Mercator family: Difference between revisions

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[[Comma list]]: {{monzo| -84 53 }}
[[Comma list]]: {{monzo| -84 53 }}


[[Mapping]]: [{{val| 53 84 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 53 84 0 | 0 0 1 }}
: mapping generators: ~531441/524288, ~5
: mapping generators: ~531441/524288, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.3137{{c}}
* [[WE]]: ~531441/524288 = 22.6419{{c}}, ~5/4 = 386.2707{{c}} (~32805/32768 = 1.3581{{c}})
* [[CWE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.2804{{c}}
: [[error map]]: {{val| +0.022 -0.034 -0.000 }}
* [[CWE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.2804{{c}} (~32805/32768 = 1.3747{{c}})
: error map: {{val| 0.000 -0.068 -0.033 }}


{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}


[[Badness]] (Sintel): 6.670
[[Badness]] (Sintel): 6.67


== Schismerc ==
== 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.
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
[[Subgroup]]: 2.3.5.7
Line 35: Line 37:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~81/80 = 22.6415{{c}}, ~8/7 = 231.1741{{c}}
* [[WE]]: ~81/80 = 22.6456{{c}}, ~7/4 = 968.3928{{c}} (~225/224 = 5.3675{{c}})
* [[CWE]]: ~81/80 = 22.641{{c}}, ~8/7 = 231.6299{{c}}
: [[error map]]: {{val| +0.216 +0.275 -0.906 -0.001 }}
* [[CWE]]: ~81/80 = 22.6415{{c}}, ~7/4 = 968.3701{{c}} (~225/224 = 5.2148{{c}})
: error map: {{val| 0.000 -0.068 -1.408 -0.456 }}


{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc, 1113ccd }}


[[Badness]] (Sintel): 2.202
[[Badness]] (Sintel): 2.20


=== Cartography ===
=== 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.
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
Subgroup: 2.3.5.7.11
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Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.641{{c}}, ~8/7 = 232.4299{{c}}
* WE: ~81/80 = 22.6478{{c}}, ~7/4 = 967.7312{{c}} (~225/224 = 6.1221{{c}})
* CWE: ~81/80 = 22.641{{c}}, ~8/7 = 232.5178{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 967.4822{{c}} (~225/224 = 6.1027{{c}})


{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}


Badness (Sintel): 1.800
Badness (Sintel): 1.80


==== 13-limit ====
==== 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
Subgroup: 2.3.5.7.11.13


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Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.4299{{c}}
* WE: ~81/80 = 22.6491{{c}}, ~7/4 = 967.7656{{c}} (~225/224 = 6.1450{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.5397{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 967.4603{{c}} (~225/224 = 6.1246{{c}})


{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}


Badness (Sintel): 1.239
Badness (Sintel): 1.24


=== Pentacontatritonic ===
=== Pentacontatritonic ===
First proposed by [[User:Xenllium|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.
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
Subgroup: 2.3.5.7.11
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Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5956{{c}}
* WE: ~81/80 = 22.6427{{c}}, ~7/4 = 969.4874{{c}} (~385/384 = 4.1496{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5697{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 969.4303{{c}} (~385/384 = 4.1546{{c}})


{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}
{{Optimal ET sequence|legend=0| 53, 212e, 265, 318 }}


Badness (Sintel): 3.804
Badness (Sintel): 3.80


==== 13-limit ====
==== 13-limit ====
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: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Mapping: {{mapping| 53 84 123 0 481 345 | 0 0 0 1 -2 1 }}
Mapping: {{mapping| 53 84 123 0 481 345 | 0 0 0 1 -2 1 }}
Optimal tuning (POTE): ~385/384 = 3.9850


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4057{{c}}
* WE: ~81/80 = 22.6416{{c}}, ~7/4 = 969.6046{{c}} (~385/384 = 3.9850{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4008{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 969.5992{{c}} (~385/384 = 3.9858{{c}})


{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}
{{Optimal ET sequence|legend=0| 53, 212ef, 265, 318 }}


Badness (Sintel): 2.527
Badness (Sintel): 2.53


=== Boiler ===
=== 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.
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
Subgroup: 2.3.5.7.11
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 230.6341{{c}}
* WE: ~2835/2816 = 11.3229{{c}}, ~7/4 = 968.8469{{c}} (~225/224 = 4.9241{{c}})
* CWE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 231.1634{{c}}
* CWE: ~2835/2816 = 11.3208{{c}}, ~7/4 = 968.8633{{c}} (~225/224 = 4.7483{{c}})


{{Optimal ET sequence|legend=0| 106, 212 }}
{{Optimal ET sequence|legend=0| 106, 212, 530c, 742ce }}


Badness (Sintel): 3.625
Badness (Sintel): 3.62


=== Auric ===
=== 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.
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
Subgroup: 2.3.5.7.11
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: mappping generators: ~243/242, ~7
: mappping generators: ~243/242, ~7


Optimal tuning (CTE): ~243/242 = 7.139{{c}}, ~7/4 = 968.726{{c}}
Optimal tunings:
* WE: ~243/242 = 7.5484{{c}}, ~7/4 = 968.4469{{c}} (~3024/3025 = 2.2569{{c}})
* CWE: ~243/242 = 7.5472{{c}}, ~7/4 = 968.4005{{c}} (~3024/3025 = 2.3627{{c}})
 
{{Optimal ET sequence|legend=0| 159, 318, 477c }}


{{Optimal ET sequence|legend=0| 159, 318, 477c, 636c, 795cd }}
Badness (Sintel): 5.23


== Joliet ==
== 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.
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
[[Subgroup]]: 2.3.5.7.11
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}
* [[CTE]]: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}
* [[CWE]]: ~50/49 = 22.641{{c}}, ~11/8 = 552.0415{{c}}
* [[CWE]]: ~50/49 = 22.6415{{c}}, ~11/8 = 552.0415{{c}}


{{Optimal ET sequence|legend=1| 53, 106, 159d }}
{{Optimal ET sequence|legend=1| 53, 106, 159d }}
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: mapping generators: ~225/224, ~5
: mapping generators: ~225/224, ~5


[[Optimal tuning]] ([[CTE]]): ~225/224 = 7.139{{c}}, ~5/4 = 386.303{{c}}
[[Optimal tuning]] ([[CTE]]): ~225/224 = 7.547{{c}}, ~5/4 = 386.303{{c}}


[[Support]]ing [[ET]]s: {{EDOs| 159, 795, 954, 1749, 2544, 2703 }}
[[Support]]ing [[ET]]s: {{EDOs| 159, 795, 954, 1749, 2544, 2703 }}
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Mapping: {{mapping| 159 252 0 -292 550 | 0 0 1 2 0 }}
Mapping: {{mapping| 159 252 0 -292 550 | 0 0 1 2 0 }}


Optimal tuning (CTE): ~225/224 = 7.139{{c}}, ~5/4 = 386.303{{c}}
Optimal tuning (CTE): ~225/224 = 7.547{{c}}, ~5/4 = 386.303{{c}}


Supporting ETs: {{EDOs| 159, 795, 954 }}
Supporting ETs: {{EDOs| 159, 795, 954 }}
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Mapping: {{mapping| 159 252 0 -292 550 -150 | 0 0 1 2 0 2 }}
Mapping: {{mapping| 159 252 0 -292 550 -150 | 0 0 1 2 0 2 }}


Optimal tuning (CTE): ~225/224 = 7.139{{c}}, ~5/4 = 386.303{{c}}
Optimal tuning (CTE): ~225/224 = 7.547{{c}}, ~5/4 = 386.303{{c}}


Supporting ETs: {{EDOs| 159, 795, 954 }}
Supporting ETs: {{EDOs| 159, 795, 954 }}
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Mapping: {{mapping| 159 252 0 -292 550 -150 1019 | 0 0 1 2 0 2 -1 }}
Mapping: {{mapping| 159 252 0 -292 550 -150 1019 | 0 0 1 2 0 2 -1 }}


Optimal tuning (CTE): ~225/224 = 7.139{{c}}, ~5/4 = 386.263{{c}}
Optimal tuning (CTE): ~225/224 = 7.547{{c}}, ~5/4 = 386.263{{c}}


Supporting ETs: {{EDOs| 159, 795g, 954 }}
Supporting ETs: {{EDOs| 159, 795g, 954 }}

Revision as of 07:36, 10 September 2026

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

← 52nd-octave temperaments 53rd-octave temperaments 54th-octave temperaments →

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: ~50/49, ~11

Optimal tunings:

  • CTE: ~50/49 = 22.6415 ¢, ~11/8 = 551.3179 ¢
  • CWE: ~50/49 = 22.6415 ¢, ~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 ¢, ~11/8 = 551.3179 ¢
  • CWE: ~50/49 = 22.6415 ¢, ~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 ¢, ~6075/3584 = 913.7347 ¢
  • CWE: ~3125/3087 = 22.6415 ¢, ~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 ¢, ~6075/3584 = 913.7322 ¢
  • CWE: ~1815/1792 = 22.6415 ¢, ~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 ¢, ~441/260 = 913.7115 ¢
  • CWE: ~78/77 = 22.6415 ¢, ~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 ¢, ~441/260 = 913.7131 ¢
  • CWE: ~78/77 = 22.6415 ¢, ~441/260 = 913.7208 ¢

Optimal ET sequence: 159, 583f, 742

Badness (Sintel): 1.568

Aemilic

Aemilic is described as the 159 & 954 temperament in the 17-limit and is named after the minor planet 159 Aemilia.

In the 7-limit, aemilic tempers out the landscape comma alongisde the Mercator comma, and also [-76 41 12 -6⟩.

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 tuning (CTE): ~225/224 = 7.547 ¢, ~5/4 = 386.303 ¢

Supporting ETs: 159, 795, 954, 1749, 2544, 2703

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 160083/160000, [34 19 8 4 -1⟩

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

Optimal tuning (CTE): ~225/224 = 7.547 ¢, ~5/4 = 386.303 ¢

Supporting ETs: 159, 795, 954

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 123201/123200, 160083/160000, 4100625/4100096

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

Optimal tuning (CTE): ~225/224 = 7.547 ¢, ~5/4 = 386.303 ¢

Supporting ETs: 159, 795, 954

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2431/2430, 3025/3024, 57375/57344, 123201/123200, 160083/160000

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

Optimal tuning (CTE): ~225/224 = 7.547 ¢, ~5/4 = 386.263 ¢

Supporting ETs: 159, 795g, 954

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