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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The mercator family of temperaments 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 0], ⟨0 0 1]]

mapping generators: ~531441/524288, ~5

Optimal tunings:

• WE: ~531441/524288 = 22.6419 ¢, ~5/4 = 386.2707 ¢ (~32805/32768 = 1.3581 ¢)

error map: ⟨+0.022 -0.034 -0.000]

• CWE: ~531441/524288 = 22.6415 ¢, ~5/4 = 386.2804 ¢ (~32805/32768 = 1.3747 ¢)

error map: ⟨0.000 -0.068 -0.033]

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

Badness (Sintel): 6.67

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 11-limit extensions are cartography, pentacontatritonic, boiler, and auric.

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:

• WE: ~81/80 = 22.6456 ¢, ~7/4 = 968.3928 ¢ (~225/224 = 5.3675 ¢)

error map: ⟨+0.216 +0.275 -0.906 -0.001]

• CWE: ~81/80 = 22.6415 ¢, ~7/4 = 968.3701 ¢ (~225/224 = 5.2148 ¢)

error map: ⟨0.000 -0.068 -1.408 -0.456]

Optimal ET sequence: 53, 159, 212, 689c, 901cc, 1113ccd

Badness (Sintel): 2.20

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.

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

Comma list: 385/384, 6250/6237, 19712/19683

Mapping: [⟨53 84 123 0 332], ⟨0 0 0 1 -1]]

Optimal tunings:

• WE: ~81/80 = 22.6478 ¢, ~7/4 = 967.7312 ¢ (~225/224 = 6.1221 ¢)
• CWE: ~81/80 = 22.6415 ¢, ~7/4 = 967.4822 ¢ (~225/224 = 6.1027 ¢)

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

Badness (Sintel): 1.80

13-limit

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:

• WE: ~81/80 = 22.6491 ¢, ~7/4 = 967.7656 ¢ (~225/224 = 6.1450 ¢)
• CWE: ~81/80 = 22.6415 ¢, ~7/4 = 967.4603 ¢ (~225/224 = 6.1246 ¢)

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

Badness (Sintel): 1.24

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.

13-limit pentacontatritonic adds the minisma 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

Comma list: 540/539, 15625/15552, 32805/32768

Mapping: [⟨53 84 123 0 481], ⟨0 0 0 1 -2]]

Optimal tunings:

• WE: ~81/80 = 22.6427 ¢, ~7/4 = 969.4874 ¢ (~385/384 = 4.1496 ¢)
• CWE: ~81/80 = 22.6415 ¢, ~7/4 = 969.4303 ¢ (~385/384 = 4.1546 ¢)

Optimal ET sequence: 53, 212e, 265, 318

Badness (Sintel): 3.80

13-limit

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 tunings:

• WE: ~81/80 = 22.6416 ¢, ~7/4 = 969.6046 ¢ (~385/384 = 3.9850 ¢)
• CWE: ~81/80 = 22.6415 ¢, ~7/4 = 969.5992 ¢ (~385/384 = 3.9858 ¢)

Optimal ET sequence: 53, 212ef, 265, 318

Badness (Sintel): 2.53

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:

• WE: ~2835/2816 = 11.3229 ¢, ~7/4 = 968.8469 ¢ (~225/224 = 4.9241 ¢)
• CWE: ~2835/2816 = 11.3208 ¢, ~7/4 = 968.8633 ¢ (~225/224 = 4.7483 ¢)

Optimal ET sequence: 106, 212, 530c, 742ce

Badness (Sintel): 3.62

Auric

Auric, named after Aura, is defined by setting the 2.3.5.11 mapping to that of 159edo and the generator for the prime 7 being independent.

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 15625/15552, 32805/32768

Mapping: [⟨159 252 369 0 550], ⟨0 0 0 1 0]]

mappping generators: ~243/242, ~7

Optimal tunings:

• WE: ~243/242 = 7.5484 ¢, ~7/4 = 968.4469 ¢ (~3024/3025 = 2.2569 ¢)
• CWE: ~243/242 = 7.5472 ¢, ~7/4 = 968.4005 ¢ (~3024/3025 = 2.3627 ¢)

Optimal ET sequence: 159, 318, 477c

Badness (Sintel): 5.23

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

Optimal tunings:

• WE: ~81/80 = 22.6365 ¢, ~11/8 = 551.1028 ¢ (~176/175 = 8.8263 ¢)

error map: ⟨-0.264 -0.487 -2.022 +4.015 -0.009]

• CWE: ~81/80 = 22.6415 ¢, ~11/8 = 552.0415 ¢ (~176/175 = 8.6452 ¢)

error map: ⟨0.000 -0.068 -1.408 +4.759 +0.724]

Optimal ET sequence: 53, 106, 159d

Badness (Sintel): 2.09

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:

• WE: ~81/80 = 22.6403 ¢, ~11/8 = 551.4931 ¢ (~176/175 = 8.1250 ¢)
• CWE: ~81/80 = 22.6415 ¢, ~11/8 = 551.4859 ¢ (~176/175 = 8.0897 ¢)

Optimal ET sequence: 53, 106

Badness (Sintel): 1.53

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. 3 generators plus 2 periods reach the 5th harmonic. 5 generators reach the 14th harmonic.

In the 11-limit, the ~11/8 can serve as a generator, being 16 periods less than the canonical generator.

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:

• WE: ~3125/3087 = 22.6417 ¢, ~6075/3584 = 913.7361 ¢ (~225/224 = 8.0675 ¢)

error map: ⟨+0.011 -0.051 +0.178 -0.156]

• CWE: ~3125/3087 = 22.6415 ¢, ~6075/3584 = 913.7301 ¢ (~225/224 = 8.0697 ¢)

error map: ⟨0.000 -0.068 +0.159 -0.176]

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

Badness (Sintel): 12.1

11-limit

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:

• WE: ~1815/1792 = 22.6414 ¢, ~6075/3584 = 913.7318 ¢ (~225/224 = 8.0750 ¢)
• CWE: ~1815/1792 = 22.6415 ¢, ~6075/3584 = 913.7345 ¢ (~225/224 = 8.0742 ¢)

Optimal ET sequence: 159, 424cd, 583, 742

Badness (Sintel): 2.89

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:

• WE: ~78/77 = 22.6415 ¢, ~441/260 = 913.7113 ¢ (~225/224 = 8.0526 ¢)
• CWE: ~78/77 = 22.6415 ¢, ~441/260 = 913.7126 ¢ (~225/224 = 8.0522 ¢)

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

Badness (Sintel): 1.97

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:

• WE: ~78/77 = 22.6412 ¢, ~441/260 = 913.7109 ¢ (~225/224 = 8.0623 ¢)
• CWE: ~78/77 = 22.6415 ¢, ~441/260 = 913.7208 ¢ (~225/224 = 8.0605 ¢)

Optimal ET sequence: 159, 583f, 742

Badness (Sintel): 1.57

Aemilic

Aemilic tempers out the landscape comma alongisde the mercator comma. It may be described as the 159 & 954 temperament. It was named after the minor planet 159 Aemilia.

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-84 53⟩

Mapping: [⟨159 252 0 -292], ⟨0 0 1 2]]

mapping generators: ~225/224, ~5

Optimal tunings

• WE: ~225/224 = 7.5473 ¢, ~5/4 = 386.2754 ¢ (~32805/32768 = 1.3629 ¢)

error map: ⟨+0.021 -0.035 +0.004 -0.003]

• CWE: ~225/224 = 7.5472 ¢, ~5/4 = 386.2798 ¢ (~32805/32768 = 1.3741 ¢)

error map: ⟨0.000 -0.068 -0.034 -0.040]

Optimal ET sequence: 159, …, 795, 954, 1749, 6201, 7950, 9699bd, 11448bd

Badness (Sintel): 4.86

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 102487/102400, [34 19 8 4 -1⟩

Mapping: [⟨159 252 0 -292 550], ⟨0 0 1 2 0]]

Optimal tunings:

• WE: ~225/224 = 7.5475 ¢, ~5/4 = 386.2382 ¢ (~32805/32768 = 1.3168 ¢)
• CWE: ~225/224 = 7.5472 ¢, ~5/4 = 386.2421 ¢ (~32805/32768 = 1.3365 ¢)

Optimal ET sequence: 159, …, 795, 954, 1749e, 2703e, 3657ee

Badness (Sintel): 2.79

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 102487/102400, 123201/123200, 4100625/4100096

Mapping: [⟨159 252 0 -292 550 -150], ⟨0 0 1 2 0 2]]

Optimal tunings:

• WE: ~225/224 = 7.5475 ¢, ~5/4 = 386.2350 ¢ (~32805/32768 = 1.3136 ¢)
• CWE: ~225/224 = 7.5472 ¢, ~5/4 = 386.2395 ¢ (~32805/32768 = 1.3338 ¢)

Optimal ET sequence: 159, …, 795, 954, 2703e, 3657eef

Badness (Sintel): 1.42

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2431/2430, 3025/3024, 57375/57344, 102487/102400, 123201/123200

Mapping: [⟨159 252 0 -292 550 -150 1019], ⟨0 0 1 2 0 2 -1]]

Optimal tunings:

• WE: ~225/224 = 7.5476 ¢, ~5/4 = 386.1835 ¢ (~32805/32768 = 1.2536 ¢)
• CWE: ~225/224 = 7.5472 ¢, ~5/4 = 386.1765 ¢ (~32805/32768 = 1.2708 ¢)

Optimal ET sequence: 159, 795g, 954

Badness (Sintel): 1.64