229edo: Difference between revisions

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~~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.~~

== 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 ===

=== 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" | [[Subgroup]]

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

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

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

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

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

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| 363 -229 }}

| [{{~~val~~| 229 363 }}]

| {{mapping| 229 363 }}

| -0.072

| 0.072

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| 2.3.5

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

| [{{~~val~~| 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

| [{{~~val~~| 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

| [{{~~val~~| 229 363 532 643 792 }}]

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

| -0.134

| 0.308

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|-

| 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

| 0.384

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|-

| 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

| 0.356

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| 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 }}]

| {{mapping| 229 363 532 643 792 847 936 973 }}

| -0.043

| 0.344

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{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

! Periods per ~~octave~~

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

|1

| 1

|16\229

| 16\229

|83.84

| 83.84

|16807/16000

| 16807/16000

|[[Sextilimeans]]

| [[Sextilimeans]]

|-

| 1

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| 387.77

| 5/4

| [[Würschmidt]]

| [[Würschmidt]] (5-limit)

|-

| 1

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| 565.94

| 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|###]] ~~

~~[[Category:Prime EDO]]~~

[[Category:Würschmidt]]

[[Category:Hemiwürschmidt]]

@@ Line 1: / Line 1: @@
 {{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 ==
-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]].
-edo is the 50th [[prime EDO]].
 === Prime harmonics ===
 {{Harmonics in equal|229|columns=11}}
+=== Subsets and supersets ===
+edo is the 50th [[prime edo]].
 == Regular temperament properties ==
 {| 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" | Optimal 8ve <br>stretch (¢)
+! rowspan="2" | Optimal 8ve <br>Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 25: / Line 26: @@
 | 2.3
 | {{monzo| 363 -229 }}
-| [{{val| 229 363 }}]
+| {{mapping| 229 363 }}
 | -0.072
 | 0.072
@@ Line 32: / Line 33: @@
 | 2.3.5
 | 393216/390625, {{monzo| 39 -29 3 }}
-| [{{val| 229 363 532 }}]
+| {{mapping| 229 363 532 }}
 | -0.258
 | 0.269
@@ Line 39: / Line 40: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 14348907/14336000
-| [{{val| 229 363 532 643 }}]
+| {{mapping| 229 363 532 643 }}
 | -0.247
 | 0.233
@@ Line 46: / Line 47: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 3136/3125, 8019/8000
-| [{{val| 229 363 532 643 792 }}]
+| {{mapping| 229 363 532 643 792 }}
 | -0.134
 | 0.308
@@ Line 52: / Line 53: @@
 |-
 | 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
 | 0.384
@@ Line 59: / Line 60: @@
 |-
 | 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
 | 0.356
@@ Line 67: / Line 68: @@
 | 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 }}]
+| {{mapping| 229 363 532 643 792 847 936 973 }}
 | -0.043
 | 0.344
@@ Line 76: / Line 77: @@
 {| class="wikitable center-all left-5"
 |+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
 |-
-|1
+| 1
-|16\229
+| 16\229
-|83.84
+| 83.84
-|16807/16000
+| 16807/16000
-|[[Sextilimeans]]
+| [[Sextilimeans]]
 |-
 | 1
@@ Line 110: / Line 111: @@
 | 387.77
 | 5/4
-| [[Würschmidt]]
+| [[Würschmidt]] (5-limit)
 |-
 | 1
@@ Line 128: / Line 129: @@
 | 565.94
 | 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:Hemiwürschmidt]]