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{{Technical data page}}
'''The Jacobins''' is a collection of microtemperaments of different ranks which all temper out the jacobin comma, [[6656/6655]].
'''The Jacobins''' is a collection of microtemperaments of different ranks which all temper out the jacobin comma, [[6656/6655]].


The main focus here will be on the 2.5.11.13 [[subgroup]], the subgorup of the comma. Besides, in the full 13-limit the jacobin comma often functions as a part of a basis of other temperaments of other families and groups, like [[vidar]].  
The main focus here will be on the 2.5.11.13 [[subgroup]], the subgroup of the comma. Besides, in the full 13-limit the jacobin comma often functions as a part of a basis of other temperaments of other families and groups, like [[vidar]].  


Quite coincidentally, [[1789edo]] supports an enormous amount of these temperaments. Since 1789edo has a bad approximation to the 3rd harmonic, 2.5.7.11.13 is also the main subgroup for many temperaments, and 7-limit extensions to 2.5.11.13 temperaments are named "septimal ..." after the original temperament.
Quite coincidentally, [[1789edo]] supports an enormous amount of these temperaments. Since 1789edo has a bad approximation to the 3rd harmonic, 2.5.7.11.13 is also the main subgroup for many temperaments, and 7-limit extensions to 2.5.11.13 temperaments are named "septimal …" after the original temperament.
 
== Jacobin ==
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 6656/6655
 
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || ]]
|}
 
: mapping generators: ~2, ~3, ~5, ~7, ~11
 
{{Optimal ET sequence|legend=1| 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}
 
=== Septendecimal jacobin ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 6656/6655, 12376/12375
 
Mapping: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || 6 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 2 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || 2 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || -2 || ]]
|}
 
Optimal ET sequence: {{Optimal ET sequence| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}


== Jacobin-naiadic ==
== Jacobin-naiadic ==
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Comma list: 6656/6655
Comma list: 6656/6655


Sval mapping: 1 2 4, 0 -3 -1
Sval mapping: [{{Val|1 2 4}}, {{Val|0 -3 -1}}]


Optimal tuning (CTE): ~16/11 = 648.608
Optimal tuning (CTE): ~16/11 = 648.608
== Barton ==
{{See also| Chromatic pairs #Barton }}
Barton may be described as the 11 &amp; 13 temperament in the 2.5.11.13 subgroup. It was named after [[Jacob Barton]] by [[Gene Ward Smith]] and [[Carl Lumma]] in 2006<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_14632.html Yahoo! Tuning Group | "father" variant?]</ref>. [[Scott Dakota]] rediscovered this same temperament in 2025 and named it "hem"{{idio}}.
[[Subgroup]]: 2.5.11.13
[[Comma list]]: [[2200/2197]], [[6656/6655]]
{{Mapping|legend=2| 1 6 3 6 | 0 -8 1 -5 }}
{{Mapping|legend=3| 1 0 6 0 3 6 | 0 0 -8 0 1 -5 }}
: gencom: [2 11/8; 2200/2197 6656/6655]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 551.699
{{Optimal ET sequence|legend=1| 11, 13, 24, 37, 50, 87, 298, 385, 472, 559, 1590cd }}
[[Tp tuning #T2 tuning|RMS error]]: 0.0822 cents
== Genojacobin ==
Described as the 1789 & 3395 temperament, and named in honor of [[Gene Ward Smith]], who named the jacobin comma, and the fact that 3395edo provides the optimal patent val for the comma. 7 generators are equal to [[55/32]].
Subgroup: 2.5.11.13
Comma list: 6656/6655, {{monzo|-177 76 -79 74}}
Sval mapping: {{Val|1 100 -99 -206}}, {{Val|0 -143 150 307}}
Optimal tuning (CTE): ~2.5.11.13 {{monzo|-106 28 -4 15}} = 819.676
{{Optimal ET sequence|legend=1|183, 1789, 3395}}, ...
=== 2.5.11.13.29 subgroup ===
An extension for this subgroup is prescribed because both 1789edo and 3395edo are good at 29th harmonic, which in this temperament is also reached in just 32 generator steps.
Subgroup: 2.5.11.13.29
Comma list: 6656/6655, 594880000/594823321, 8091203119330852077568/8090590952301025390625
Sval mapping: {{Val|1 100 -99 -206 -17}}, {{Val|0 -143 150 307 32}}
Optimal tuning (CTE): ~55115776/34328125 = 819.676
{{Optimal ET sequence|legend=1|183, 1057f, 1240, 1423, 1606, 1789, 3395 }}


== Onzonic ==
== Onzonic ==
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=== Pure onzonic ===
=== Pure onzonic ===
Pure onzonic is the temperament that was initially referred to as "jacobin" before it was pointed out that the same name would be reserved for the rank-5 temperamnet tempering out 6656/6655 alone (see jacobin-naiadic above).   
Pure onzonic is the temperament that was initially referred to as "jacobin" before it was pointed out that the same name would be reserved for the rank-5 temperamnet tempering out 6656/6655 alone (see above).   


[[Subgroup]]: 2.5.11.13
[[Subgroup]]: 2.5.11.13
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[[Optimal tuning]] ([[CTE]]): ~11/8 = 551.370
[[Optimal tuning]] ([[CTE]]): ~11/8 = 551.370


Vals: {{EDOs| 37, 1789 }}
{{Optimal ET sequence|legend=1| 37, 1789 }}


=== Septimal onzonic ===
=== Septimal onzonic ===
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Optimal tuning (CTE): ~11/8 = 551.369
Optimal tuning (CTE): ~11/8 = 551.369


Vals: {{EDOs| 37, 1789 }}
{{Optimal ET sequence|legend=1| 37, 1789 }}


== Estates general ==
== Estates general ==
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[[Optimal tuning]] ([[CTE]]): ~2588443885831192576/1914932769775390625 = 521.856
[[Optimal tuning]] ([[CTE]]): ~2588443885831192576/1914932769775390625 = 521.856
{{Optimal ET sequence|legend=1|23, 430fhhh, 453h, 1336, 1789, 3125}}


=== 2.5.11.13.19.23 subgroup ===
=== 2.5.11.13.19.23 subgroup ===
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Optimal tuning (CTE): ~2592407900127232/1918105439453125 = 521.856
Optimal tuning (CTE): ~2592407900127232/1918105439453125 = 521.856
{{Optimal ET sequence|legend=1|23, 430fhhhiiii, 453hi, 1336, 1789, 4914h}}


=== 2.5.11.13.19.23.29 subgroup ===
=== 2.5.11.13.19.23.29 subgroup ===
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Optimal tuning (CTE): ~184000/136097 = 521.856
Optimal tuning (CTE): ~184000/136097 = 521.856
{{Optimal ET sequence|legend=1|23, 430fhhhiiiij, 453hi, 1336, 1789, 3125}}


=== 2.5.11.13.19.23.29.31 subgroup ===
=== 2.5.11.13.19.23.29.31 subgroup ===
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Optimal tuning (CTE): ~80275/59392 = 521.856
Optimal tuning (CTE): ~80275/59392 = 521.856
{{Optimal ET sequence|legend=1|23, 430fhhhiiiijk, 453hi, 1336, 1789, 4914h}}


== Sextilimeans ==
== Sextilimeans ==
Sextilimeans is like sextilififths, but the fourth that is divided into 6 in sextilififths is tuned to a meantone fourth in the optimal tuning, or about 1/4.26-commma meantone. It should be noted, however, that this meantone fourth is not ~4/3 despite that the name may suggest so. In fact, the 3rd harmonic is not mapped in this temperament at all. It is described as the 229 & 1789 temperament.  
Sextilimeans is like [[sextilifourths]], but the fourth that is divided into 6 in sextilifourths is tuned to a meantone fourth in the optimal tuning, or about 1/4.26-commma meantone. It should be noted, however, that this meantone fourth is not ~4/3 despite that the name may suggest so. In fact, the 3rd harmonic is not mapped in this temperament at all. It is described as the 229 & 1789 temperament.  


[[Subgroup]]: 2.5.7.11.13
[[Subgroup]]: 2.5.7.11.13
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[[Optimal tuning]] ([[CTE]]): ~16807/16000 = 83.846
[[Optimal tuning]] ([[CTE]]): ~16807/16000 = 83.846


Vals: {{EDOs|229, 1789}}, ...
{{Optimal ET sequence|legend=1|229, 1789}}, ...


== Double Bastille ==
== Pure bastille ==
Described as the 1789 & 2814 temperament, and named because 2814 divided in two is 1407, and Bastille storming happened on 14 July 1789. Unfortunately the 1407 & 1789 temperament in the patent val does not temper out the jacobin comma, so it is not included here.
{{Main| Bastille }}


[[Subgroup]]: 2.5.7.11.13
Subgroup: 2.5.11.13


[[Comma list]]: 6656/6655, {{monzo|43 -18 0  5 -5}}, {{monzo|6 -30 -3  8 12}}
Comma list: 6656/6655, [1156  -812 336 -117⟩


[[Sval]] [[mapping]]: [{{Val|1 26 -938 -51 -136}}, {{Val|0 -30 1192 69 177}}]
Sval mapping: {{Val|1 11 -534 -1600}}, {{Val|0 -15 929 2772}}


[[Optimal tuning]] ([[CTE]]): ~91750400/53094899 = 947.121
Optimal tuning (CTE): ~2.5.11.13 {{Monzo|103 -57 14 -5}} = 694.243


Vals: 1789, 2814, ...
{{Optimal ET sequence|legend=1|1407eff, 1789, 4985eff}}


== French deck ==
== Double bastille ==
A period-52 temperament described as the 988 & 2444 temperament for the 2.5.11.13.29.31 subgroup, and tempers out the comma 2.29.31 {{monzo|-5 -52 52}}, which means 5 periods are equal to 31/29. Called so because there's 52 playing cards in the traditional deck. 1789edo does not support it as 1789 is a prime number, and therefore is not divisible by 52.
{{See also| No-threes subgroup temperaments #Bastille }}


[[Subgroup]]: 2.5.11.13.29.31
Described as the 1789 & 2814 temperament, and named because 2814 divided in two is 1407.


[[Comma list]]: 6656/6655, 2177736704/2177265625, 17179869184/17174157715, 57949573168357/57940459520000
[[Subgroup]]: 2.5.7.11.13


[[Sval]] [[mapping]]: [{{val|52 1 197 124 475 480}}, {{val|0 7 -1 4 -13 13}}]
[[Comma list]]: 6656/6655, {{monzo|43 -18  0  5 -5}}, {{monzo|6 -30 -3  8 12}}
 
[[Sval]] [[mapping]]: [{{Val|1 26 -938 -51 -136}}, {{Val|0 -30 1192 69 177}}]


Sval mapping generators: ~1460875/1441792, ~134560000/107132311
[[Optimal tuning]] ([[CTE]]): ~91750400/53094899 = 947.121


[[Optimal tuning]] ([[CTE]]): ~134560000/107132311 = 394.757
{{Optimal ET sequence|legend=1|1789, 2814, }} ...


Vals: {{EDOs|988, 1456, 2444}}, ...
== Acrosextilifourths ==
== Acrosextilififths ==
Discovered by [[Aura]] and defined as the 159 & 1619 temperament, with prefix acro- denoting the fact that it's a more precise version of sextilifourths, with fourth divided into 6 parts in 1619edo just as it is in 159edo.
Discovered by [[Aura]] and defined as the 159 & 1619 temperament, with prefix acro- denoting the fact that it's a more precise version of sextilififths, with fourth divided into 6 parts in 1619edo just as it is in 159edo.


[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13
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[[Optimal tuning]] ([[CTE]]): ~1573/1500 = 83.014
[[Optimal tuning]] ([[CTE]]): ~1573/1500 = 83.014


Vals: {{EDOs|159, 1460, 1619, 1778, 3079}}, ...
{{Optimal ET sequence|legend=1|159, 1460, 1619, 1778, 3079}}, ...


=== 17-limit ===
=== 17-limit ===
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[[Optimal tuning]] ([[CTE]]): ~1573/1500 = 83.014
[[Optimal tuning]] ([[CTE]]): ~1573/1500 = 83.014


Vals: {{EDOs|159, 1460, 1619, 1778}}, ..
{{Optimal ET sequence|legend=1|159, 1460, 1619, 1778}}, ..


== Declaration of Rights ==
== Declaration of rights ==
Defined as the 1789 & 1793 temperament, and called so because that's what both these years have in common.
Defined as the 1789 & 1793 temperament, and called so because that's what both these years have in common.


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Optimal tuning (CTE): ~2552639375/2147483648 = 299.162
Optimal tuning (CTE): ~2552639375/2147483648 = 299.162


Vals: {{EDOs|353, 357, 361, 710, 718, 1789, 1793}}, ...
{{Optimal ET sequence|legend=1|353, 357, 361, 710, 718, 1789, 1793}}, ...


== Eternal Revolutionary ==
== Eternal revolutionary ==
Described as the 1789 & 1880 temperament, and is named after a [[Wikipedia:ua:Вічний революціонер|poem by Ivan Franko]] <sup>[UA, no EN]</sup> which was written in the year 1880, hence the name.
Described as the 91 & 1880 temperament, or 1789bd & 1880 temperament, and is named after a [[Wikipedia:ua:Вічний революціонер|poem by Ivan Franko]] <sup>[UA, no EN]</sup> which was written in the year 1880, hence the name.


Subgroup: 2.5.11.13
Subgroup: 2.5.11.13
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Optimal tuning (CTE): ~2.5.11.13 {{monzo|294 -46 7 -57}} = 580.212
Optimal tuning (CTE): ~2.5.11.13 {{monzo|294 -46 7 -57}} = 580.212


Vals: {{EDOs|91, 1698, 1789, 1880, 3669}}, ...
[[Support]]ing [[ET]]s: {{EDOs|91, 1698, 1789, 1880, 3487, 3669, 5458, 7247}}, ...
=== Septimal Eternal Revolutionary===
 
Subgroup: 2.5.7.11.13
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 6656/6655, 768320/767637, {{Monzo|17  -6 13 -7 -2 -3}}
 
{{Mapping|legend=1|1 224 261 437 -159 -225|0 -460 -535 -898 336 473}}
 
: mapping generators: ~2 = 1\1, ~6875/4914 = 580.213
 
[[Optimal tuning]] ([[CTE]]): ~6875/4914 = 580.213
 
[[Support]]ing [[ET]]s: {{EDOs|91, 1698bdd, 1789bd, 1880, 1971c}}, ...
 
=== Hymn (rank-3) ===
An expansion of eternal revolutionary resulting from the 31 & 91 maximal evenness scale. Described as the 31f & 91 & 1880 temperament. It contains as a subset a rank-2 extension of the [[tritoni]] temperament into the 13-limit.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 6656/6655, {{monzo|-17 -12 6 4 1 2}}, {{monzo|-12 2 17 -11 -1 1}}


Comma list: 6656/6655, {{Monzo|-26 5 1 -2 5}}, {{Monzo|-43 35 35 -17 -21}}
{{Mapping|legend=2| 1 4 14 19 -15 40 | 0 -5 -6 -10 4 6 | 0 0 -17 22 32 79 }}


Sval mapping: [{{Val|1 261 -472 -159 -225}}, {{Val|0 -535 982 336 473}}]
Sval mapping generators: ~2 = 1\1, ~3773/2700 = 579.594, ~290304/203125 = 619.783


Optima tuning (CTE): ~482745786865234375/344859973188583424 = 580.213
[[Support]]ing [[ET]]s: {{EDOs|31f, 60f, 91, 122, 1789bd, 1880, 1911f, 2002c}}, ...


Vals: {{Val list|91d, 1698d, 1789, 1880, 1971cd, 3669}}
[[Category:Commatic realms]]
[[Category:Commatic realms]]
[[Category:Jacobin]]
[[Category:Jacobin]]
{{Todo| review }}
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