Octagar temperaments: Difference between revisions

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This is a collection of [[~~Rank~~-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the octagar comma, {{monzo| 5 -4 3 -2 }} = [[4000/3969]].

This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).

Temperaments considered below are slendi and slithy. Discussed elsewhere are:

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* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]

* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]

* ''[[Superkleismic]]'' (+875/864 or 1029/1024) → [[~~Shibboleth family~~ #Superkleismic|Shibboleth family]] and [[Gamelismic clan #Superkleismic|Gamelismic clan]]

* ''[[Superkleismic]]'' (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Shibboleth family]] and [[Gamelismic clan #Superkleismic|Gamelismic clan]]

* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]

* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]

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~~{{Multival|legend=1| 17 28 8 5 -35 -60 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 29.181

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~49/48 = 29.181

{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}

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Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~49/48 = 29.180

Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.180

Badness: 0.063217

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Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -1 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~49/48 = 29.125

Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.125

Badness: 0.043644

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~~{{Multival|legend=1| 19 36 16 13 -28 -64 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 89.282

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~21/20 = 89.282

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Mapping: {{mapping| 1 3 5 4 8 | 0 -19 -36 -16 -61 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~21/20 = 89.291

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.291

Badness: 0.045895

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Mapping: {{mapping| 1 3 5 4 8 4 | 0 -19 -36 -16 -61 -4 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~21/20 = 89.286

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286

Badness: 0.028492

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 540/539, 595/594

Mapping: {{mapping| 1 3 5 4 8 4 9 | 0 -19 -36 -16 -61 -4 -66 }}

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286

Badness: 0.019811

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594

Mapping: {{mapping| 1 3 5 4 8 4 9 7 | 0 -19 -36 -16 -61 -4 -66 -37 }}

Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.288

Badness: 0.016408

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483

Mapping: {{mapping| 1 3 5 4 8 4 9 7 11 | 0 -19 -36 -16 -61 -4 -66 -37 -87 }}

Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.297

Badness: 0.014946

== Tridecatonic ==

The tridecatonic temperament (26 &~~amp;~~ 39) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26 &~~amp;~~ 39d, tempering out 4000/3969), [[~~Ragismic~~ microtemperaments #Trideci|trideci]] (26 &~~amp;~~ 39, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] (26 &~~amp;~~ 130, tempering out 321489/320000).

The tridecatonic temperament ({{nowrap| 26 & 39 }}) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic ({{nowrap| 26 & 39d }}, tempering out 4000/3969), [[ragismic microtemperaments #Trideci|trideci]] ({{nowrap| 26 & 39 }}, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] ({{nowrap| 26 & 130 }}, tempering out 321489/320000).

[[Subgroup]]: 2.3.5.7

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[[Optimal tuning]] ([[POTE]]): ~21/20 = ~~1\13~~, ~3/2 = 701.9605

[[Optimal tuning]] ([[POTE]]): ~21/20 = 92.3077, ~3/2 = 701.9605

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Mapping: {{mapping| 13 21 31 37 45 | 0 -1 -2 -1 0 }}

Optimal tuning (POTE): ~21/20 = ~~1\13~~, ~3/2 = 701.7028

Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.7028

Badness: 0.063257

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Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}

Optimal tuning (POTE): ~21/20 = ~~1\13~~, ~3/2 = 701.3573

Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.3573

Badness: 0.037851

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[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~6/5 = 319.198

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~6/5 = 319.198

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Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~6/5 = 319.197

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.197

~~Optimal ET sequence:~~ {{Optimal ET sequence| 15, 64c, 79, 94, 203, 297c }}

Badness: 0.051216

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Mapping: {{Mapping| 1 -4 -3 6 0 -8 | 0 21 20 -12 13 44 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~6/5 = 319.165

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.165

~~Optimal ET sequence:~~ {{Optimal ET sequence| 15, 49bceff, 64cf, 79, 94 }}

Badness: 0.037468

[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Octagar temperaments| ]]

[[Category:Octagar| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the octagar comma, {{monzo| 5 -4 3 -2 }} = [[4000/3969]].
+This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).
 Temperaments considered below are slendi and slithy. Discussed elsewhere are:
@@ Line 10: / Line 10: @@
 * ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
 * ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
-* ''[[Superkleismic]]'' (+875/864 or 1029/1024) → [[Shibboleth family #Superkleismic|Shibboleth family]] and [[Gamelismic clan #Superkleismic|Gamelismic clan]]
+* ''[[Superkleismic]]'' (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Shibboleth family]] and [[Gamelismic clan #Superkleismic|Gamelismic clan]]
 * ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]
 * ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
@@ Line 25: / Line 25: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -17 -28 -8 }}
-{{Multival|legend=1| 17 28 8 5 -35 -60 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 29.181
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 29.181
 {{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
@@ Line 40: / Line 38: @@
 Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.180
+Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.180
-{{Optimal ET sequence|legend=1| 41, 206e }}
+{{Optimal ET sequence|legend=0| 41, 206e }}
 Badness: 0.063217
@@ Line 53: / Line 51: @@
 Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.125
+Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.125
-{{Optimal ET sequence|legend=1| 41, 124ef, 165ef }}
+{{Optimal ET sequence|legend=0| 41, 124ef, 165ef }}
 Badness: 0.043644
@@ Line 66: / Line 64: @@
 {{Mapping|legend=1| 1 3 5 4 | 0 -19 -36 -16 }}
-{{Multival|legend=1| 19 36 16 13 -28 -64 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 89.282
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 89.282
 {{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
@@ Line 81: / Line 77: @@
 Mapping: {{mapping| 1 3 5 4 8 | 0 -19 -36 -16 -61 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.291
+Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.291
-{{Optimal ET sequence|legend=1| 94, 121, 215, 336de, 551bde }}
+{{Optimal ET sequence|legend=0| 94, 121, 215, 336de, 551bde }}
 Badness: 0.045895
@@ Line 94: / Line 90: @@
 Mapping: {{mapping| 1 3 5 4 8 4 | 0 -19 -36 -16 -61 -4 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.286
+Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286
-{{Optimal ET sequence|legend=1| 94, 121, 215, 336def }}
+{{Optimal ET sequence|legend=0| 94, 121, 215, 336def }}
 Badness: 0.028492
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 325/324, 352/351, 375/374, 540/539, 595/594
+Mapping: {{mapping| 1 3 5 4 8 4 9 | 0 -19 -36 -16 -61 -4 -66 }}
+Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286
+{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}
+Badness: 0.019811
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594
+Mapping: {{mapping| 1 3 5 4 8 4 9 7 | 0 -19 -36 -16 -61 -4 -66 -37 }}
+Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.288
+{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}
+Badness: 0.016408
+=== 23-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483
+Mapping: {{mapping| 1 3 5 4 8 4 9 7 11 | 0 -19 -36 -16 -61 -4 -66 -37 -87 }}
+Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.297
+{{Optimal ET sequence|legend=0| 94, 121i, 215 }}
+Badness: 0.014946
 == Tridecatonic ==
-{{See also| High badness temperaments #Tridecatonic }}
+{{See also| 13th-octave temperaments #Tridecatonic }}
-The tridecatonic temperament (26 &amp; 39) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26 &amp; 39d, tempering out 4000/3969), [[Ragismic microtemperaments #Trideci|trideci]] (26 &amp; 39, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] (26 &amp; 130, tempering out 321489/320000).
+The tridecatonic temperament ({{nowrap| 26 & 39 }}) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic ({{nowrap| 26 & 39d }}, tempering out 4000/3969), [[ragismic microtemperaments #Trideci|trideci]] ({{nowrap| 26 & 39 }}, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] ({{nowrap| 26 & 130 }}, tempering out 321489/320000).
 [[Subgroup]]: 2.3.5.7
@@ Line 111: / Line 146: @@
 {{Mapping|legend=1| 13 21 31 37 | 0 -1 -2 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~21/20 = 1\13, ~3/2 = 701.9605
+[[Optimal tuning]] ([[POTE]]): ~21/20 = 92.3077, ~3/2 = 701.9605
 {{Optimal ET sequence|legend=1| 26, 39d, 65d }}
@@ Line 124: / Line 159: @@
 Mapping: {{mapping| 13 21 31 37 45 | 0 -1 -2 -1 0 }}
-Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.7028
+Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.7028
-{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
+{{Optimal ET sequence|legend=0| 26, 39d, 65d }}
 Badness: 0.063257
@@ Line 137: / Line 172: @@
 Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}
-Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.3573
+Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.3573
-{{Optimal ET sequence|legend=1| 26, 39df, 65d }}
+{{Optimal ET sequence|legend=0| 26, 39df, 65d }}
 Badness: 0.037851
@@ Line 150: / Line 185: @@
 {{Mapping|legend=1| 1 -4 -3 6 | 0 21 20 -12 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 319.198
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~6/5 = 319.198
 {{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297 }}
@@ Line 163: / Line 198: @@
 Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 319.197
+Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.197
-Optimal ET sequence: {{Optimal ET sequence| 15, 64c, 79, 94, 203, 297c }}
+{{Optimal ET sequence|legend=0| 15, 64c, 79, 94, 203, 297c }}
 Badness: 0.051216
@@ Line 176: / Line 211: @@
 Mapping: {{Mapping| 1 -4 -3 6 0 -8 | 0 21 20 -12 13 44 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 319.165
+Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.165
-Optimal ET sequence: {{Optimal ET sequence| 15, 49bceff, 64cf, 79, 94 }}
+{{Optimal ET sequence|legend=0| 15, 49bceff, 64cf, 79, 94 }}
 Badness: 0.037468
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Octagar temperaments| ]] <!-- main article -->
 [[Category:Octagar| ]] <!-- key article -->
 [[Category:Rank 2]]