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Schismic–countercommatic equivalence continuum: Difference between revisions

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The '''schismic-countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with the [[41-comma|Pythagorean countercomma ({{monzo| 65 -41 }})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
The '''schismic-countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with the [[41-comma|Pythagorean countercomma ({{monzo| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]].


All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| 65 -41 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[41edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 10.1575233481..., and temperaments having ''n'' near this value tend to be the most accurate ones.  
All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| 65 -41 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 41edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones.  


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-Pythagorean equivalence continuum]].


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ Temperaments in the continuum
|+ Temperaments of integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
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|-
|-
| -7
| -7
| [[Marvel temperaments #Merman|Merman]]
| [[Merman]]
| 1121008359375 / 1099511627776
| 1121008359375 / 1099511627776
| {{monzo|-40 15 7}}
| {{monzo| -40 15 7 }}
|-
|-
| -6
| -6
| [[Syntonic-31 equivalence continuum|Ampersand]]
| [[Syntonic-31 equivalence continuum|Ampersand]]
| 34171875 / 33554432
| 34171875 / 33554432
| {{monzo|-25 7 6}}
| {{monzo| -25 7 6 }}
|-
|-
| -5
| -5
| [[Magic family|Magic]]
| [[Magic]]
| 3125 / 3072
| 3125 / 3072
| {{monzo|-10 -1 5}}
| {{monzo| -10 -1 5 }}
|-
|-
| -4
| -4
| [[Tetracot family|Tetracot]]
| [[Tetracot]]
| 20000 / 19683
| 20000 / 19683
| {{monzo|5 -9 4}}
| {{monzo| 5 -9 4 }}
|-
|-
| -3
| -3
| [[Gamelismic clan #Rodan|Rodan]]
| [[Rodan]]
| 131072000 / 129140163
| 131072000 / 129140163
| {{monzo|20 -17 3}}
| {{monzo| 20 -17 3 }}
|-
|-
| -2
| -2
| [[Breedsmic temperaments #Hemififths|Hemififths]]
| [[Hemififths]]
| 858993459200 / 847288609443
| 858993459200 / 847288609443
| {{monzo|35 -25 2}}
| {{monzo| 35 -25 2 }}
|-
|-
| -1
| -1
| [[Mirkwai clan #Kwai|Kwai]]
| [[Kwai]]
|  
| (32 digits)
| {{monzo|50 -33 1}}
| {{monzo| 50 -33 1 }}
|-
|-
| 0
| 0
| [[Countercomp family #Countercomp|Countercomp]]
| [[Countercomp]]
|  
| (40 digits)
| {{monzo|65 -41}}
| {{monzo| 65 -41 }}
|-
|-
| 1
| 1
| [[Hemimage temperaments #Cotoneum|Cotoneum]]
| [[Cotoneum]]
|  
| (50 digits)
| {{monzo|80 -49 -1}}
| {{monzo| 80 -49 -1 }}
|-
|-
| 2
| 2
| [[Breedsmic temperaments #Newt|Newt]]
| [[Newt]]
|  
| (58 digits)
| {{monzo|95 -57 -2}}
| {{monzo| 95 -57 -2 }}
|-
|-
| 3
| 3
| 41&amp;282
| 41 &amp; 282
|  
| (68 digits)
| {{monzo|110 -65 -3}}
| {{monzo| 110 -65 -3 }}
|-
|-
| 4
| 4
| 41&amp;335
| 41 &amp; 335
|  
| (76 digits)
| {{monzo|125 -73 -4}}
| {{monzo| 125 -73 -4 }}
|-
|-
| 5
| 5
| 41&amp;388
| 41 &amp; 388
|  
| (86 digits)
| {{monzo|140 -81 -5}}
| {{monzo| 140 -81 -5 }}
|-
|-
| 6
| 6
| 41&amp;441
| 41 &amp; 441
|  
| (94 digits)
| {{monzo|155 -89 -6}}
| {{monzo| 155 -89 -6 }}
|-
|-
| 7
| 7
| 41&amp;453
| 41 &amp; 453
|  
| (104 digits)
| {{monzo|170 -97 -7}}
| {{monzo| 170 -97 -7 }}
|-
|-
| 8
| 8
| 41&amp;506
| 41 &amp; 506
|  
| (112 digits)
| {{monzo|185 -105 -8}}
| {{monzo| 185 -105 -8 }}
|-
|-
| 9
| 9
| 41&amp;559
| 41 &amp; 559
|  
| (122 digits)
| {{monzo|200 -113 -9}}
| {{monzo| 200 -113 -9 }}
|-
|-
| 10
| 10
| 41&amp;571
| 41 &amp; 571
|  
| (130 digits)
| {{monzo|215 -121 -10}}
| {{monzo| 215 -121 -10 }}
|-
|-
| 11
| 11
| 41&amp;624
| 41 &amp; 624
|  
| (140 digits)
| {{monzo|-230 129 11}}
| {{monzo| -230 129 11 }}
|-
|-
| 12
| 12
| 41&amp;677
| 41 &amp; 677
|  
| (148 digits)
| {{monzo|-245 137 12}}
| {{monzo| -245 137 12 }}
|-
|-
| 13
| 13
| 41&amp;730
| 41 &amp; 730
|  
| (158 digits)
| {{monzo|-260 145 13}}
| {{monzo| -260 145 13 }}
|-
|-
| …
| …
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| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1}}
| {{monzo| -15 8 1 }}
|}
|}


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[[Comma list]]: 131072000/129140163
[[Comma list]]: 131072000/129140163


[[Mapping]]: [{{val| 1 1 -1 }}, {{val| 0 3 17 }}]
{{Mapping|legend=1| 1 1 -1 | 0 3 17 }}


[[Optimal tuning]] ([[POTE]]): ~729/640 = 234.528
[[Optimal tuning]] ([[POTE]]): ~729/640 = 234.528
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[[Comma list]]: 858993459200/847288609443
[[Comma list]]: 858993459200/847288609443


[[Mapping]]: [{{val| 1 1 -5 }}, {{val| 0 2 25 }}]
{{Mapping|legend=1| 1 1 -5 | 0 2 25 }}


[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476
[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476
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[[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523
[[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523


[[Mapping]]: [{{val| 1 0 -50 }}, {{val| 0 1 33 }}]
{{Mapping|legend=1| 1 0 -50 | 0 1 33 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630
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[[Comma list]]: {{monzo| 65 -41 }}
[[Comma list]]: {{monzo| 65 -41 }}


[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 41 65 0 | 0 0 1 }}


[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668
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== Cotoneum (5-limit) ==
== Cotoneum (5-limit) ==
{{See also| Hemimage temperaments #Cotoneum }}
{{See also| Garischismic clan #Cotoneum }}


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Comma list]]: {{monzo| 80 -49 -1 }}
[[Comma list]]: {{monzo| 80 -49 -1 }}


[[Mapping]]: [{{val| 1 0 80 }}, {{val| 0 1 -49 }}]
{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315
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[[Comma list]]: {{monzo| 95 -57 -2 }}
[[Comma list]]: {{monzo| 95 -57 -2 }}


[[Mapping]]: [{{val| 1 1 19 }}, {{val| 0 2 -57 }}]
{{Mapping|legend=1| 1 1 19 | 0 2 -57 }}


[[Optimal tuning]] ([[POTE]]): ~{{monzo| 47 -28 -1 }} = 351.114
[[Optimal tuning]] ([[POTE]]): ~{{monzo| 47 -28 -1 }} = 351.114
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