Xenharmonic Wiki

Mercator family: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,376 edits
newer nicer template, discussed elsewhere
m Name for 4096/4095
 
(11 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{Technical data page}}
<div class="toccolours" style="float: right">
<div class="toccolours" style="float: right">
<center>'''[[Fractional-octave temperaments]]'''</center>
<center>'''[[Fractional-octave temperaments]]'''</center>
----
----
<small>← [[{{Ordinal|{{#expr:53-1}}}}-octave temperaments]]</small> {{PAGENAME}} <small>[[{{Ordinal|{{#expr:53+1}}}}-octave temperaments]] →</small>
<small>← [[{{Ordinal|{{#expr:53-1}}}}-octave temperaments]]</small> 53rd-octave temperaments <small>[[{{Ordinal|{{#expr:53+1}}}}-octave temperaments]] →</small>
</div>
</div>


Line 20: 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
: mapping generators: ~531441/524288, ~5


{{Multival|legend=1| 0 53 84 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.3137¢
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264
* [[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 41: 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]]s:
 
* [[CTE]]: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.1741¢
[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666
* [[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 60: 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 77: 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¢
Optimal tuning (POTE): ~225/224 = 6.1430
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5397¢


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


Badness: 0.029980
Badness (Sintel): 1.239


=== 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 94: 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 111: 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 128: 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 145: 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 160: 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 177: 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 194: 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 207: 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 220: 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"