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→Barton: + Scott Dakota rediscovered this same temperament in 2025 and named it "hem"{{idio}}.
 
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The Jacobins is a family of microtemperaments which temper out the jacobin comma, 6656/6655.
{{Technical data page}}
'''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, as jacobin comma can be entrenched in other temperaments 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.
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 ==
Since 6656/6655 is the difference between a stack of three 11/8's and 13/10, it is natural to choose a rank-2 temperament that uses 11/8 as the generator to exploit the comma. Such a mapping is realized through the fractional subgroup 2.13/10.11, which produces a basis with just one comma - namely the 6656/6655. Name given because the 13/10 interval is sometimes referred to as a "naiadic", and this name separates it from the standard diatonic framework.
 
Subgroup: 2.13/10.11
 
Comma list: 6656/6655
 
Sval mapping: [{{Val|1 2 4}}, {{Val|0 -3 -1}}]
 
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 ==
Named for the French word for eleven, ''onze'', since the generator is 11/8. Initially defined for 2.5.11.13, but it can be extended.
Named for the French word for eleven, ''onze'', since the generator is 11/8. Initially defined for 2.5.11.13, but it can be extended.


===Pure onzonic===
=== Pure onzonic ===
Pure onzonic is the temperament that was initially defined as "jacobin" before pointing out that the name "jacobin temperament" would refer to a rank 5 temperamnet tempering out 6656/6655. Since jacobin comma is the difference between a stack of three 11/8s and 13/10, it was natural to choose 11/8 as the generator for the rank 2 "jacobin temperament". Name "pure onzonic" is thus reserved for the pure 2.5.11.13 subgroup.
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
 
[[Comma list]]: 6656/6655, {{monzo| -119 -46 15 47 }}
 
[[Sval]] [[mapping]]: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
 
[[Optimal tuning]] ([[CTE]]): ~11/8 = 551.370
 
{{Optimal ET sequence|legend=1| 37, 1789 }}
 
=== Septimal onzonic ===
Septimal onzonic in between the 2.5.11.13 subgroup adds the mapping for 7.


Subgroup: 2.5.11.13
Subgroup: 2.5.7.11.13


Comma list: 6656/6655, {{monzo|-119 -46 15 47}}
Comma list: 6656/6655, 200126927/200000000, 41322093568/41259765625


Mapping: [{{val|1 74 3 74}}, {{val|0 -156 1 -153}}]
Sval mapping: [{{val| 1 74 114 3 74 }}, {{val| 0 -156 -242 1 -153 }}]


Optimal tuning (CTE): ~11/8 = 551.370
Optimal tuning (CTE): ~11/8 = 551.369


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


== Estates general ==
== Estates general ==
Named so because it is defined as the 1789 & 3125 temperament due to 3125 providing optimal patent val for the jacobin comma, 3125 is 5 to the 5th power, and Estates General were called by Louis XVI on 5th May 1789 (05/05). Defined starting with the 2.5.11.13.19 subgroup, upwards to the 2.5.11.13.19.23.29.31 subgroup.
Named so because it is described as the 1789 & 3125 temperament due to 3125 providing the optimal patent val for the jacobin comma, 3125 is 5 to the 5th power, and Estates General were called by Louis XVI on 5th May 1789 (05/05). Defined starting with the 2.5.11.13.19 subgroup, upwards to the 2.5.11.13.19.23.29.31 subgroup.


Subgroup: 2.5.11.13.19
3 generators below 600 cents lead to 25289/10240, and octave reduced to [[247/200]] since the jacobin comma is tempered out. 24 generators below 600 cents lead to [[88/65]].


Comma list: 6656/6655, 40960000000/40943078891, {{monzo| -133 50 -7 18 -6 }}
[[Subgroup]]: 2.5.11.13.19


Mapping: [{{val| 1 118 -107 -212 450 }}, {{val| 0 -266 254 496 -1025 }}]
[[Comma list]]: 6656/6655, 40960000000/40943078891, {{monzo| -133 50 -7 18 -6 }}


Optimal tuning (CTE): ~2588443885831192576/1914932769775390625 = 521.856
[[Sval]] [[mapping]]: [{{val| 1 118 -107 -212 450 }}, {{val| 0 -266 254 496 -1025 }}]
 
[[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 ===
Line 37: Line 157:
Comma list: 6656/6655, 62500/62491, 190676992/190653125, {{Monzo|-92 23 -2 14 -10  8}}
Comma list: 6656/6655, 62500/62491, 190676992/190653125, {{Monzo|-92 23 -2 14 -10  8}}


Mapping: [{{val| 1 118 -107 -212 450 579}}, {{val| 0 -266 254 496 -1025 -1321}}]
Sval mapping: [{{val| 1 118 -107 -212 450 579}}, {{val| 0 -266 254 496 -1025 -1321}}]


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 ===
Line 46: Line 168:
Comma list: 6656/6655, 62500/62491, 190676992/190653125, 7592198144/7591796875, 897740062375/897648164864
Comma list: 6656/6655, 62500/62491, 190676992/190653125, 7592198144/7591796875, 897740062375/897648164864


Mapping: [{{val| 1 118 -107 -212 450 579 251}}, {{val| 0 -266 254 496 -1025 -1321 -566}}]
Sval mapping: [{{val| 1 118 -107 -212 450 579 251}}, {{val| 0 -266 254 496 -1025 -1321 -566}}]


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 ===
[[31/26]] can be reached in 73 generators.
Subgroup: 2.5.11.13.19.23.29.31
Subgroup: 2.5.11.13.19.23.29.31


Comma list: 6656/6655, 62500/62491, 9425/9424, 190676992/190653125, 507528125/507510784, 519411073024/519363934375
Comma list: 6656/6655, 62500/62491, 9425/9424, 190676992/190653125, 507528125/507510784, 519411073024/519363934375


Mapping: [{{val| 1 118 -107 -212 450 579 251 -179}}, {{val| 0 -266 254 496 -1025 -1321 -566 423}}]
Sval mapping: [{{val| 1 118 -107 -212 450 579 251 -179}}, {{val| 0 -266 254 496 -1025 -1321 -566 423}}]


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 ==
It's like sextilififths, but the fourth that is divided into 6 is tuned meantone, corresponding to a fifth of 1039\1789, or about 1/4.26-commma meantone.
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.  
Defined as the 229 & 1789 temperament.


Subgroup: 2.5.7.11.13
[[Subgroup]]: 2.5.7.11.13
 
[[Comma list]]: 6656/6655, 8122034375/8120172544, {{monzo|-12 -29 36 -2 -4}}
 
[[Sval]] [[mapping]]: [{{val|1 36 23 -24 -45}}, {{val|0 -482 -289 393 697}}]
 
[[Optimal tuning]] ([[CTE]]): ~16807/16000 = 83.846
 
{{Optimal ET sequence|legend=1|229, 1789}}, ...
 
== Pure bastille ==
{{Main| Bastille }}
 
Subgroup: 2.5.11.13
 
Comma list: 6656/6655, [1156  -812  336 -117⟩
 
Sval mapping: {{Val|1 11 -534 -1600}}, {{Val|0 -15 929 2772}}
 
Optimal tuning (CTE): ~2.5.11.13 {{Monzo|103 -57 14 -5}} = 694.243
 
{{Optimal ET sequence|legend=1|1407eff, 1789, 4985eff}}
 
== Double bastille ==
{{See also| No-threes subgroup temperaments #Bastille }}
 
Described as the 1789 & 2814 temperament, and named because 2814 divided in two is 1407.
 
[[Subgroup]]: 2.5.7.11.13
 
[[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}}]
 
[[Optimal tuning]] ([[CTE]]): ~91750400/53094899 = 947.121
 
{{Optimal ET sequence|legend=1|1789, 2814, }} ...
 
== Acrosextilifourths ==
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.
 
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 6656/6655, 123201/123200, 759375/758912, 2250423/2249728
 
[[Mapping]]: [{{val|1 2 21 43 11 45}}, {{val|0 -6 -270 -581 -109 -597}}]
 
[[Optimal tuning]] ([[CTE]]): ~1573/1500 = 83.014
 
{{Optimal ET sequence|legend=1|159, 1460, 1619, 1778, 3079}}, ...
 
=== 17-limit ===
 
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[Comma list]]: 2500/2499, 6656/6655, 61965/61952, 123201/123200, 1285956/1285625
 
[[Mapping]]: [{{val|1 2 21 43 11 45 -2}}, {{val|0 -6 -270 -581 -109 -597 88}}]
 
[[Optimal tuning]] ([[CTE]]): ~1573/1500 = 83.014
 
{{Optimal ET sequence|legend=1|159, 1460, 1619, 1778}}, ..
 
== Declaration of rights ==
Defined as the 1789 & 1793 temperament, and called so because that's what both these years have in common.
 
Subgroup: 2.5.11.13
 
Comma list: 6656/6655, {{monzo|-176 23 -2 35}}
 
Sval mapping: [{{val|1 28 -11 -14}}, {{val|0 -103 58 71}}]
 
Optimal tuning (CTE): ~2552639375/2147483648 = 299.162
 
{{Optimal ET sequence|legend=1|353, 357, 361, 710, 718, 1789, 1793}}, ...
 
== Eternal revolutionary ==
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
 
Comma list: 6656/6655, {{Monzo|-966 151 -20 185}}
 
Sval mapping: [{{Val|1 261 -159 -225}}, {{Val|0 -535 336 473}}]
 
Optimal tuning (CTE): ~2.5.11.13 {{monzo|294 -46 7 -57}} = 580.212
 
[[Support]]ing [[ET]]s: {{EDOs|91, 1698, 1789, 1880, 3487, 3669, 5458, 7247}}, ...
 
=== 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, 8122034375/8120172544, {{monzo|-12 -29 36 -2 -4}}
{{Mapping|legend=2| 1 4 14 19 -15 40 | 0 -5 -6 -10 4 6 | 0 0 -17 22 32 79 }}


Mapping: [{{val|1 36 23 -24 -45}}, {{val|0 -482 -289 393 697}}]
Sval mapping generators: ~2 = 1\1, ~3773/2700 = 579.594, ~290304/203125 = 619.783


Optimal tuning (CTE): ~16807/16000 = 83.846
[[Support]]ing [[ET]]s: {{EDOs|31f, 60f, 91, 122, 1789bd, 1880, 1911f, 2002c}}, ...


Vals: {{EDOs|229, 1789}}, ...
[[Category:Commatic realms]]
[[Category:Temperament collections]]
[[Category:Jacobin]]
[[Category:Jacobin]]
[[Category:Rank 2]]
{{Todo| review }}
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