# 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, 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.

## Jacobin

Subgroup: 2.3.5.7.11.13

Comma list: 6656/6655

[⟨ | 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: 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:

[⟨ | 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: 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: [⟨1 2 4], ⟨0 -3 -1]]

Optimal tuning (CTE): ~16/11 = 648.608

## 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, [357 -96 19 -54⟩

Sval mapping: ⟨1 100 -99 -206], ⟨0 -143 150 307]

Optimal tuning (CTE): ~2.5.11.13 [-106 28 -4 15⟩ = 819.676

Optimal ET sequence: 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: ⟨1 100 -99 -206 -17], ⟨0 -143 150 307 32]

Optimal tuning (CTE): ~55115776/34328125 = 819.676

Optimal ET sequence: 183, 1057f, 1240, 1423, 1606, 1789, 3395

## 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 (see above).

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

### 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: [⟨1 74 114 3 74], ⟨0 -156 -242 1 -153]]

Optimal tuning (CTE): ~11/8 = 551.369

## 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.

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

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

Optimal ET sequence: 23, 430fhhh, 453h, 1336, 1789, 3125

### 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

Optimal ET sequence: 23, 430fhhhiiii, 453hi, 1336, 1789, 4914h

### 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

Optimal ET sequence: 23, 430fhhhiiiij, 453hi, 1336, 1789, 3125

### 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

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

Optimal ET sequence: 23, 430fhhhiiiijk, 453hi, 1336, 1789, 4914h

## 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

Optimal ET sequence: 229, 1789, ...

## Pure bastille

Subgroup: 2.5.11.13

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

Sval mapping: ⟨1 11 -534 -1600], ⟨0 -15 929 2772]

Optimal tuning (CTE): ~2.5.11.13 [103 -57 14 -5⟩ = 694.243

Optimal ET sequence: 1407eff, 1789, 4985eff

## Double bastille

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

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

Optimal ET sequence: 1789, 2814, ...

## 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 sextilififths, with fourth divided into 6 parts in 1619edo just as it is in 159edo.

Subgroup: 2.3.5.7.11.13

Comma list: 6656/6655, 123201/123200, 759375/758912, 2250423/2249728

Mapping: [⟨1 2 21 43 11 45], ⟨0 -6 -270 -581 -109 -597]]

Optimal tuning (CTE): ~1573/1500 = 83.014

Optimal ET sequence: 159, 1460, 1619, 1778, 3079, ...

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2500/2499, 6656/6655, 61965/61952, 123201/123200, 1285956/1285625

Mapping: [⟨1 2 21 43 11 45 -2], ⟨0 -6 -270 -581 -109 -597 88]]

Optimal tuning (CTE): ~1573/1500 = 83.014

Optimal ET sequence: 159, 1460, 1619, 1778, ..

## Declaration of rights

Defined as the 1789 & 1793 temperament, and called so because that's what both these years have in common.

Subgroup: 2.5.11.13

Comma list: 6656/6655, [-176 23 -2 35⟩

Sval mapping: [⟨1 28 -11 -14], ⟨0 -103 58 71]]

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

Optimal ET sequence: 353, 357, 361, 710, 718, 1789, 1793, ...

## Eternal revolutionary

Described as the 91 & 1880 temperament, or 1789bd & 1880 temperament, and is named after a poem by Ivan Franko ^{[UA, no EN]} which was written in the year 1880, hence the name.

Subgroup: 2.5.11.13

Comma list: 6656/6655, [-966 151 -20 185⟩

Sval mapping: [⟨1 261 -159 -225], ⟨0 -535 336 473]]

Optimal tuning (CTE): ~2.5.11.13 [294 -46 7 -57⟩ = 580.212

Supporting ETs: 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, [17 -6 13 -7 -2 -3⟩

Mapping: [⟨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

Supporting ETs: 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, [-17 -12 6 4 1 2⟩, [-12 2 17 -11 -1 1⟩

Sval mapping: [⟨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

Supporting ETs: 31f, 60f, 91, 122, 1789bd, 1880, 1911f, 2002c, ...