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<div style="
{{Technical data page}}
border: 1px solid #999;
<div class="toccolours" style="float: right">
margin: 0;
<center>'''[[Fractional-octave temperaments]]'''</center>
margin-left: 1em;
----
margin-bottom: 0.5em;
<small>← [[{{Ordinal|{{#expr:53-1}}}}-octave temperaments]]</small> 53rd-octave temperaments <small>[[{{Ordinal|{{#expr:53+1}}}}-octave temperaments]] →</small>
padding: 0.5em;
background-color: #f0f0f0;
min-width: 15em;
float: right;
max-width: 100%;
overflow: auto;
">
{| class="wikitable" width="100%" style="border-collapse: collapse;"
|+ [[Fractional-octave temperaments]]
| [[{{Ordinal|{{#expr:53-1}}}}-octave temperaments]]
| '''53rd-octave temperaments'''<br>({{PAGENAME}})
| [[{{Ordinal|{{#expr:53+1}}}}-octave temperaments]]
|}
</div>
</div>
[[Category:53edo]]
[[Category:53edo]]
[[Category:Fractional-octave temperaments]]
[[Category:Fractional-octave temperaments]]
[[Category:Temperament collections]]
[[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.  
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:
* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]


== Mercator ==
== Mercator ==
Line 28: Line 21:
[[Comma list]]: {{monzo| -84 53 }}
[[Comma list]]: {{monzo| -84 53 }}


[[Mapping]]: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]
[[Mapping]]: [{{val| 53 84 0 }}, {{val| 0 0 1 }}]
 
Mapping generators: ~531441/524288, ~5/1


{{Multival|legend=1| 0 53 84 }}
: mapping generators: ~531441/524288, ~5


[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264
[[Optimal tuning]]s:
* [[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|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]]: 0.284323
[[Badness]] (Sintel): 6.670


== 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
Line 49: Line 42:
[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]
[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]


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


[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666
[[Optimal tuning]]s:
* [[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|legend=1| 53, 159, 212, 689c, 901cc }}
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}


[[Badness]]: 0.087022
[[Badness]] (Sintel): 2.202


=== Cartography ===
=== Cartography ===
Line 68: Line 61:
Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}]
Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}]


Mapping generators: ~81/80, ~7/1
Optimal tunings:
 
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
Optimal tuning (POTE): ~225/224 = 6.1204
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5178¢


{{Optimal ET sequence|legend=1| 53, 106d, 159, 212, 371d, 583cde }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}


Badness: 0.054452
Badness (Sintel): 1.800


==== 13-limit ====
==== 13-limit ====
Line 85: Line 78:
Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}
Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}


Mapping generators: ~81/80, ~7/1
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 tuning (POTE): ~225/224 = 6.1430
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}


{{Optimal ET sequence|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}
Badness (Sintel): 1.239
 
Badness: 0.029980


=== 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 [[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.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 102: Line 95:
Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]
Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]


Mapping generators: ~81/80, ~7/1
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 tuning (POTE): ~385/384 = 4.1494
{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}


{{Optimal ET sequence|legend=1| 53, 159e, 212e, 265, 318, 583c }}
Badness (Sintel): 3.804
 
Badness: 0.115066


==== 13-limit ====
==== 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.
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
Line 119: Line 112:
Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}
Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}


Mapping generators: ~81/80, ~7/1
Optimal tuning (POTE): ~385/384 = 3.9850


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|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }}
{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}


Badness: 0.061158
Badness (Sintel): 2.527


=== Boiler ===
=== Boiler ===
Line 136: Line 131:
Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}]
Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}]


Mapping generators: ~2835/2816, ~7
: mapping generators: ~2835/2816, ~7


Optimal tuning (POTE): ~225/224 = 6.3976 or ~441/440 = 4.9232
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|legend=1| 106, 212 }}
{{Optimal ET sequence|legend=0| 106, 212 }}


Badness: 0.109648
Badness (Sintel): 3.625


== Joliet ==
== Joliet ==
Line 153: Line 150:
[[Mapping]]: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}]
[[Mapping]]: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}]


Mapping generators: ~81/80, ~11/1
: mapping generators: ~50/49, ~11


[[Optimal tuning]] ([[POTE]]): ~176/175 = 8.8283
[[Optimal tuning]]s:
* [[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|legend=1| 53, 106, 159d }}
{{Optimal ET sequence|legend=1| 53, 106, 159d }}


[[Badness]]: 0.063254
[[Badness]] (Sintel): 2.091


=== 13-limit ===
=== 13-limit ===
Line 168: Line 167:
Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]


Mapping generators: ~81/80, ~11/1
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 tuning (POTE): ~176/175 = 8.1254
{{Optimal ET sequence|legend=0| 53, 106, 159d }}


{{Optimal ET sequence|legend=1| 53, 106, 159d }}
Badness (Sintel): 1.528
 
Badness: 0.036988


== Iodine ==
== Iodine ==
Line 185: Line 184:
[[Mapping]]: [{{val| 53 84 2 -53 }}, {{val| 0 0 3 5 }}]
[[Mapping]]: [{{val| 53 84 2 -53 }}, {{val| 0 0 3 5 }}]


Mapping generators: ~3125/3087, 6075/3584
: mapping generators: ~3125/3087, ~6075/3584


[[Optimal tuning]] ([[CTE]]): ~6075/3584 = 913.7347
[[Optimal tuning]]s:
* [[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|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}


[[Badness]]: 0.477
[[Badness]] (Sintel): 12.075


=== 11-limit ===
=== 11-limit ===
Line 202: Line 203:
Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]
Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]


Optimal tuning (CTE): ~6075/3584 = 913.7322
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|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742, 2385d, 3127d }}


Badness: 0.0875
Badness (Sintel): 2.893


=== 13-limit ===
=== 13-limit ===
Line 215: Line 218:
Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]
Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]


Optimal tuning (CTE): ~441/260 = 913.7115
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|legend=1| 159, 424cdff, 583f, 742, 1643 }}
{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}


Badness: 0.0476
Badness (Sintel): 1.967


=== 17-limit ===
=== 17-limit ===
Line 228: Line 233:
Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]
Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]


Optimal tuning (CTE): ~441/260 = 913.7131
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|legend=0| 159, 583f, 742 }}


{{Optimal ET sequence|legend=1| 159, 583f, 742 }}
Badness (Sintel): 1.568


Badness: 0.0328
{{Navbox fractional-octave|53}}


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Mercator family]] <!-- main article -->
[[Category:Mercator family]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"