2814 | 252.878 | 53094899/45875200 | Double bastille |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct