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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]] [[microtemperament]]s [[support]]ed by [[41edo]].
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 [[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.  
All temperaments in the continuum satisfy {{nowrap|(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.  


The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example:
The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example:
* [[Cotoneum]] (''n'' = 1) is generated by a fifth;
* [[Cotoneum]] ({{nowrap|''n'' {{=}} 1}}) is generated by a fifth;
* [[Newt]] (''n'' = 2) splits its fifth in two;
* [[Newt]] ({{nowrap|''n'' {{=}} 2}}) splits its fifth in two;
* 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"
|+ Temperaments of integer ''n''
|+ style="font-size: 105%;" | Temperaments of integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 20: Line 20:
! Monzo
! Monzo
|-
|-
| -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 }}
|-
|-
| -1
| −1
| [[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 &amp; 282
| 41 & 282
| (68 digits)
| (68 digits)
| {{monzo| 110 -65 -3 }}
| {{Monzo| 110 -65 -3 }}
|-
|-
| 4
| 4
| 41 &amp; 335
| 41 & 335
| (76 digits)
| (76 digits)
| {{monzo| 125 -73 -4 }}
| {{Monzo| 125 -73 -4 }}
|-
|-
| 5
| 5
| 41 &amp; 388
| 41 & 388
| (86 digits)
| (86 digits)
| {{monzo| 140 -81 -5 }}
| {{Monzo| 140 -81 -5 }}
|-
|-
| 6
| 6
| 41 &amp; 441
| 41 & 441
| (94 digits)
| (94 digits)
| {{monzo| 155 -89 -6 }}
| {{Monzo| 155 -89 -6 }}
|-
|-
| 7
| 7
| 41 &amp; 453
| 41 & 453
| (104 digits)
| (104 digits)
| {{monzo| 170 -97 -7 }}
| {{Monzo| 170 -97 -7 }}
|-
|-
| 8
| 8
| 41 &amp; 506
| 41 & 506
| (112 digits)
| (112 digits)
| {{monzo| 185 -105 -8 }}
| {{Monzo| 185 -105 -8 }}
|-
|-
| 9
| 9
| 41 &amp; 559
| 41 & 559
| (122 digits)
| (122 digits)
| {{monzo| 200 -113 -9 }}
| {{Monzo| 200 -113 -9 }}
|-
|-
| 10
| 10
| 41 &amp; 571
| 41 & 571
| (130 digits)
| (130 digits)
| {{monzo| 215 -121 -10 }}
| {{Monzo| 215 -121 -10 }}
|-
|-
| 11
| 11
| 41 &amp; 624
| 41 & 624
| (140 digits)
| (140 digits)
| {{monzo| -230 129 11 }}
| {{Monzo| -230 129 11 }}
|-
|-
| 12
| 12
| 41 &amp; 677
| 41 & 677
| (148 digits)
| (148 digits)
| {{monzo| -245 137 12 }}
| {{Monzo| -245 137 12 }}
|-
|-
| 13
| 13
| 41 &amp; 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]] (''n'' = -11/2 = -5.5)
* [[Septimin]] ({{nowrap|''n'' {{=}} −11/2}})
* [[Shibboleth family|Shibboleth]] (''n'' = -9/2 = -4.5)
* [[Shibboleth]] ({{nowrap|''n'' {{=}} −9/2}})
* [[Mirkwai clan #Pluto|Pluto]] (''n'' = -7/2 = -3.5)
* [[Pluto]] ({{nowrap|''n'' {{=}} −7/2}})
* 3737 &amp; 5585 (''n'' = 31/3 = 10.{{overline|3}})
* 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
* 1277 &amp; 2513 (''n'' = 21/2 = 10.5)
* 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})
 
== Rodan (5-limit) ==
{{See also| Gamelismic clan #Rodan }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 131072000/129140163
 
{{Mapping|legend=1| 1 1 -1 | 0 3 17 }}
 
[[Optimal tuning]] ([[POTE]]): ~729/640 = 234.528
 
{{Optimal ET sequence|legend=1| 5, 31c, 36c, 41, 46, 87, 220, 307 }}
 
[[Badness]]: 0.168264
 
== Hemififths (5-limit) ==
{{See also| Breedsmic temperaments #Hemififths }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 858993459200/847288609443
 
{{Mapping|legend=1| 1 1 -5 | 0 2 25 }}
 
[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476
 
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }}
 
[[Badness]]: 0.372848


== Kwai (5-limit) ==
== Kwai (5-limit) ==
{{See also| Mirkwai clan #Kwai }}
: ''For extensions, see [[Hemifamity temperaments #Kwai]].''


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


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


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


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630
[[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, 111, 152 }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 2017bbc, 2169bbc }}


[[Badness]]: 0.636715
[[Badness]] (Sintel): 14.9


== Countercomp ==
== Cotoneum (5-limit) ==
:''See also: [[Countercomp family]] and [[41-comma]]''
: ''For extensions, see [[Garischismic clan #Cotoneum]].''


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


[[Comma list]]: {{monzo| 65 -41 }}
[[Comma list]]: {{monzo| 80 -49 -1 }}


{{Mapping|legend=1| 41 65 0 | 0 0 1 }}
{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668
[[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, 123, 164, 205, 369, 574, 779, 2132bc }}
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b, 3759bbc }}


[[Badness]]: 0.934310
[[Badness]] (Sintel): 29.1


== Cotoneum (5-limit) ==
== Hemififths (5-limit) ==
{{See also| 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) ==
{{See also| Breedsmic temperaments #Newt }}
: ''For extensions, see [[Garischismic clan #Newt]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 226: 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]]
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