Mercator family: Difference between revisions

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[[Category:53edo]]
The '''mercator family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[Mercator's comma]], {{monzo| -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.  
[[Category:Fractional-octave temperaments]]
[[Category:Temperament collections]]
 
The '''Mercator family''' tempers out [[Mercator's comma]], {{monzo| -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:
Discussed elsewhere are:
* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]
* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]


Line 22: Line 17:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.3137¢
* [[CTE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.3137{{c}}
* [[CWE]]: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.2804¢
* [[CWE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.2804{{c}}


{{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 }}
Line 34: Line 28:


== 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 known 11-limit extensions are cartography, pentacontatritonic and boiler.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 32805/32768
[[Comma list]]: 15625/15552, 32805/32768


[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 53 84 123 0 | 0 0 0 1 }}
 
: mapping generators: ~81/80, ~7
: mapping generators: ~81/80, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.1741¢
* [[CTE]]: ~81/80 = 22.6415{{c}}, ~8/7 = 231.1741{{c}}
* [[CWE]]: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.6299¢
* [[CWE]]: ~81/80 = 22.641{{c}}, ~8/7 = 231.6299{{c}}


{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}
Line 59: Line 52:
Comma list: 385/384, 6250/6237, 19712/19683
Comma list: 385/384, 6250/6237, 19712/19683


Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}]
Mapping: {{mapping| 53 84 123 0 332 | 0 0 0 1 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CTE: ~81/80 = 22.641{{c}}, ~8/7 = 232.4299{{c}}
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5178¢
* CWE: ~81/80 = 22.641{{c}}, ~8/7 = 232.5178{{c}}


{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}
Line 76: Line 69:
Comma list: 325/324, 385/384, 625/624, 19712/19683
Comma list: 325/324, 385/384, 625/624, 19712/19683


Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}
Mapping: {{mapping| 53 84 123 0 332 196 | 0 0 0 1 -1 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.4299{{c}}
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5397¢
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.5397{{c}}


{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}
Line 93: Line 86:
Comma list: 540/539, 15625/15552, 32805/32768
Comma list: 540/539, 15625/15552, 32805/32768


Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]
Mapping: {{mapping| 53 84 123 0 481 | 0 0 0 1 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5956¢
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5956{{c}}
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5697¢
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5697{{c}}


{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}
{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}
Line 110: Line 103:
Comma list: 540/539, 729/728, 4096/4095, 13750/13689
Comma list: 540/539, 729/728, 4096/4095, 13750/13689


Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 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 tuning (POTE): ~385/384 = 3.9850


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4057¢
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4057{{c}}
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4008¢
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4008{{c}}


{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}
{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}
Line 129: Line 122:
Comma list: 9801/9800, 15625/15552, 32805/32768
Comma list: 9801/9800, 15625/15552, 32805/32768


Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}]
Mapping: {{mapping| 106 168 246 0 69 | 0 0 0 1 1 }}
 
: mapping generators: ~2835/2816, ~7
: mapping generators: ~2835/2816, ~7


Optimal tunings:  
Optimal tunings:  
* CTE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 230.6341¢
* CTE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 230.6341{{c}}
* CWE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 231.1634¢
* CWE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 231.1634{{c}}


{{Optimal ET sequence|legend=0| 106, 212 }}
{{Optimal ET sequence|legend=0| 106, 212 }}
Line 142: Line 134:


== Joliet ==
== Joliet ==
Joliet can be characterized as the 53 &amp; 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


[[Comma list]]: 225/224, 1728/1715, 3125/3087
[[Comma list]]: 225/224, 1728/1715, 3125/3087


[[Mapping]]: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=1| 53 84 123 149 0 | 0 0 0 0 1 }}
 
: mapping generators: ~50/49, ~11
: mapping generators: ~50/49, ~11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
* [[CTE]]: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}
* [[CWE]]: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 552.0415¢
* [[CWE]]: ~50/49 = 22.641{{c}}, ~11/8 = 552.0415{{c}}


{{Optimal ET sequence|legend=1| 53, 106, 159d }}
{{Optimal ET sequence|legend=1| 53, 106, 159d }}
Line 165: Line 156:
Comma list: 169/168, 225/224, 325/324, 640/637
Comma list: 169/168, 225/224, 325/324, 640/637


Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: {{mapping| 53 84 123 149 0 196 | 0 0 0 0 1 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
* CTE: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}
* CWE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.4859¢
* CWE: ~50/49 = 22.6415{{c}}, ~11/8 = 551.4859{{c}}


{{Optimal ET sequence|legend=0| 53, 106, 159d }}
{{Optimal ET sequence|legend=0| 53, 106, 159d }}
Line 182: Line 173:
[[Comma list]]: {{monzo| -19 14 -5 3 }}, {{monzo| 8 3 -20 12 }}
[[Comma list]]: {{monzo| -19 14 -5 3 }}, {{monzo| 8 3 -20 12 }}


[[Mapping]]: [{{val| 53 84 2 -53 }}, {{val| 0 0 3 5 }}]
{{Mapping|legend=1| 53 84 2 -53 | 0 0 3 5 }}
 
: mapping generators: ~3125/3087, ~6075/3584
: mapping generators: ~3125/3087, ~6075/3584


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7347¢
* [[CTE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7347{{c}}
* [[CWE]]: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7301¢
* [[CWE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7301{{c}}


{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
Line 201: Line 191:
Comma list: 160083/160000, 820125/819896, 4302592/4296875
Comma list: 160083/160000, 820125/819896, 4302592/4296875


Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]
Mapping: {{mapping| 53 84 2 -53 143 | 0 0 3 5 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7322¢
* CTE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7322{{c}}
* CWE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7345¢
* CWE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7345{{c}}


{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742, 2385d, 3127d }}
Line 216: Line 206:
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488


Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]
Mapping: {{mapping| 53 84 2 -53 143 -46 | 0 0 3 5 1 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7115¢
* CTE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7115{{c}}
* CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7126¢
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7126{{c}}


{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}
{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}
Line 231: Line 221:
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872


Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]
Mapping: {{mapping| 53 84 2 -53 143 -46 257 | 0 0 3 5 1 6 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7131¢
* CTE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7131{{c}}
* CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7208¢
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7208{{c}}


{{Optimal ET sequence|legend=0| 159, 583f, 742 }}
{{Optimal ET sequence|legend=0| 159, 583f, 742 }}
Line 243: Line 233:
{{Navbox fractional-octave|53}}
{{Navbox fractional-octave|53}}


[[Category:Mercator family]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Mercator family]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
 
<!-- as 53rd-octave temperaments page redirects here -->
[[Category:53edo]]
[[Category:Fractional-octave temperaments]]

Revision as of 06:26, 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.

Discussed elsewhere are:

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 ¢, ~5/4 = 386.3137 ¢
  • CWE: ~531441/524288 = 22.6415 ¢, ~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 ¢, ~8/7 = 231.1741 ¢
  • CWE: ~81/80 = 22.641 ¢, ~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 = 22.641 ¢, ~8/7 = 232.4299 ¢
  • CWE: ~81/80 = 22.641 ¢, ~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 ¢, ~8/7 = 232.4299 ¢
  • CWE: ~81/80 = 22.6415 ¢, ~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 ¢, ~8/7 = 230.5956 ¢
  • CWE: ~81/80 = 22.6415 ¢, ~8/7 = 230.5697 ¢

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

Badness (Sintel): 3.804

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

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 ¢, ~8/7 = 230.4057 ¢
  • CWE: ~81/80 = 22.6415 ¢, ~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 ¢, ~8/7 = 230.6341 ¢
  • CWE: ~2835/2816 = 11.3208 ¢, ~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 ¢, ~11/8 = 551.3179 ¢
  • CWE: ~50/49 = 22.641 ¢, ~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

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