The Jacobins: Difference between revisions
→Barton: + Scott Dakota rediscovered this same temperament in 2025 and named it "hem"{{idio}}. |
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'''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 | 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. | ||
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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 }} | 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. | 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. | ||
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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 & 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 == | == Genojacobin == | ||
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Subgroup: 2.5.11.13 | Subgroup: 2.5.11.13 | ||
Comma list: 6656/6655, {{monzo| | Comma list: 6656/6655, {{monzo|-177 76 -79 74}} | ||
Sval mapping: {{Val|1 100 -99 -206}}, {{Val|0 -143 150 307}} | Sval mapping: {{Val|1 100 -99 -206}}, {{Val|0 -143 150 307}} | ||
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Optimal tuning (CTE): ~55115776/34328125 = 819.676 | 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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== Sextilimeans == | == Sextilimeans == | ||
Sextilimeans is like | 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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== Pure bastille == | == Pure bastille == | ||
{{Main| Bastille }} | |||
Subgroup: 2.5.11.13 | Subgroup: 2.5.11.13 | ||
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== Double bastille == | == Double bastille == | ||
{{See also| No-threes subgroup temperaments #Bastille }} | |||
Described as the 1789 & 2814 temperament, and named because 2814 divided in two is 1407. | Described as the 1789 & 2814 temperament, and named because 2814 divided in two is 1407. | ||
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{{Optimal ET sequence|legend=1|1789, 2814, }} ... | {{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 | 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 | [[Subgroup]]: 2.3.5.7.11.13 | ||
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== Eternal revolutionary == | == Eternal revolutionary == | ||
Described as the | 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 | ||
{{ | [[Support]]ing [[ET]]s: {{EDOs|91, 1698, 1789, 1880, 3487, 3669, 5458, 7247}}, ... | ||
=== | === 13-limit === | ||
Subgroup: 2.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 6656/6655, {{Monzo|- | 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) === | === Hymn (rank-3) === | ||
An expansion of | 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: 2.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 6656/6655, 2 | 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, ~ | 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 }} | {{Todo| review }} | ||