The Jacobins

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.

Quite coincidentally, 1789edo supports an enormous amount of these temperaments.

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.

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.

Subgroup: 2.5.11.13

Comma list: 6656/6655, [-119 -46 15 47⟩

Sval mapping: [⟨1 74 3 74], ⟨0 -156 1 -153]]

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

Vals: 37, 1789

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.

Subgroup: 2.5.11.13.19

Comma list: 6656/6655, 40960000000/40943078891, [-133 50 -7 18 -6⟩

Sval mapping: [⟨1 118 -107 -212 450], ⟨0 -266 254 496 -1025]]

Optimal tuning (CTE): ~2588443885831192576/1914932769775390625 = 521.856

2.5.11.13.19.23 subgroup

Subgroup: 2.5.11.13.19.23

Comma list: 6656/6655, 62500/62491, 190676992/190653125, [-92 23 -2 14 -10  8⟩

Sval mapping: [⟨1 118 -107 -212 450 579], ⟨0 -266 254 496 -1025 -1321]]

Optimal tuning (CTE): ~2592407900127232/1918105439453125 = 521.856

2.5.11.13.19.23.29 subgroup

Subgroup: 2.5.11.13.19.23.29

Comma list: 6656/6655, 62500/62491, 190676992/190653125, 7592198144/7591796875, 897740062375/897648164864

Sval mapping: [⟨1 118 -107 -212 450 579 251], ⟨0 -266 254 496 -1025 -1321 -566]]

Optimal tuning (CTE): ~184000/136097 = 521.856

2.5.11.13.19.23.29.31 subgroup

Subgroup: 2.5.11.13.19.23.29.31

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

Sval mapping: [⟨1 118 -107 -212 450 579 251 -179], ⟨0 -266 254 496 -1025 -1321 -566 423]]

Optimal tuning (CTE): ~80275/59392 = 521.856

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.

Subgroup: 2.5.7.11.13

Comma list: 6656/6655, 8122034375/8120172544, [-12 -29 36 -2 -4⟩

Sval mapping: [⟨1 36 23 -24 -45], ⟨0 -482 -289 393 697]]

Optimal tuning (CTE): ~16807/16000 = 83.846

Vals: 229, 1789, ...

Double Bastille

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, [43 -18 0 5 -5⟩, [6 -30 -3 8 12⟩

Sval mapping: [⟨1 26 -938 -51 -136], ⟨0 -30 1192 69 177]]

Optimal tuning (CTE): ~91750400/53094899 = 947.121

Vals: 1789, 2814, ...