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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 }}
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== 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]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"