229edo: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ Line 1: / Line 1: @@
-'''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.
-EDO 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]].
+edo 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:Prime EDO]]
+[[Category:Würschmidt]]
+[[Category:Hemiwürschmidt]]