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Schismic–Mercator equivalence continuum: Difference between revisions

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| [[Mercator]]
| [[Mercator]]
| (52 digits)
| (52 digits)
| {{monzo| -84 53 }}
| {{Monzo| -84 53 }}
|-
|-
| 1
| 1
| Counterschismic
| [[Counterschismic]]
| (44 digits)
| (44 digits)
| [[Counterschisma|{{Monzo| -69 45 -1 }}]]
| [[Counterschisma|{{Monzo| -69 45 -1 }}]]
Line 57: Line 57:
|-
|-
| 8
| 8
| [[Submajor (temperament)|Submajor]]
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]]
| (22 digits)
| (22 digits)
| {{Monzo| -36 11 8 }}
| {{Monzo| -36 11 8 }}
|-
|-
| 9
| 9
| [[High badness temperaments #Untriton|Untriton]]
| [[Miscellaneous 5-limit temperaments #Untriton|Untriton]]
| (32 digits)
| (32 digits)
| {{Monzo| -51 19 9 }}
| {{Monzo| -51 19 9 }}
Line 118: Line 118:
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|-
|-
! Temperament !! ''n'' !! ''m''
! Temperament !! ''n'' !! ''m'' !! Comma
|-
|-
| 53 & 3684 || 11/6 = 1.8{{overline|3}} || 11/5 = 2.2
| 53 & 3684 || 11/6 = 1.8{{overline|3}} || 11/5 = 2.2 || {{Monzo| -339 230 -11 }}
|-
|-
| 53 & 4190 || 13/7 = 1.{{overline|857142}} || 13/6 = 2.1{{overline|6}}
| 53 & 4296 || 13/7 = 1.{{overline|857142}} || 13/6 = 2.1{{overline|6}} || {{Monzo| 393 -267 13 }}
|-
|-
| [[Countritonic]] || 9/2 = 4.5 || 9/7 = 1.{{overline|285714}}
| [[Countritonic]] || 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || {{Monzo| 33 -34 9 }}
|-
|-
| [[Quartonic]] || 11/2 = 5.5 || 11/9 = 1.{{overline|2}}
| [[Quartonic]] || 11/2 = 5.5 || 11/9 = 1.{{overline|2}} || {{Monzo| 3 -18 11 }}
|-
|-
| [[Ditonic]] || 13/2 = 6.5 || 13/11 = 1.{{overline|18}}
| [[Maja]] || 17/3 = 5.{{overline|6}} || 17/14 = 1.2{{overline|142857}} || {{Monzo| -3 -23 17 }}
|-
| [[Ditonic]] || 13/2 = 6.5 || 13/11 = 1.{{overline|18}} || {{Monzo| -27 -2 13 }}
|}
|}


== Mercator ==
== Counterschismic ==
{{See also| Mercator's comma | Mercator family }}
{{See also| Counterschisma }}


Comma list: {{monzo| -84 53 }}
Counterschismic is generated by a [[3/2|perfect fifth]], like [[schismic]], but the [[5/1|5th]] [[harmonic]] is located at +45 fifths instead of schismic's -8. They unite at [[53edo]], of course. [[730edo]] may be recommended as a tuning.


Mapping: {{mapping| 53 84 123 | 0 0 1 }}
[[Subgroup]]: 2.3.5


{{Multival|legend=1| 0 53 84 }}
[[Comma list]]: {{monzo| -69 45 -1 }}


Optimal tuning (CTE): ~531441/524288 = 1\53, ~5/4 = 386.264
{{Mapping|legend=1| 1 0 -69 | 0 1 45 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0116{{c}}, ~3/2 = 701.9243{{c}}
: [[error map]]: {{val| +0.012 -0.019 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9177{{c}}
: error map: {{val| 0.000 -0.037 -0.017 }}


Badness: 0.2843
{{Optimal ET sequence|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056, 16219b }}


== Counterschismic ==
[[Badness]] (Sintel): 2.14
{{See also| Counterschisma }}


Counterschismic is much like [[schismic]], but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in [[53edo]], of course.  
== Demibuzzard (5-limit) ==
: ''For extensions, see [[Buzzardsmic clan #Demibuzzard]].''


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


Comma list: {{monzo| -69 45 -1 }}
[[Comma list]]: 69198046875/68719476736


Mapping: {{mapping| 1 2 21 | 0 -1 -45 }}
{{Mapping|legend=1| 1 -4 10 | 0 8 -11 }}
: mapping generators: ~2, ~16384/10125


{{Multival|legend=1| 1 45 69 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~16384/10125 = 837.8342{{c}}
: [[error map]]: {{val| +0.222 -0.171 -0.267 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16384/10125 = 837.6794{{c}}
: error map: {{val| 0.000 -0.520 -0.787 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9175
{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}


{{Optimal ET sequence|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}}
[[Badness]] (Sintel): 3.06
 
Badness: 0.09123


== Countritonic ==
== Countritonic ==
{{See also| High badness temperaments #Countritonic }}
: ''For extensions, see [[Hemifamity temperaments #Countriton]], [[Ragismic microtemperaments #Ragitritonic]], and [[Garischismic clan #Garitritonic]].''
: ''For extensions, see [[Ragismic microtemperaments #Countritonic]] and [[Hemifamity temperaments #Countriton]].''


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


Comma list: {{monzo| 33 -34 9 }}
[[Comma list]]: {{monzo| 33 -34 9 }}


Mapping: {{mapping| 1 6 19 | 0 -9 -34 }}
{{Mapping|legend=1| 1 -3 -15 | 0 9 34 }}
: mapping generators: ~2, ~20480000/14348907


Optimal tuning (CTE): ~2 = 1\1, ~14348907/10240000 = 588.636
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9228{{c}}, ~20480000/14348907 = 611.3248{{c}}
: [[error map]]: {{val| -0.077 +0.200 -0.113 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20480000/14348907 = 611.3614{{c}}
: error map: {{val| 0.000 +0.297 -0.027 }}


{{Optimal ET sequence|legend=1| 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b }}
{{Optimal ET sequence|legend=1| 51c, 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b }}


Badness: 0.256
[[Badness]] (Sintel): 6.00


== 53 & 3684 ==
== 53 & 3684 ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5
 
Comma list: {{monzo| -339 230 -11 }}
 
Mapping: {{mapping| 1 2 11 | 0 -11 -230 }}


{{Multival|legend=1| 11 230 339 }}
[[Comma list]]: {{monzo| -339 230 -11 }}


Optimal tuning (CTE): ~2 = 1\1, ~10737418240/10460353203 = 45.2769
{{Mapping|legend=1| 1 2 11 | 0 -11 -230 }}
: mapping generators: ~2, ~10737418240/10460353203


{{Optimal ET sequence|legend=1| 53, 3684, 11105 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.000272{{c}}, ~10737418240/10460353203 = 45.276910{{c}}
: [[error map]]: {{val| +0.0003 -0.0005 +0.0000 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~10737418240/10460353203 = 45.276898{{c}}
: error map: {{val| 0.0000 -0.0009 -0.0003 }}


Badness: 0.276036
{{Optimal ET sequence|legend=1| 53, …, 3313, 3366, 3419, 3472, 3525, 3578, 3631, 3684, 7421, 11105, 25894, 36999 }}


== 53 & 4190 ==
[[Badness]] (Sintel): 6.48
Subgroup: 2.3.5


Comma list: {{monzo| 393 -267 13 }}
== 53 & 4296 ==
[[Subgroup]]: 2.3.5


Mapping: {{mapping| 1 6 93 | 0 -13 -267 }}
[[Comma list]]: {{monzo| 393 -267 13 }}


{{Multival|legend=1| 13 267 393 }}
{{Mapping|legend=1| 1 -7 -174 | 0 13 267 }}
: mapping generators: ~2, ~{{monzo| 61 -41 2 }}


Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| -60 41 -2 }} = 407.5419
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.999891{{c}}, ~{{monzo| 61 -41 2 }} = 792.458032{{c}}
: [[error map]]: {{val| -0.0001 +0.0002 -0.0000 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 61 -41 2 }} = 792.458104{{c}}
: error map: {{val| 0.0000 +0.0004 +0.0002 }}


{{Optimal ET sequence|legend=1| 53, 4190, 4243, 4296 }}
{{Optimal ET sequence|legend=1| 53, …, 3872, 3925, 3978, 4031, 4084, 4137, 4190, 4243, 4296, 34315, 38611, 42907, 47203, 51499, 55795, 60091, 64387, 68683 }}


Badness: 0.173433
[[Badness]] (Sintel): 4.07


[[Category:53edo]]
[[Category:53edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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