Mercator family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

Fractional-octave temperaments

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← 52nd-octave temperaments 53rd-octave temperaments 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.

Discussed elsewhere are:

• Aemilic (+250047/250000) → 159th-octave temperaments

Mercator

Subgroup: 2.3.5

Comma list: [-84 53⟩

Mapping: [⟨53 84 0], ⟨0 0 1]]

mapping generators: ~531441/524288, ~5

Optimal tunings:

• CTE: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.3137¢
• CWE: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.2804¢

Optimal ET sequence: 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650

Badness (Sintel): 6.670

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

Optimal tunings:

• CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.1741¢
• CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.6299¢

Optimal ET sequence: 53, 159, 212, 689c, 901cc

Badness (Sintel): 2.202

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]]

Optimal tunings:

• CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
• CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5178¢

Optimal ET sequence: 53, 106d, 159, 212, 371d, 583cde

Badness (Sintel): 1.800

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]

Optimal tunings:

• CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
• CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5397¢

Optimal ET sequence: 53, 106d, 159, 212, 371df, 583cdeff

Badness (Sintel): 1.239

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]]

Optimal tunings:

• CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5956¢
• CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5697¢

Optimal ET sequence: 53, 159e, 212e, 265, 318, 583c

Badness (Sintel): 3.804

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]

Optimal tuning (POTE): ~385/384 = 3.9850

Optimal tunings:

• CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4057¢
• CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4008¢

Optimal ET sequence: 53, 159ef, 212ef, 265, 318, 583cf

Badness (Sintel): 2.527

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 tunings:

• CTE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 230.6341¢
• CWE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 231.1634¢

Optimal ET sequence: 106, 212

Badness (Sintel): 3.625

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: ~50/49, ~11

Optimal tunings:

• CTE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
• CWE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 552.0415¢

Optimal ET sequence: 53, 106, 159d

Badness (Sintel): 2.091

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]]

Optimal tunings:

• CTE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
• CWE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.4859¢

Optimal ET sequence: 53, 106, 159d

Badness (Sintel): 1.528

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 tunings:

• CTE: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7347¢
• CWE: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7301¢

Optimal ET sequence: 159, 424cd, 583, 742, 2385d, 3127d

Badness (Sintel): 12.075

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 tunings:

• CTE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7322¢
• CWE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7345¢

Optimal ET sequence: 159, 424cd, 583, 742, 2385d, 3127d

Badness (Sintel): 2.893

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 tunings:

• CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7115¢
• CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7126¢

Optimal ET sequence: 159, 424cdff, 583f, 742, 1643

Badness (Sintel): 1.967

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 tunings:

• CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7131¢
• CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7208¢

Optimal ET sequence: 159, 583f, 742

Badness (Sintel): 1.568