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The ''229 equal division'' divides the octave into 229 equal parts of 5.240 cents each. In the 5-limit it tempers out the wuerschmidt comma, 393216/390625, and in the 7-limit 3136/3125, 6144/6125 and 2401/2400, so that it supports wuerschmidt and hemiwuerschmidt temperament. 229 is the 50th prime EDO.
{{Infobox ET}}
[[Category:hemiwuerschmidt]]
{{ED intro}}
[[Category:hemiwurschmidt]]
 
[[Category:wuerschmidt]]
== Theory ==
[[Category:wurschmidt]]
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:Würschmidt]]

Latest revision as of 12:02, 3 July 2025

← 228edo 229edo 230edo →
Prime factorization 229 (prime)
Step size 5.24017 ¢ 
Fifth 134\229 (702.183 ¢)
Semitones (A1:m2) 22:17 (115.3 ¢ : 89.08 ¢)
Consistency limit 11
Distinct consistency limit 11

229 equal divisions of the octave (abbreviated 229edo or 229ed2), also called 229-tone equal temperament (229tet) or 229 equal temperament (229et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 229 equal parts of about 5.24 ¢ each. Each step represents a frequency ratio of 21/229, or the 229th root of 2.

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 [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 supporting hemiwürschmidt, newt, and alphatrident.

It extends less well to the 13-limit. Using the patent val ⟨229 363 532 643 792 847], it tempers out 351/350, 1573/1568, 2080/2079, and 4096/4095. Using the alternative 229f val ⟨229 363 532 643 792 848], it tempers out 352/351, 729/728, 1001/1000, and 1716/1715.

Higher harmonics like 17, 19, and 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

Approximation of prime harmonics in 229edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.23 +1.46 +0.61 -1.10 -2.10 -0.15 +1.18 +0.55 -2.50 +2.56
Relative (%) +0.0 +4.4 +27.8 +11.6 -21.0 -40.1 -2.9 +22.5 +10.4 -47.8 +48.9
Steps
(reduced) 		229
(0) 		363
(134) 		532
(74) 		643
(185) 		792
(105) 		847
(160) 		936
(20) 		973
(57) 		1036
(120) 		1112
(196) 		1135
(219)

Subsets and supersets

229edo is the 50th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [363 -229⟩ [⟨229 363]] −0.072 0.072 1.38
2.3.5 393216/390625, [39 -29 3⟩ [⟨229 363 532]] −0.258 0.269 5.13
2.3.5.7 2401/2400, 3136/3125, 14348907/14336000 [⟨229 363 532 643]] −0.247 0.233 4.46
2.3.5.7.11 2401/2400, 3025/3024, 3136/3125, 8019/8000 [⟨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 [⟨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 [⟨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 [⟨229 363 532 643 792 936 973 1036]] −0.129 0.256 4.88
2.3.5.7.11.13 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125 [⟨229 363 532 643 792 847]] (229) −0.017 0.384 7.32
2.3.5.7.11.13 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024 [⟨229 363 532 643 792 848]] (229f) −0.253 0.387 7.39

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
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