229edo: Difference between revisions

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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|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]].
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 5.2402 [[cent]]s each.  


229EDO is the 50th [[prime EDO]].
== Theory ==
While not highly accurate for its size, 229et 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]], [[2080/2079]], and [[4096/4095]] in the 13-limit, notably supporting [[hemiwürschmidt]], [[newt]], and [[trident]].  


229edo is the 50th [[prime EDO]].
=== Prime harmonics ===
{{Primes in edo|229}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 363 -229 }}
| [{{val| 229 363 }}]
| -0.072
| 0.072
| 1.38
|-
| 2.3.5
| 393216/390625, {{monzo| 39 -29 3 }}
| [{{val| 229 363 532 }}]
| -0.258
| 0.269
| 5.13
|-
| 2.3.5.7
| 2401/2400, 3136/3125, 14348907/14336000
| [{{val| 229 363 532 643 }}]
| -0.247
| 0.233
| 4.46
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 3136/3125, 8019/8000
| [{{val| 229 363 532 643 792 }}]
| -0.134
| 0.308
| 5.87
|-
| 2.3.5.7.11.13
| 351/350, 2080/2079, 3025/3024, 3136/3125, 4096/4095
| [{{val| 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, 3136/3125, 4096/4095
| [{{val| 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
| [{{val| 229 363 532 643 792 847 936 973 }}]
| -0.043
| 0.344
| 6.57
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 19\229
| 99.56
| 18/17
| [[Quintagar]] / [[quintasandra]] / [[quintasandroid]]
|-
| 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]]
|-
| 1
| 95\229
| 497.82
| 4/3
| [[Gary]]
|-
| 1
| 75\229
| 503.06
| 147/110
| [[Quadrawürschmidt]]
|-
| 1
| 108\229
| 565.94
| 18/13
| [[Tricot]] / [[trident]]
|}


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

Revision as of 07:05, 8 August 2021

The 229 equal divisions of the octave (229edo), or the 229(-tone) equal temperament (229tet, 229et), is the equal division of the octave into 229 parts of 5.2402 cents each.

Theory

While not highly accurate for its size, 229et 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⟩ (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, 2080/2079, and 4096/4095 in the 13-limit, notably supporting hemiwürschmidt, newt, and trident.

229edo is the 50th prime EDO.

Prime harmonics

Script error: No such module "primes_in_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, 2080/2079, 3025/3024, 3136/3125, 4096/4095 [⟨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, 3136/3125, 4096/4095 [⟨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 octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 19\229 99.56 18/17 Quintagar / quintasandra / quintasandroid
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
1 95\229 497.82 4/3 Gary
1 75\229 503.06 147/110 Quadrawürschmidt
1 108\229 565.94 18/13 Tricot / trident
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