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
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[1].
Subgroup: 2.5.11.13
Comma list: 2200/2197, 6656/6655
Sval mapping: [⟨1 6 3 6], ⟨0 -8 1 -5]]
Gencom mapping: [⟨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: 11, 13, 24, 37, 50, 87, 298, 385, 472, 559, 1590cd
RMS error: 0.0822 cents
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, [-177 76 -79 74⟩
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, ...