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{{Infobox ET}}
{{Infobox ET}}
The '''229 equal divisions of the octave''' ('''229edo'''), or the '''229(-tone) equal temperament''' ('''229tet''', '''229et'''), is the [[EDO|equal division of the octave]] into 229 parts of about 5.24 [[cent]]s each.
{{ED intro}}


== Theory ==
== Theory ==
While not highly accurate for its size, 229edo is the point where a few important temperaments meet, and is distinctly [[consistent]] in the [[11-odd-limit]]. It tempers out 393216/390625 ([[würschmidt comma]]) and {{monzo| 39 -29 3 }} ([[tricot 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, and using the [[patent val]], [[351/350]], [[1573/1568]], [[2080/2079]], and [[4096/4095]] in the 13-limit, notably [[support|supporting]] [[hemiwürschmidt]], [[newt]], and [[trident]].  
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]].  


The 229b val supports a [[septimal meantone]] close to the [[CTE tuning]].  
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]].  


229edo is the 50th [[prime EDO]].
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 ===
=== Prime harmonics ===
{{Harmonics in equal|229|columns=11}}
{{Harmonics in equal|229}}
 
=== Subsets and supersets ===
229edo is the 50th [[prime edo]].


== 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" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 24: Line 30:
|-
|-
| 2.3
| 2.3
| {{monzo| 363 -229 }}
| {{Monzo| 363 -229 }}
| [{{val| 229 363 }}]
| {{Mapping| 229 363 }}
| -0.072
| −0.072
| 0.072
| 0.072
| 1.38
| 1.38
Line 32: Line 38:
| 2.3.5
| 2.3.5
| 393216/390625, {{monzo| 39 -29 3 }}
| 393216/390625, {{monzo| 39 -29 3 }}
| [{{val| 229 363 532 }}]
| {{Mapping| 229 363 532 }}
| -0.258
| −0.258
| 0.269
| 0.269
| 5.13
| 5.13
Line 39: Line 45:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 3136/3125, 14348907/14336000
| 2401/2400, 3136/3125, 14348907/14336000
| [{{val| 229 363 532 643 }}]
| {{Mapping| 229 363 532 643 }}
| -0.247
| −0.247
| 0.233
| 0.233
| 4.46
| 4.46
Line 46: Line 52:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 3136/3125, 8019/8000
| 2401/2400, 3025/3024, 3136/3125, 8019/8000
| [{{val| 229 363 532 643 792 }}]
| {{Mapping| 229 363 532 643 792 }}
| -0.134
| −0.134
| 0.308
| 0.308
| 5.87
| 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
| 2.3.5.7.11.13
| 351/350, 1573/1568, 2080/2079, 3136/3125, 4096/4095
| 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125
| [{{val| 229 363 532 643 792 847 }}]
| {{Mapping| 229 363 532 643 792 847 }} (229)
| -0.017
| −0.017
| 0.384
| 0.384
| 7.32
| 7.32
|-
|- style="border-top: double;"
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13
| 351/350, 442/441, 561/560, 715/714, 3136/3125, 4096/4095
| 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024
| [{{val| 229 363 532 643 792 847 936 }}]
| {{Mapping| 229 363 532 643 792 848 }} (229f)
| -0.009
| −0.253
| 0.356
| 0.387
| 6.79
| 7.39
|-
| 2.3.5.7.11.13.17.19
| 286/285, 351/350, 442/441, 476/475, 561/560, 1216/1215, 1729/1728
| [{{val| 229 363 532 643 792 847 936 973 }}]
| -0.043
| 0.344
| 6.57
|}
|}


=== 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 octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
| 1
| 16\229
| 83.84
| 16807/16000
| [[Sextilimeans]]
|-
|-
| 1
| 1
Line 86: Line 113:
| 99.56
| 99.56
| 18/17
| 18/17
| [[Quintagar]] / [[quinsandra]] / [[quinsandric]]
| [[Quintagar]] / [[quinsandra]] (229) / [[quinsandric]] (229)
|-
|-
| 1
| 1
Line 98: Line 125:
| 351.09
| 351.09
| 49/40
| 49/40
| [[Newt]]
| [[Newt]] (229)
|-
|-
| 1
| 1
Line 104: Line 131:
| 387.77
| 387.77
| 5/4
| 5/4
| [[Würschmidt]]
| [[Würschmidt]] (5-limit)
|-
|-
| 1
| 1
Line 122: Line 149:
| 565.94
| 565.94
| 18/13
| 18/13
| [[Tricot]] / [[trident]]
| [[Alphatrident]] (229)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Music ==
[[Category:Prime EDO]]
; [[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:Würschmidt]]
[[Category:Würschmidt]]
[[Category:Hemiwürschmidt]]
Retrieved from "https://en.xen.wiki/w/229edo"