# 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 subgorup 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-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

## Gene's jacobin

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

### 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

## 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 jacobin-naiadic 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

### 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

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

## 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

## 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

## 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

Vals: 1789, 2814, ...

## French deck

A period-52 temperament described as the 988 & 2444 temperament for the 2.5.11.13.19.29.31 subgroup, and tempers out the comma 2.29.31 [-5 -52 52⟩, which means 5 periods are equal to 31/29. Called so because there's 52 playing cards in the traditional deck. 1789edo does not support it as 1789 is a prime number, and therefore is not divisible by 52.

Subgroup: 2.5.11.13.19.29.31

Comma list: 6656/6655, 600704/600625, 1308736/1308625, 35934301/35932160, 17179869184/17174157715

Sval mapping: [⟨52 1 197 124 238 475 480], ⟨0 7 -1 4 -1 -13 13]]

Sval mapping generators: ~1460875/1441792, ~1045/832

Optimal tuning (CTE): ~134560000/107132311 = 394.757

## 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

Vals: 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

Vals: 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

Vals: 353, 357, 361, 710, 718, 1789, 1793, ...

## Eternal Revolutionary

Described as the 1789 & 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

Vals: 91, 1698, 1789, 1880, 3669, ...

### Septimal Eternal Revolutionary

Subgroup: 2.5.7.11.13

Comma list: 6656/6655, [-26 5 1 -2 5⟩, [-43 35 35 -17 -21⟩

Sval mapping: [⟨1 261 -472 -159 -225], ⟨0 -535 982 336 473]]

Optima tuning (CTE): ~482745786865234375/344859973188583424 = 580.213

Vals: 91d, 1698d, 1789, 1880, 1971cd, 3669

### Hymn (rank-3)

An expansion of Eternal Revolutionary resulting from the 31 & 91 maximal evenness scale. Described as the 31f & 91d & 1789 temperament.

Subgroup: 2.5.7.11.13

Comma list: 6656/6655, [-533 190 124 -119 42⟩

Sval mapping: [⟨1 0 3 77 222], ⟨0 1 4 -104 -311], ⟨0 0 7 -124 -372]]

Sval mapping generators: ~2, ~5, ~? = -1625.1946

Vals: 31f, 91d, 1789, 3669, ...

## Silicon

While 14edo is poor in simple harmonics, some of its multiples (such as 224edo and 742edo) are members of zeta edo list. The name of silicon temperament comes from the 14th element. Defined upwards to the 13-limit. When tuned in 742edo, it is generated by a 53edo fifth intermingled with 14edo periods.

Subgroup: 2.3.5.7

Comma list: 14348907/14336000, 56358560858112/56296884765625

Mapping: [⟨14 0 -145 239], ⟨0 1 8 -9]]

Mapping generators: ~6125/5832, ~3

Optimal tuning (CTE): ~3/2 = 701.870

Optimal GPV sequence: 224, 518, 742, 966, 1708, 2674

Badness: 0.196

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 1240029/1239040, 2359296/2358125

Mapping: [⟨14 0 -145 239 115], ⟨0 1 8 -9 -3]]

Optimal tuning (CTE): ~3/2 = 701.872

Optimal GPV sequence: 224, 518, 742, 966, 1708, 4382e

Badness: 0.0450

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 6656/6655, 9801/9800, 24192/24167

Mapping: [⟨14 0 -145 239 115 74], ⟨0 1 8 -9 -3 -1]]

Optimal tuning (CTE): ~3/2 = 701.8733

Optimal GPV sequence: 224, 518, 742, 966, 1708f

Badness: 0.0269