193edo: Difference between revisions

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== Theory ==

193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.

193edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong [[23-limit]] system.

As an equal temperament, ~~193et~~ [[tempering out|tempers out]] the [[15625/15552|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[460/459]], [[507/506]], [[529/528]] in the 23-limit.

As an equal temperament, it [[tempering out|tempers out]] the [[15625/15552|kleisma]] in the [[5-limit]]; [[5120/5103]] and [[16875/16807]] in the [[7-limit]]; [[540/539]], [[1375/1372]], [[3025/3024]], and 4375/4356 in the [[11-limit]]; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], and [[4096/4095]] in the [[13-limit]]; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]; and [[460/459]], [[507/506]], and [[529/528]] in the 23-limit.

It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra|vish]] temperaments.

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== Regular temperament properties ==

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

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

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

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

! rowspan="2" | Optimal<br />8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{monzo| 306 -193 }}

| {{mapping| 193 306 }}

| ~~−0~~.2005

| −0.2005

| 0.2005

| 3.23

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| 15625/15552, {{monzo| 50 -33 1 }}

| {{mapping| 193 306 448 }}

| ~~−0~~.0158

| −0.0158

| 0.3084

| 4.96

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| 5120/5103, 15625/15552, 16875/16807

| {{mapping| 193 306 448 542 }}

| ~~−0~~.1118

| −0.1118

| 0.3146

| 5.06

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| 540/539, 1375/1372, 4375/4356, 5120/5103

| {{mapping| 193 306 448 542 668 }}

| ~~−0~~.2080

| −0.2080

| 0.3408

| 5.48

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| 325/324, 364/363, 540/539, 625/624, 4096/4095

| {{mapping| 193 306 448 542 668 714 }}

| ~~−0~~.1216

| −0.1216

| 0.3662

| 5.89

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| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095

| {{mapping| 193 306 448 542 668 714 789 }}

| ~~−0~~.1302

| −0.1302

| 0.3397

| 5.46

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| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215

| {{mapping| 193 306 448 542 668 714 789 820 }}

| ~~−0~~.1414

| −0.1414

| 0.3191

| 5.13

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| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528

| {{mapping| 193 306 448 542 668 714 789 820 873 }}

| ~~−0~~.1184

| −0.1184

| 0.3078

| 4.95

~~{{comma basis end}~~}

|}

* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo|190g]] and followed by [[217edo|217]].

* 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by [[212edo|212gh]].

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

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

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

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 1

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| 4/3

| [[Kwai]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|193}}
+{{ED intro}}
 == Theory ==
-edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.
+edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong [[23-limit]] system.
-As an equal temperament, 193et [[tempering out|tempers out]] the [[15625/15552|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[460/459]], [[507/506]], [[529/528]] in the 23-limit.
+As an equal temperament, it [[tempering out|tempers out]] the [[15625/15552|kleisma]] in the [[5-limit]]; [[5120/5103]] and [[16875/16807]] in the [[7-limit]]; [[540/539]], [[1375/1372]], [[3025/3024]], and 4375/4356 in the [[11-limit]]; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], and [[4096/4095]] in the [[13-limit]]; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]; and [[460/459]], [[507/506]], and [[529/528]] in the 23-limit.
 It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra|vish]] temperaments.
@@ Line 16: / Line 16: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
 | {{monzo| 306 -193 }}
 | {{mapping| 193 306 }}
-| &minus;0.2005
+| −0.2005
 | 0.2005
 | 3.23
@@ Line 28: / Line 37: @@
 | 15625/15552, {{monzo| 50 -33 1 }}
 | {{mapping| 193 306 448 }}
-| &minus;0.0158
+| −0.0158
 | 0.3084
 | 4.96
@@ Line 35: / Line 44: @@
 | 5120/5103, 15625/15552, 16875/16807
 | {{mapping| 193 306 448 542 }}
-| &minus;0.1118
+| −0.1118
 | 0.3146
 | 5.06
@@ Line 42: / Line 51: @@
 | 540/539, 1375/1372, 4375/4356, 5120/5103
 | {{mapping| 193 306 448 542 668 }}
-| &minus;0.2080
+| −0.2080
 | 0.3408
 | 5.48
@@ Line 49: / Line 58: @@
 | 325/324, 364/363, 540/539, 625/624, 4096/4095
 | {{mapping| 193 306 448 542 668 714 }}
-| &minus;0.1216
+| −0.1216
 | 0.3662
 | 5.89
@@ Line 56: / Line 65: @@
 | 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
 | {{mapping| 193 306 448 542 668 714 789 }}
-| &minus;0.1302
+| −0.1302
 | 0.3397
 | 5.46
@@ Line 63: / Line 72: @@
 | 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215
 | {{mapping| 193 306 448 542 668 714 789 820 }}
-| &minus;0.1414
+| −0.1414
 | 0.3191
 | 5.13
@@ Line 70: / Line 79: @@
 | 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528
 | {{mapping| 193 306 448 542 668 714 789 820 873 }}
-| &minus;0.1184
+| −0.1184
 | 0.3078
 | 4.95
-{{comma basis end}}
+|}
 * 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo|190g]] and followed by [[217edo|217]].
 * 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by [[212edo|212gh]].
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 133: / Line 149: @@
 | 4/3
 | [[Kwai]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Scales ==