226edo: Difference between revisions
Created page with "'''226edo''' is the equal division of the octave into 226 parts of 5.3097 cents each. It tempers out 2029/2024 in the 7-limit; 243/242 and [..." |
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{{Infobox ET}} | |||
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[[ | == 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 === | |||
{{Harmonics in equal|226}} | |||
=== 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 == | |||
{| 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]] (%) | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 17 1 -8 }}, {{monzo| -32 29 -6 }} | |||
| {{mapping| 226 358 525 }} | |||
| +0.0386 | |||
| 0.5044 | |||
| 9.50 | |||
|} | |||
=== 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 | |||
|- | |||
| 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]] | |||
|} | |||
<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 | |||