Mercator family
| 52nd-octave temperaments | 53rd-octave temperaments (Mercator family) |
54th-octave temperaments |
The Mercator family tempers out Mercator's comma, [-84 53⟩, and hence the fifths form a closed 53-note circle of fifths, identical to 53edo. While the tuning of the fifth will be that of 53edo, 0.069 cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.
Mercator
Subgroup: 2.3.5
Comma list: [-84 53⟩
Mapping: [⟨53 84 123], ⟨0 0 1]]
Mapping generators: ~531441/524288, ~5/1
Wedgie: ⟨⟨0 53 84]]
Optimal tuning (POTE): ~5/4 = 386.264
Optimal ET sequence: 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650
Badness: 0.284323
Schismerc
As per the name, Schismerc is characterized by the addition of the schisma, 32805/32768, to Mercator's comma, which completely reduces all commas in the Schismic-Mercator equivalence continuum to the unison, and thus, the 5-limit part is exactly the same as the 5-limit of 53edo, with the addition of harmonic 7 represented by an independent generator. Among the known 11-limit extensions are cartography, pentacontatritonic and boiler.
Subgroup: 2.3.5.7
Comma list: 15625/15552, 32805/32768
Mapping: [⟨53 84 123 0], ⟨0 0 0 1]]
Mapping generators: ~81/80, ~7/1
Wedgie: ⟨⟨0 0 53 0 84 123]]
Optimal tuning (POTE): ~225/224 = 5.3666
Optimal ET sequence: 53, 159, 212, 689c, 901cc
Badness: 0.087022
Cartography
Cartography is a strong extension to Schismerc that nails down both the 7-limit and the 11-limit by adding the symbiotic comma to Schismerc's list of tempered commas. The name for this temperament comes from how good the mappings are, and also from the idea of "Mercator" being a dual reference to both Nicolas Mercator and Gerardus Mercator.
Subgroup: 2.3.5.7.11
Comma list: 385/384, 6250/6237, 19712/19683
Mapping: [⟨53 84 123 0 332], ⟨0 0 0 1 -1]]
Mapping generators: ~81/80, ~7/1
Optimal tuning (POTE): ~225/224 = 6.1204
Optimal ET sequence: 53, 106d, 159, 212, 371d, 583cde
Badness: 0.054452
13-limit
13-limit Cartography adds the island comma to the list of tempered commas – a development which fits well with the ideas of mapmaking and geography. The harmonic 13 in this extension is part of the period and independent of the generator for harmonics 7 and 11.
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 625/624, 19712/19683
Mapping: [⟨53 84 123 0 332 196], ⟨0 0 0 1 -1 0]
Mapping generators: ~81/80, ~7/1
Optimal tuning (POTE): ~225/224 = 6.1430
Optimal ET sequence: 53, 106d, 159, 212, 371df, 583cdeff
Badness: 0.029980
Pentacontatritonic
First proposed by Xenllium, this temperament nails down both the 7-limit and the 11-limit by tempering out the swetisma. Like Cartography, Pentacontatritonic is a strong extension to Schismerc.
Subgroup: 2.3.5.7.11
Comma list: 540/539, 15625/15552, 32805/32768
Mapping: [⟨53 84 123 0 481], ⟨0 0 0 1 -2]]
Mapping generators: ~81/80, ~7/1
Optimal tuning (POTE): ~385/384 = 4.1494
Optimal ET sequence: 53, 159e, 212e, 265, 318, 583c
Badness: 0.115066
13-limit
13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out – in this extension the harmonic 13 is connected to the generator.
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4096/4095, 13750/13689
Mapping: [⟨53 84 123 0 481 345], ⟨0 0 0 1 -2 1]
Mapping generators: ~81/80, ~7/1
Optimal tuning (POTE): ~385/384 = 3.9850
Optimal ET sequence: 53, 159ef, 212ef, 265, 318, 583cf
Badness: 0.061158
Boiler
Boiler nails down both the 7-limit and the 11-limit by adding the kalisma to Schismerc's list of tempered commas, though unlike with the other extensions of Schismerc, this temperament is not only a weak extension, but lacks a clear 13-limit extension of its own. The name for this temperament is a reference to how 212 degrees Fahrenheit is the boiling point of water, as well as to a number of mechanical devices that boil water for various purposes.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 15625/15552, 32805/32768
Mapping: [⟨106 168 246 0 69], ⟨0 0 0 1 1]]
Mapping generators: ~2835/2816, ~7
Optimal tuning (POTE): ~225/224 = 6.3976 or ~441/440 = 4.9232
Badness: 0.109648
Joliet
Joliet can be characterized as the 53 & 106 temperament, having 7-limit representation akin to 53EDO with the addition of harmonic 11 represented by an independent generator. The name for this temperament is a reference to 106 being the maximum number of characters in the Joliet extension to the ISO 9660 file system.
Subgroup: 2.3.5.7.11
Comma list: 225/224, 1728/1715, 3125/3087
Mapping: [⟨53 84 123 149 0], ⟨0 0 0 0 1]]
Mapping generators: ~81/80, ~11/1
Optimal tuning (POTE): ~176/175 = 8.8283
Optimal ET sequence: 53, 106, 159d
Badness: 0.063254
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 325/324, 640/637
Mapping: [⟨53 84 123 149 0 196], ⟨0 0 0 0 1 0]]
Mapping generators: ~81/80, ~11/1
Optimal tuning (POTE): ~176/175 = 8.1254
Optimal ET sequence: 53, 106, 159d
Badness: 0.036988
Iodine
Proposed by Eliora, the name of iodine is taken from the convention of naming some fractional-octave temperaments after elements, in this case the 53rd chemical element. It can be expressed as the 159 & 742 temperament. 2 periods + 3 less than 600 cent generators correspond to 8/5. 5 less than 600 cent generators (minus 1 octave) correspond to 8/7.
Subgroup: 2.3.5.7
Comma list: [-19 14 -5 3⟩, [8 3 -20 12⟩
Mapping: [⟨53 84 2 -53], ⟨0 0 3 5]]
Mapping generators: ~3125/3087, 6075/3584
Optimal tuning (CTE): ~6075/3584 = 913.7347
Optimal ET sequence: 159, 424cd, 583, 742, 2385d, 3127d
Badness: 0.477
11-limit
24 periods plus the reduced generator correspond to 11/8.
Subgroup: 2.3.5.7.11
Comma list: 160083/160000, 820125/819896, 4302592/4296875
Mapping: [⟨53 84 2 -53 143], ⟨0 0 3 5 1]]
Optimal tuning (CTE): ~6075/3584 = 913.7322
Optimal ET sequence: 159, 424cd, 583, 742, 2385d, 3127d
Badness: 0.0875
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488
Mapping: [⟨53 84 2 -53 143 -46], ⟨0 0 3 5 1 6]]
Optimal tuning (CTE): ~441/260 = 913.7115
Optimal ET sequence: 159, 424cdff, 583f, 742, 1643
Badness: 0.0476
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872
Mapping: [⟨53 84 2 -53 143 -46 257], ⟨0 0 3 5 1 6 -1]]
Optimal tuning (CTE): ~441/260 = 913.7131
Optimal ET sequence: 159, 583f, 742
Badness: 0.0328