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'''229EDO''' is the [[EDO|equal division of the octave]] into 229 parts of 5.2402 [[cent]]s each. It tempers out 393216/390625 ([[würschmidt comma]]) and 68719476736000/68630377364883 ([[tricot comma]]) in the 5-limit; 2401/2400, 3136/3125, and 14348907/14336000 in the 7-limit; 3025/3024, 3388/3375, 8019/8000, and 15488/15435 in the 11-limit, so that it supports the [[Würschmidt family|hemiwürschmidt temperament]].
{{Infobox ET}}
{{ED intro}}


229EDO is the 50th [[prime EDO]].
== Theory ==
While not highly accurate for its size, 229edo is the point where a few important temperaments meet, and is [[consistency|distinctly consistent]] in the [[11-odd-limit]]. It [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 39 -29 3 }} ([[alphatricot comma]]) in the [[5-limit]]; [[2401/2400]], [[3136/3125]], [[6144/6125]], and [[14348907/14336000]] in the [[7-limit]]; [[3025/3024]], [[3388/3375]], [[8019/8000]], [[14641/14580]] and 15488/15435 in the [[11-limit]], notably [[support]]ing [[hemiwürschmidt]], [[newt]], and [[alphatrident]].  


It extends less well to the 13-limit. Using the [[patent val]] {{val| 229 363 532 643 792 '''847''' }}, it tempers out [[351/350]], [[1573/1568]], [[2080/2079]], and [[4096/4095]]. Using the alternative 229f val {{val| 229 363 532 643 792 '''848''' }}, it tempers out [[352/351]], [[729/728]], [[1001/1000]], and [[1716/1715]].
Higher [[harmonic]]s like [[17/1|17]], [[19/1|19]], and [[23/1|23]] are well-approximated, so it shows great potential in the no-13 23-limit. It tempers out [[561/560]], [[1089/1088]], and [[1701/1700]] in the 17-limit; [[476/475]], [[1216/1215]], [[1445/1444]], and [[1540/1539]] in the 19-limit; and [[484/483]], [[576/575]] and [[736/735]] in the 23-limit.
The 229b [[val]] supports a [[septimal meantone]] close to the [[CTE tuning]].
=== Prime harmonics ===
{{Harmonics in equal|229}}
=== Subsets and supersets ===
229edo is the 50th [[prime edo]].
== 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
| {{Monzo| 363 -229 }}
| {{Mapping| 229 363 }}
| −0.072
| 0.072
| 1.38
|-
| 2.3.5
| 393216/390625, {{monzo| 39 -29 3 }}
| {{Mapping| 229 363 532 }}
| −0.258
| 0.269
| 5.13
|-
| 2.3.5.7
| 2401/2400, 3136/3125, 14348907/14336000
| {{Mapping| 229 363 532 643 }}
| −0.247
| 0.233
| 4.46
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 3136/3125, 8019/8000
| {{Mapping| 229 363 532 643 792 }}
| −0.134
| 0.308
| 5.87
|-
| 2.3.5.7.11.17
| 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125
| {{Mapping| 229 363 532 643 792 936 }}
| −0.106
| 0.288
| 5.50
|-
| 2.3.5.7.11.17.19
| 476/475, 561/560, 1089/1088, 1216/1215, 1445/1444, 2401/2400
| {{Mapping| 229 363 532 643 792 936 973 }}
| −0.130
| 0.273
| 5.22
|-
| 2.3.5.7.11.17.19.23
| 476/475, 484/483, 561/560, 576/575, 736/735, 1089/1088, 1216/1215
| {{Mapping| 229 363 532 643 792 936 973 1036 }}
| −0.129
| 0.256
| 4.88
|- style="border-top: double;"
| 2.3.5.7.11.13
| 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125
| {{Mapping| 229 363 532 643 792 847 }} (229)
| −0.017
| 0.384
| 7.32
|- style="border-top: double;"
| 2.3.5.7.11.13
| 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024
| {{Mapping| 229 363 532 643 792 848 }} (229f)
| −0.253
| 0.387
| 7.39
|}
=== 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
| 16\229
| 83.84
| 16807/16000
| [[Sextilimeans]]
|-
| 1
| 19\229
| 99.56
| 18/17
| [[Quintagar]] / [[quinsandra]] (229) / [[quinsandric]] (229)
|-
| 1
| 37\229
| 193.87
| 28/25
| [[Didacus]] / [[hemiwürschmidt]]
|-
| 1
| 67\229
| 351.09
| 49/40
| [[Newt]] (229)
|-
| 1
| 74\229
| 387.77
| 5/4
| [[Würschmidt]] (5-limit)
|-
| 1
| 95\229
| 497.82
| 4/3
| [[Gary]]
|-
| 1
| 75\229
| 503.06
| 147/110
| [[Quadrawürschmidt]]
|-
| 1
| 108\229
| 565.94
| 18/13
| [[Alphatrident]] (229)
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Francium]]
* "Don't Think About Mimes" from ''Don't'' (2025) – [https://open.spotify.com/track/4jGvn8IFTQeJwNc0y17MpQ Spotify] | [https://francium223.bandcamp.com/track/dont-think-about-mimes Bandcamp] | [https://www.youtube.com/watch?v=MNHUrF4Ff0A YouTube]


[[Category:Hemiwürschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
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