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Schismic–Mercator equivalence continuum: Difference between revisions

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{{Technical data page}}
The '''schismic–Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo| -84 53 }})]]. This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.
The '''schismic–Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo| -84 53 }})]]. This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.


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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 }}]]
|-
|-
| 2
| 2
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
| (36 digits)
| (36 digits)
| [[Monzisma|{{monzo| 54 -37 2 }}]]
| [[Monzisma|{{Monzo| 54 -37 2 }}]]
|-
|-
| 3
| 3
| [[Tricot]]
| [[Alphatricot]]
| (28 digits)
| (28 digits)
| [[Tricot comma|{{monzo| 39 -29 3 }}]]
| [[Alphatricot comma|{{Monzo| 39 -29 3 }}]]
|-
|-
| 4
| 4
| [[Vulture]]
| [[Vulture]]
| (22 digits)
| (22 digits)
| [[Vulture comma|{{monzo| 24 -21 4 }}]]
| [[Vulture comma|{{Monzo| 24 -21 4 }}]]
|-
|-
| 5
| 5
| [[Amity]]
| [[Amity]]
| [[1600000/1594323]]
| [[1600000/1594323]]
| {{monzo| 9 -13 5 }}
| {{Monzo| 9 -13 5 }}
|-
|-
| 6
| 6
| [[Kleismic]]
| [[Kleismic]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo|-6 -5 6}}
| {{Monzo|-6 -5 6}}
|-
|-
| 7
| 7
| [[Orson]]
| [[Orson]]
| [[Semicomma|2109375/2097152]]
| [[Semicomma|2109375/2097152]]
| {{monzo|-21 3 7 }}
| {{Monzo|-21 3 7 }}
|-
|-
| 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 }}
|-
|-
| …
| …
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| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
|}
|}


We may invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''counterschismic–Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600…  
We may invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''counterschismic–Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600… While the counterschisma is of comparable size as the schisma, it is way more complex, so this continuum does not contain as many useful temperaments at integer points.


{| class="wikitable center-1"
{| class="wikitable center-1"
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| [[Mercator]]
| [[Mercator]]
| (52 digits)
| (52 digits)
| {{monzo| -84 53 }}
| {{Monzo| -84 53 }}
|-
|-
| 1
| 1
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
|-
|-
| 2
| 2
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
| [[Monzismic]]
| (36 digits)
| (36 digits)
| [[Monzisma|{{monzo| 54 -37 2 }}]]
| [[Monzisma|{{Monzo| 54 -37 2 }}]]
|-
|-
| …
| …
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|-
|-
| ∞
| ∞
| Counterschismic
| [[Counterschismic]]
| (44 digits)
| (44 digits)
| [[Counterschisma|{{monzo| -69 45 -1 }}]]
| [[Counterschisma|{{Monzo| -69 45 -1 }}]]
|}
|}


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|+ 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 }} ''and [[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.  


Subgroup: 2.3.5
== Demibuzzard (5-limit) ==
: ''For extensions, see [[Buzzardsmic clan #Demibuzzard]].''


Comma list: {{monzo| -69 45 -1 }}
[[Subgroup]]: 2.3.5


Mapping: {{mapping| 1 2 21 | 0 -1 -45 }}
[[Comma list]]: 69198046875/68719476736


{{Multival|legend=1| 1 45 69 }}
{{Mapping|legend=1| 1 -4 10 | 0 8 -11 }}
: mapping generators: ~2, ~16384/10125


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9175
[[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 ET sequence|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}}
{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}


Badness: 0.09123
[[Badness]] (Sintel): 3.06


== 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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