229edo: Difference between revisions

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]]. ~~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, 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 [[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, notably [[support|supporting]] [[hemiwürschmidt]], [[newt]], and [[trident]].

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]].

Higher [[harmonic]]s like [[17/1|17]], [[19/1|19]], and [[23/1|23]] are well approximated, so it shows great potential in the no-13 23-limit. It tempers out [[561/560]], [[1089/1088]], and [[1701/1700]] in the 17-limit; [[476/475]], [[1216/1215]], [[1445/1444]], and [[1540/1539]] in the 19-limit; and [[484/483]], [[576/575]] and [[736/735]] in the 23-limit.

Higher [[harmonic]]s like [[17/1|17]], [[19/1|19]], and [[23/1|23]] are well-approximated, so it shows great potential in the no-13 23-limit. It tempers out [[561/560]], [[1089/1088]], and [[1701/1700]] in the 17-limit; [[476/475]], [[1216/1215]], [[1445/1444]], and [[1540/1539]] in the 19-limit; and [[484/483]], [[576/575]] and [[736/735]] in the 23-limit.

The 229b [[val]] supports a [[septimal meantone]] close to the [[CTE tuning]].

Line 32:

| {{monzo| 363 -229 }}

| {{mapping| 229 363 }}

| ~~−0~~.072

| −0.072

| 0.072

| 1.38

Line 39:

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

| {{mapping| 229 363 532 }}

| ~~−0~~.258

| −0.258

| 0.269

| 5.13

Line 46:

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

| {{mapping| 229 363 532 643 }}

| ~~−0~~.247

| −0.247

| 0.233

| 4.46

Line 53:

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

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

| ~~−0~~.134

| −0.134

| 0.308

| 5.87

Line 60:

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

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

| ~~−0~~.106

| −0.106

| 0.288

| 5.50

Line 67:

| 476/475, 561/560, 1089/1088, 1216/1215, 1445/1444, 2401/2400

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

| ~~−0~~.130

| −0.130

| 0.273

| 5.22

Line 74:

| 476/475, 484/483, 561/560, 576/575, 736/735, 1089/1088, 1216/1215

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

| ~~−0~~.129

| −0.129

| 0.256

| 4.88

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

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

| ~~−0~~.017

| −0.017

| 0.384

| 7.32

Line 88:

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

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

| ~~−0~~.253

| −0.253

| 0.387

| 7.39

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| [[Trident]] (229)

|}

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

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

[[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]]. 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, 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 [[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, notably [[support|supporting]] [[hemiwürschmidt]], [[newt]], and [[trident]].
 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]].
-Higher [[harmonic]]s like [[17/1|17]], [[19/1|19]], and [[23/1|23]] are well approximated, so it shows great potential in the no-13 23-limit. It tempers out [[561/560]], [[1089/1088]], and [[1701/1700]] in the 17-limit; [[476/475]], [[1216/1215]], [[1445/1444]], and [[1540/1539]] in the 19-limit; and [[484/483]], [[576/575]] and [[736/735]] in the 23-limit.
+Higher [[harmonic]]s like [[17/1|17]], [[19/1|19]], and [[23/1|23]] are well-approximated, so it shows great potential in the no-13 23-limit. It tempers out [[561/560]], [[1089/1088]], and [[1701/1700]] in the 17-limit; [[476/475]], [[1216/1215]], [[1445/1444]], and [[1540/1539]] in the 19-limit; and [[484/483]], [[576/575]] and [[736/735]] in the 23-limit.
 The 229b [[val]] supports a [[septimal meantone]] close to the [[CTE tuning]].
@@ Line 32: / Line 32: @@
 | {{monzo| 363 -229 }}
 | {{mapping| 229 363 }}
-| &minus;0.072
+| −0.072
 | 0.072
 | 1.38
@@ Line 39: / Line 39: @@
 | 393216/390625, {{monzo| 39 -29 3 }}
 | {{mapping| 229 363 532 }}
-| &minus;0.258
+| −0.258
 | 0.269
 | 5.13
@@ Line 46: / Line 46: @@
 | 2401/2400, 3136/3125, 14348907/14336000
 | {{mapping| 229 363 532 643 }}
-| &minus;0.247
+| −0.247
 | 0.233
 | 4.46
@@ Line 53: / Line 53: @@
 | 2401/2400, 3025/3024, 3136/3125, 8019/8000
 | {{mapping| 229 363 532 643 792 }}
-| &minus;0.134
+| −0.134
 | 0.308
 | 5.87
@@ Line 60: / Line 60: @@
 | 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125
 | {{mapping| 229 363 532 643 792 936 }}
-| &minus;0.106
+| −0.106
 | 0.288
 | 5.50
@@ Line 67: / Line 67: @@
 | 476/475, 561/560, 1089/1088, 1216/1215, 1445/1444, 2401/2400
 | {{mapping| 229 363 532 643 792 936 973 }}
-| &minus;0.130
+| −0.130
 | 0.273
 | 5.22
@@ Line 74: / Line 74: @@
 | 476/475, 484/483, 561/560, 576/575, 736/735, 1089/1088, 1216/1215
 | {{mapping| 229 363 532 643 792 936 973 1036 }}
-| &minus;0.129
+| −0.129
 | 0.256
 | 4.88
@@ Line 81: / Line 81: @@
 | 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125
 | {{mapping| 229 363 532 643 792 847 }} (229)
-| &minus;0.017
+| −0.017
 | 0.384
 | 7.32
@@ Line 88: / Line 88: @@
 | 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024
 | {{mapping| 229 363 532 643 792 848 }} (229f)
-| &minus;0.253
+| −0.253
 | 0.387
 | 7.39
@@ Line 151: / Line 151: @@
 | [[Trident]] (229)
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Hemiwürschmidt]]
 [[Category:Würschmidt]]