The Jacobins: Difference between revisions
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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]]. | 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 ― and it is consistent to distance 2 in the 2.5.11.13 subgroup, subgroup of the comma. 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 == | == Jacobin == | ||
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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 == | === 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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{{See also| Chromatic pairs #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>. | 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 | [[Subgroup]]: 2.5.11.13 | ||
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[[Tp tuning #T2 tuning|RMS error]]: 0.0822 cents | [[Tp tuning #T2 tuning|RMS error]]: 0.0822 cents | ||
== Huguenot == | |||
Huguenot, named after French calvinists who fled to Ireland, tempers out the [[4225/4224|leprechaun comma]] together with the jacobin comma. | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 4225/4224, 6656/6655 | |||
{{Mapping|legend=1| 1 1 4 2 2 1 | 0 1 4 0 -3 -5 | 0 0 -5 0 4 7 | 0 0 0 1 0 0 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.009{{c}}, ~3/2 = 701.954{{c}}, ~96/55 = 964.312{{c}}, 7/4 = 968.809{{c}} | |||
: [[error map]]: {{val| 0.009, 0.008, -0.025, -0.000, 0.087, -0.102 }} | |||
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.953{{c}}, ~96/55 = 964.309{{c}}, 7/4 = 968.807{{c}} | |||
: error map: {{val| 0.000, -0.002, -0.045, -0.019, 0.058, -0.132 }} | |||
[[Support]]ing [[ET]]s: {{EDOs|87, 224, 270, 311, 494, 581, 764, 814, 1084, 1308}} | |||
[[Badness]]: 0.508 | |||
== Genojacobin == | == Genojacobin == | ||
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{{Optimal ET sequence|legend=1|183, 1057f, 1240, 1423, 1606, 1789, 3395 }} | {{Optimal ET sequence|legend=1|183, 1057f, 1240, 1423, 1606, 1789, 3395 }} | ||
=== Acroquanharuk === | |||
Described as 183 & 1240 temperament in the full 13-limit, and named by analogy with acrosextilifourths (see below) as a more large-edo expansion upon quanharuk. 1789d val is chosen as it is simpler. | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 729/728, 6656/6655, 1016064/1015625, 14085981/14080000 | |||
{{Mapping|legend=1| 1 0 -43 -104 51 101 | 0 5 143 337 -150 -307 }} | |||
Optimal tuning (CWE): ~1755/1408 = 380.323 | |||
[[Support]]ing [[ET]]s: {{EDOs|183, 1057f, 1240, 1403, 1789d}} | |||
== Onzonic == | == Onzonic == | ||
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== Eternal revolutionary == | == Eternal revolutionary == | ||
Described as the 91 & 1880 temperament, or 1789bd & 1880 temperament, and is named after a [[Wikipedia: | Described as the 91 & 1880 temperament, or 1789bd & 1880 temperament, and is named after a [[Wikipedia:uk: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 | ||