229edo: Difference between revisions

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== Theory ==

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 }} ([[~~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]], notably [[support]]ing [[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]].

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]].

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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|-

| 2.3

| {{~~monzo~~| 363 -229 }}

| {{Monzo| 363 -229 }}

| {{~~mapping~~| 229 363 }}

| {{Mapping| 229 363 }}

| −0.072

| 0.072

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| 2.3.5

| 393216/390625, {{monzo| 39 -29 3 }}

| {{~~mapping~~| 229 363 532 }}

| {{Mapping| 229 363 532 }}

| −0.258

| 0.269

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| 2.3.5.7

| 2401/2400, 3136/3125, 14348907/14336000

| {{~~mapping~~| 229 363 532 643 }}

| {{Mapping| 229 363 532 643 }}

| −0.247

| 0.233

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 3136/3125, 8019/8000

| {{~~mapping~~| 229 363 532 643 792 }}

| {{Mapping| 229 363 532 643 792 }}

| −0.134

| 0.308

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| 2.3.5.7.11.17

| 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125

| {{~~mapping~~| 229 363 532 643 792 936 }}

| {{Mapping| 229 363 532 643 792 936 }}

| −0.106

| 0.288

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| 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 }}

| {{Mapping| 229 363 532 643 792 936 973 }}

| −0.130

| 0.273

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| 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 }}

| {{Mapping| 229 363 532 643 792 936 973 1036 }}

| −0.129

| 0.256

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| 2.3.5.7.11.13

| 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125

| {{~~mapping~~| 229 363 532 643 792 847 }} (229)

| {{Mapping| 229 363 532 643 792 847 }} (229)

| −0.017

| 0.384

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| 2.3.5.7.11.13

| 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024

| {{~~mapping~~| 229 363 532 643 792 848 }} (229f)

| {{Mapping| 229 363 532 643 792 848 }} (229f)

| −0.253

| 0.387

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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| 565.94

| 18/13

| [[~~Trident~~]] (229)

| [[Alphatrident]] (229)

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal~~ lists|minimal form]] in parentheses if distinct

<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]]

@@ Line 3: / Line 3: @@
 == Theory ==
-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 }} ([[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]], notably [[support]]ing [[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]].
 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]].
@@ Line 23: / Line 23: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 30: / Line 30: @@
 |-
 | 2.3
-| {{monzo| 363 -229 }}
+| {{Monzo| 363 -229 }}
-| {{mapping| 229 363 }}
+| {{Mapping| 229 363 }}
 | −0.072
 | 0.072
@@ Line 38: / Line 38: @@
 | 2.3.5
 | 393216/390625, {{monzo| 39 -29 3 }}
-| {{mapping| 229 363 532 }}
+| {{Mapping| 229 363 532 }}
 | −0.258
 | 0.269
@@ Line 45: / Line 45: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 14348907/14336000
-| {{mapping| 229 363 532 643 }}
+| {{Mapping| 229 363 532 643 }}
 | −0.247
 | 0.233
@@ Line 52: / Line 52: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 3136/3125, 8019/8000
-| {{mapping| 229 363 532 643 792 }}
+| {{Mapping| 229 363 532 643 792 }}
 | −0.134
 | 0.308
@@ Line 59: / Line 59: @@
 | 2.3.5.7.11.17
 | 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125
-| {{mapping| 229 363 532 643 792 936 }}
+| {{Mapping| 229 363 532 643 792 936 }}
 | −0.106
 | 0.288
@@ Line 66: / Line 66: @@
 | 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 }}
+| {{Mapping| 229 363 532 643 792 936 973 }}
 | −0.130
 | 0.273
@@ Line 73: / Line 73: @@
 | 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 }}
+| {{Mapping| 229 363 532 643 792 936 973 1036 }}
 | −0.129
 | 0.256
@@ Line 80: / Line 80: @@
 | 2.3.5.7.11.13
 | 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125
-| {{mapping| 229 363 532 643 792 847 }} (229)
+| {{Mapping| 229 363 532 643 792 847 }} (229)
 | −0.017
 | 0.384
@@ Line 87: / Line 87: @@
 | 2.3.5.7.11.13
 | 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024
-| {{mapping| 229 363 532 643 792 848 }} (229f)
+| {{Mapping| 229 363 532 643 792 848 }} (229f)
 | −0.253
 | 0.387
@@ Line 97: / Line 97: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 149: / Line 149: @@
 | 565.94
 | 18/13
-| [[Trident]] (229)
+| [[Alphatrident]] (229)
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<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]]