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Schismic–countercommatic equivalence continuum: Difference between revisions

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* Etc.
* Etc.


For a similar but perhaps more intuitive and practical concept, see [[Schismic–Pythagorean equivalence continuum]].
For a similar but perhaps more intuitive and practical concept, see [[Schismic–commatic equivalence continuum]].


{| class="wikitable center-1"
{| class="wikitable center-1"
Line 22: Line 22:
| −7
| −7
| [[Merman]]
| [[Merman]]
| 1121008359375 / 1099511627776
| 1121008359375/1099511627776
| {{monzo| -40 15 7 }}
| {{Monzo| -40 15 7 }}
|-
|-
| −6
| −6
| [[Syntonic–31 equivalence continuum|Ampersand]]
| [[Ampersand]]
| 34171875 / 33554432
| 34171875/33554432
| {{monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
|-
|-
| −5
| −5
| [[Magic]]
| [[Magic]]
| 3125 / 3072
| 3125/3072
| {{monzo| -10 -1 5 }}
| {{Monzo| -10 -1 5 }}
|-
|-
| −4
| −4
| [[Tetracot]]
| [[Tetracot]]
| 20000 / 19683
| 20000/19683
| {{monzo| 5 -9 4 }}
| {{monzo| 5 -9 4 }}
|-
|-
| −3
| −3
| [[Rodan]]
| [[Rodan]]
| 131072000 / 129140163
| 131072000/129140163
| {{monzo| 20 -17 3 }}
| {{Monzo| 20 -17 3 }}
|-
|-
| −2
| −2
| [[Hemififths]]
| [[Hemififths]]
| 858993459200 / 847288609443
| 858993459200/847288609443
| {{monzo| 35 -25 2 }}
| {{monzo| 35 -25 2 }}
|-
|-
Line 53: Line 53:
| [[Kwai]]
| [[Kwai]]
| (32 digits)
| (32 digits)
| {{monzo| 50 -33 1 }}
| {{Monzo| 50 -33 1 }}
|-
|-
| 0
| 0
| [[Countercomp]]
| [[Countercomp]]
| (40 digits)
| (40 digits)
| {{monzo| 65 -41 }}
| {{Monzo| 65 -41 }}
|-
|-
| 1
| 1
| [[Cotoneum]]
| [[Cotoneum]]
| (50 digits)
| (50 digits)
| {{monzo| 80 -49 -1 }}
| {{Monzo| 80 -49 -1 }}
|-
|-
| 2
| 2
| [[Newt]]
| [[Newt]]
| (58 digits)
| (58 digits)
| {{monzo| 95 -57 -2 }}
| {{Monzo| 95 -57 -2 }}
|-
|-
| 3
| 3
| 41 & 282
| 41 & 282
| (68 digits)
| (68 digits)
| {{monzo| 110 -65 -3 }}
| {{Monzo| 110 -65 -3 }}
|-
|-
| 4
| 4
| 41 & 335
| 41 & 335
| (76 digits)
| (76 digits)
| {{monzo| 125 -73 -4 }}
| {{Monzo| 125 -73 -4 }}
|-
|-
| 5
| 5
| 41 & 388
| 41 & 388
| (86 digits)
| (86 digits)
| {{monzo| 140 -81 -5 }}
| {{Monzo| 140 -81 -5 }}
|-
|-
| 6
| 6
| 41 & 441
| 41 & 441
| (94 digits)
| (94 digits)
| {{monzo| 155 -89 -6 }}
| {{Monzo| 155 -89 -6 }}
|-
|-
| 7
| 7
| 41 & 453
| 41 & 453
| (104 digits)
| (104 digits)
| {{monzo| 170 -97 -7 }}
| {{Monzo| 170 -97 -7 }}
|-
|-
| 8
| 8
| 41 & 506
| 41 & 506
| (112 digits)
| (112 digits)
| {{monzo| 185 -105 -8 }}
| {{Monzo| 185 -105 -8 }}
|-
|-
| 9
| 9
| 41 & 559
| 41 & 559
| (122 digits)
| (122 digits)
| {{monzo| 200 -113 -9 }}
| {{Monzo| 200 -113 -9 }}
|-
|-
| 10
| 10
| 41 & 571
| 41 & 571
| (130 digits)
| (130 digits)
| {{monzo| 215 -121 -10 }}
| {{Monzo| 215 -121 -10 }}
|-
|-
| 11
| 11
| 41 & 624
| 41 & 624
| (140 digits)
| (140 digits)
| {{monzo| -230 129 11 }}
| {{Monzo| -230 129 11 }}
|-
|-
| 12
| 12
| 41 & 677
| 41 & 677
| (148 digits)
| (148 digits)
| {{monzo| -245 137 12 }}
| {{Monzo| -245 137 12 }}
|-
|-
| 13
| 13
| 41 & 730
| 41 & 730
| (158 digits)
| (158 digits)
| {{monzo| -260 145 13 }}
| {{Monzo| -260 145 13 }}
|-
|-
| …
| …
Line 137: Line 137:


Examples of temperaments with fractional values of ''n'':  
Examples of temperaments with fractional values of ''n'':  
* [[Marvel temperaments #Septimin|Septimin]] ({{nowrap|''n'' {{=}} −11/2}})
* [[Septimin]] ({{nowrap|''n'' {{=}} −11/2}})
* [[Shibboleth family|Shibboleth]] ({{nowrap|''n'' {{=}} −9/2}})
* [[Shibboleth]] ({{nowrap|''n'' {{=}} −9/2}})
* [[Mirkwai clan #Pluto|Pluto]] ({{nowrap|''n'' {{=}} −7/2}})
* [[Pluto]] ({{nowrap|''n'' {{=}} −7/2}})
* 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
* 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
* 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})
* 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})


== Hemififths (5-limit) ==
== Kwai (5-limit) ==
: ''For extensions, see [[Breedsmic temperaments #Hemififths]].''
: ''For extensions, see [[Hemifamity temperaments #Kwai]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 858993459200/847288609443
[[Comma list]]: {{monzo| 50 -33 1 }}


{{Mapping|legend=1| 1 1 -5 | 0 2 25 }}
{{Mapping|legend=1| 1 0 -50 | 0 1 33 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.792{{c}}, ~3/2 = 702.5077{{c}}
: [[error map]]: {{val| -0.208 +0.345 -0.023 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6243{{c}}
: error map: {{val| 0.000 +0.669 +0.288 }}


{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 2017bbc, 2169bbc }}


[[Badness]]: 0.372848
[[Badness]] (Sintel): 14.9


== Kwai (5-limit) ==
== Cotoneum (5-limit) ==
: ''For extensions, see [[Mirkwai clan #Kwai]].''
: ''For extensions, see [[Garischismic clan #Cotoneum]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523
[[Comma list]]: {{monzo| 80 -49 -1 }}


{{Mapping|legend=1| 1 0 -50 | 0 1 33 }}
{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8849{{c}}, ~3/2 = 702.2471{{c}}
: [[error map]]: {{val| -0.115 +0.177 +0.008 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3162{{c}}
: error map: {{val| 0.000 +0.361 +0.190 }}


{{Optimal ET sequence|legend=1| 41, 111, 152 }}
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b, 3759bbc }}


[[Badness]]: 0.636715
[[Badness]] (Sintel): 29.1


== Cotoneum (5-limit) ==
== Hemififths (5-limit) ==
: ''For extensions, see [[Garischismic clan #Cotoneum]].''
: ''For extensions, see [[Breedsmic temperaments #Hemififths]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 80 -49 -1 }}
[[Comma list]]: 858993459200/847288609443


{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}
{{Mapping|legend=1| 1 1 -5 | 0 2 25 }}
: mapping generators: ~2, ~655360/531441


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7047{{c}}, ~655360/531441 = 351.3898{{c}}
: [[error map]]: {{val| -0.295 +0.529 -0.091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~655360/531441 = 351.4654{{c}}
: error map: {{val| 0.000 +0.976 +0.322 }}


{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b }}
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338, 915b, 1253bc }}


[[Badness]]: 1.240078
[[Badness]] (Sintel): 8.75


== Newt (5-limit) ==
== Newt (5-limit) ==
: ''For extensions, see [[Breedsmic temperaments #Newt]].''
: ''For extensions, see [[Garischismic clan #Newt]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 196: Line 211:


{{Mapping|legend=1| 1 1 19 | 0 2 -57 }}
{{Mapping|legend=1| 1 1 19 | 0 2 -57 }}
: mapping generators: ~2, ~{{monzo| 47 -28 -1 }}


[[Optimal tuning]] ([[POTE]]): ~{{monzo| 47 -28 -1 }} = 351.114
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9120{{c}}, ~{{monzo| 47 -28 -1 }} = 351.0878{{c}}
: [[error map]]: {{val| -0.088 +0.133 +0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 47 -28 -1 }} = 351.1146{{c}}
: error map: {{val| 0.000 +0.274 +0.152 }}


{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }}
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }}


[[Badness]]: 1.528465
[[Badness]] (Sintel): 35.9


[[Category:41edo]]
[[Category:41edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
Retrieved from "https://en.xen.wiki/w/Schismic–countercommatic_equivalence_continuum"