229edo: Difference between revisions

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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.
{{EDO 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]]. The equal temperament [[tempering out|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]].  


The 229b val supports a [[septimal meantone]] close to the [[CTE tuning]].
The 229b [[val]] supports a [[septimal meantone]] close to the [[CTE tuning]].  
 
229edo is the 50th [[prime EDO]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|229|columns=11}}
{{Harmonics in equal|229|columns=11}}
=== 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|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 25: Line 26:
| 2.3
| 2.3
| {{monzo| 363 -229 }}
| {{monzo| 363 -229 }}
| [{{val| 229 363 }}]
| {{mapping| 229 363 }}
| -0.072
| -0.072
| 0.072
| 0.072
Line 32: Line 33:
| 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
Line 39: Line 40:
| 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
Line 46: Line 47:
| 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
Line 52: Line 53:
|-
|-
| 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 }}
| -0.017
| -0.017
| 0.384
| 0.384
Line 59: Line 60:
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 351/350, 442/441, 561/560, 715/714, 3136/3125, 4096/4095
| 351/350, 442/441, 561/560, 715/714, 2197/2187, 3136/3125
| [{{val| 229 363 532 643 792 847 936 }}]
| {{mapping| 229 363 532 643 792 847 936 }}
| -0.009
| -0.009
| 0.356
| 0.356
Line 67: Line 68:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 286/285, 351/350, 442/441, 476/475, 561/560, 1216/1215, 1729/1728
| 286/285, 351/350, 442/441, 476/475, 561/560, 1216/1215, 1729/1728
| [{{val| 229 363 532 643 792 847 936 973 }}]
| {{mapping| 229 363 532 643 792 847 936 973 }}
| -0.043
| -0.043
| 0.344
| 0.344
Line 76: Line 77:
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|16\229
| 16\229
|83.84
| 83.84
|16807/16000
| 16807/16000
|[[Sextilimeans]]
| [[Sextilimeans]]
|-
|-
| 1
| 1
Line 110: Line 111:
| 387.77
| 387.77
| 5/4
| 5/4
| [[Würschmidt]]
| [[Würschmidt]] (5-limit)
|-
|-
| 1
| 1
Line 128: Line 129:
| 565.94
| 565.94
| 18/13
| 18/13
| [[Tricot]] / [[trident]]
| [[Trident]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Hemiwürschmidt]]

Revision as of 05:47, 2 April 2024

← 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

Template:EDO intro

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. The equal temperament tempers out 393216/390625 (würschmidt comma) and [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 supporting hemiwürschmidt, newt, and trident.

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
2.3.5.7.11.13 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125 [⟨229 363 532 643 792 847]] -0.017 0.384 7.32
2.3.5.7.11.13.17 351/350, 442/441, 561/560, 715/714, 2197/2187, 3136/3125 [⟨229 363 532 643 792 847 936]] -0.009 0.356 6.79
2.3.5.7.11.13.17.19 286/285, 351/350, 442/441, 476/475, 561/560, 1216/1215, 1729/1728 [⟨229 363 532 643 792 847 936 973]] -0.043 0.344 6.57

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 / quinsandric
1 37\229 193.87 28/25 Didacus / hemiwürschmidt
1 67\229 351.09 49/40 Newt
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 Trident

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

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