The Jacobins: Difference between revisions
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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>. | |||
[[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 == | ||