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→Barton: + Scott Dakota rediscovered this same temperament in 2025 and named it "hem"{{idio}}.
 
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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]], as the 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 ==
Line 9: Line 112:


=== 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. 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.
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
Line 19: Line 122:
[[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 in between the 2.5.11.13 subgroup adds the mapping for 7.
 
Subgroup: 2.5.7.11.13
 
Comma list: 6656/6655, 200126927/200000000, 41322093568/41259765625
 
Sval mapping: [{{val| 1 74 114 3 74 }}, {{val| 0 -156 -242 1 -153 }}]
 
Optimal tuning (CTE): ~11/8 = 551.369
 
{{Optimal ET sequence|legend=1| 37, 1789 }}


== Estates general ==
== Estates general ==
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.
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.
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]].


[[Subgroup]]: 2.5.11.13.19
[[Subgroup]]: 2.5.11.13.19
Line 31: Line 149:


[[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 ===
Line 40: Line 160:


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 49: Line 171:


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


Line 58: Line 184:


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
Line 70: Line 198:
[[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}}, ...
 
== Pure bastille ==
{{Main| Bastille }}


== Double Bastille ==
Subgroup: 2.5.11.13
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.


[[Subgroup]]: 2.5.7.11.13
Comma list: 6656/6655, [1156  -812  336 -117⟩


[[Comma list]]: 6656/6655, {{monzo|43 -18  0  5 -5}}, {{monzo|6 -30 -3  8 12}}
Sval mapping: {{Val|1 11 -534 -1600}}, {{Val|0 -15 929 2772}}


[[Sval]] [[mapping]]: [{{Val|1 26 -938 -51 -136}}, {{Val|0 -30 1192 69 177}}]
Optimal tuning (CTE): ~2.5.11.13 {{Monzo|103 -57 14 -5}} = 694.243


[[Optimal tuning]] ([[CTE]]): ~91750400/53094899 = 947.121
{{Optimal ET sequence|legend=1|1407eff, 1789, 4985eff}}


Vals: 1789, 2814, ...
== Double bastille ==
{{See also| No-threes subgroup temperaments #Bastille }}


== French deck ==
Described as the 1789 & 2814 temperament, and named because 2814 divided in two is 1407.
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.


[[Subgroup]]: 2.5.11.13.29.31
[[Subgroup]]: 2.5.7.11.13


[[Comma list]]: 6656/6655, 2177736704/2177265625, 17179869184/17174157715, 57949573168357/57940459520000
[[Comma list]]: 6656/6655, {{monzo|43 -18  0  5 -5}}, {{monzo|6 -30 -3  8 12}}


[[Sval]] [[mapping]]: [{{val|52 1 197 124 475 480}}, {{val|0 7 -1 4 -13 13}}]
[[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
Line 110: Line 239:
[[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 ===
Line 122: Line 251:
[[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.


Line 131: Line 260:
Comma list: 6656/6655, {{monzo|-176 23 -2 35}}
Comma list: 6656/6655, {{monzo|-176 23 -2 35}}


Sval mapping: 1 28 -11 -14, 0 -103 58 71
Sval mapping: [{{val|1 28 -11 -14}}, {{val|0 -103 58 71}}]


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


Vals: 353, 357, 361, 710, 718, 1789, 1793, ...
{{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}}
 
{{Mapping|legend=2| 1 4 14 19 -15 40 | 0 -5 -6 -10 4 6 | 0 0 -17 22 32 79 }}
 
Sval mapping generators: ~2 = 1\1, ~3773/2700 = 579.594, ~290304/203125 = 619.783
 
[[Support]]ing [[ET]]s: {{EDOs|31f, 60f, 91, 122, 1789bd, 1880, 1911f, 2002c}}, ...
 
[[Category:Commatic realms]]
[[Category:Commatic realms]]
[[Category:Jacobin]]
[[Category:Jacobin]]
{{Todo| review }}
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