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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
226edo is closely related to [[113edo]], but its mapping of [[harmonic]] [[5/1|5]] is sharp instead of flat. Unlike 113, 226 is only [[consistent]] to the [[5-odd-limit]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[1029/1024]] and [[19683/19600]] in the [[7-limit]]; [[243/242]], [[9801/9800]] and notably the [[quartisma]] in the [[11-limit]]; and [[364/363]] and [[729/728]] in the [[13-limit]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|226}} | {{Harmonics in equal|226}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
226 factors into 2 × 113, with [[2edo]] and [[113edo]] as its subset edos. [[904edo]], which quadruples it, gives a good correction to the harmonic 7. | 226 factors into 2 × 113, with [[2edo]] and [[113edo]] as its subset edos. [[904edo]], which quadruples it, gives a good correction to the harmonic 7. | ||
==Regular temperament properties== | |||
== 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 | ||
| | |||
|- | |- | ||
|2.3.5 | ! [[TE error|Absolute]] (¢) | ||
|{{monzo|17 1 -8}}, {{monzo|-32 29 -6}} | ! [[TE simple badness|Relative]] (%) | ||
|{{ | |- | ||
| 0.0386 | | 2.3.5 | ||
| {{monzo| 17 1 -8 }}, {{monzo| -32 29 -6 }} | |||
| {{mapping| 226 358 525 }} | |||
| +0.0386 | |||
| 0.5044 | | 0.5044 | ||
| 9.50 | | 9.50 | ||
|} | |} | ||
=== 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 | ||
|73\226 | | 73\226 | ||
|387.61 | | 387.61 | ||
|5/4 | | 5/4 | ||
|[[Würschmidt]] | | [[Würschmidt]] (5-limit) | ||
|- | |||
| 1 | |||
| 91\226 | |||
| 483.19 | |||
| 320/243 | |||
| [[Hemiseven]] (7-limit) | |||
|- | |- | ||
|2 | | 2 | ||
|23\226 | | 23\226 | ||
|122.12 | | 122.12 | ||
|15/14 | | 15/14 | ||
|[[Lagaca]] | | [[Lagaca]] | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
[[ | |||
Latest revision as of 13:32, 13 March 2026
| ← 225edo | 226edo | 227edo → |
226 equal divisions of the octave (abbreviated 226edo or 226ed2), also called 226-tone equal temperament (226tet) or 226 equal temperament (226et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 226 equal parts of about 5.31 ¢ each. Each step represents a frequency ratio of 21/226, or the 226th root of 2.
Theory
226edo is closely related to 113edo, but its mapping of harmonic 5 is sharp instead of flat. Unlike 113, 226 is only consistent to the 5-odd-limit. Using the patent val, the equal temperament tempers out 1029/1024 and 19683/19600 in the 7-limit; 243/242, 9801/9800 and notably the quartisma in the 11-limit; and 364/363 and 729/728 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.07 | +1.30 | -2.45 | -2.14 | +0.89 | -1.59 | +0.23 | +1.24 | -0.17 | +1.79 | -1.73 |
| Relative (%) | -20.2 | +24.4 | -46.2 | -40.3 | +16.8 | -29.9 | +4.3 | +23.3 | -3.2 | +33.6 | -32.5 | |
| Steps (reduced) |
358 (132) |
525 (73) |
634 (182) |
716 (38) |
782 (104) |
836 (158) |
883 (205) |
924 (20) |
960 (56) |
993 (89) |
1022 (118) | |
Subsets and supersets
226 factors into 2 × 113, with 2edo and 113edo as its subset edos. 904edo, which quadruples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [17 1 -8⟩, [-32 29 -6⟩ | [⟨226 358 525]] | +0.0386 | 0.5044 | 9.50 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 73\226 | 387.61 | 5/4 | Würschmidt (5-limit) |
| 1 | 91\226 | 483.19 | 320/243 | Hemiseven (7-limit) |
| 2 | 23\226 | 122.12 | 15/14 | Lagaca |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct