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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 these are all 5-limit microtemperaments.
{{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.


All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}. 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 [[53edo]] (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 1.8503390493..., 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|-84 53}}}}. 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 [[53edo]] (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 1.8503390493…, 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 [[Syntonic-chromatic equivalence continuum]].
For a similar but perhaps more intuitive and practical concept, see [[Syntonic–chromatic equivalence continuum]].


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 16: Line 17:
|-
|-
| 0
| 0
| [[Mercator's comma|Mercator]]
| [[Mercator]]
|  
| (52 digits)
| {{monzo|-84 53}}
| {{Monzo| -84 53 }}
|-
|-
| 1
| 1
| Counterschismic
| [[Counterschismic]]
|  
| (44 digits)
| {{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)
| [[Monzisma|{{monzo|54 -37 2}}]]
| [[Monzisma|{{Monzo| 54 -37 2 }}]]
|-
|-
| 3
| 3
| [[Tricot]]
| [[Alphatricot]]
|  
| (28 digits)
| {{monzo| 39 -29 3}}
| [[Alphatricot comma|{{Monzo| 39 -29 3 }}]]
|-
|-
| 4
| 4
| [[Vulture]]
| [[Vulture]]
|  
| (22 digits)
| {{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
| [[Marvel temperaments|Submajor]]
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]]
|  
| (22 digits)
| {{monzo|-36 11 8 }}
| {{Monzo| -36 11 8 }}
|-
|-
| 9
| 9
| [[Marvel temperaments|Untriton]]
| [[Miscellaneous 5-limit temperaments #Untriton|Untriton]]
|  
| (32 digits)
| {{monzo|-51 19 9 }}
| {{Monzo| -51 19 9 }}
|-
|-
| …
| …
Line 73: Line 74:
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1}}
| {{Monzo| -15 8 1 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
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.  
* 53 & 3684 (''n'' = 11/6 = 1.8{{overline|3}})
* 53 & 4190 (''n'' = 13/7 = 1.{{overline|857142}})


== Mercator ==
{| class="wikitable center-1"
{{see also| Mercator's comma }} ''and [[Mercator family]]''
|+ style="font-size: 105%;" | Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Mercator]]
| (52 digits)
| {{Monzo| -84 53 }}
|-
| 1
| [[Schismic]]
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
|-
| 2
| [[Monzismic]]
| (36 digits)
| [[Monzisma|{{Monzo| 54 -37 2 }}]]
|-
| …
| …
| …
| …
|-
| ∞
| [[Counterschismic]]
| (44 digits)
| [[Counterschisma|{{Monzo| -69 45 -1 }}]]
|}


Comma list: {{monzo| -84 53 }}
{| class="wikitable"
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|-
! Temperament !! ''n'' !! ''m'' !! Comma
|-
| 53 & 3684 || 11/6 = 1.8{{overline|3}} || 11/5 = 2.2 || {{Monzo| -339 230 -11 }}
|-
| 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}} || {{Monzo| 33 -34 9 }}
|-
| [[Quartonic]] || 11/2 = 5.5 || 11/9 = 1.{{overline|2}} || {{Monzo| 3 -18 11 }}
|-
| [[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 }}
|}
 
== Counterschismic ==
{{See also| Counterschisma }}
 
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.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -69 45 -1 }}
 
{{Mapping|legend=1| 1 0 -69 | 0 1 45 }}
: mapping generators: ~2, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056, 16219b }}


POTE generator: ~5/4 = 386.264
[[Badness]] (Sintel): 2.14


Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]
== Demibuzzard (5-limit) ==
: ''For extensions, see [[Buzzardsmic clan #Demibuzzard]].''


Wedgie: {{wedgie| 0 53 84 }}
[[Subgroup]]: 2.3.5


{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
[[Comma list]]: 69198046875/68719476736


Badness: 0.2843
{{Mapping|legend=1| 1 -4 10 | 0 8 -11 }}
: mapping generators: ~2, ~16384/10125


== Counterschismic ==
[[Optimal tuning]]s:
{{see also| Counterschisma }}
* [[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| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
 
[[Badness]] (Sintel): 3.06
 
== Countritonic ==
: ''For extensions, see [[Hemifamity temperaments #Countriton]], [[Ragismic microtemperaments #Ragitritonic]], and [[Garischismic clan #Garitritonic]].''


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


POTE generator: ~3/2 = 701.9175
[[Comma list]]: {{monzo| 33 -34 9 }}


Mapping: [{{val| 1 2 21 }}, {{val| 0 -1 -45 }}]
{{Mapping|legend=1| 1 -3 -15 | 0 9 34 }}
: mapping generators: ~2, ~20480000/14348907


Wedgie: {{wedgie| 1 45 69 }}
[[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 }}


{{Val list|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}}
{{Optimal ET sequence|legend=1| 51c, 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b }}


Badness: 0.09123
[[Badness]] (Sintel): 6.00


== 53 & 3684 ==
== 53 & 3684 ==
Comma list: {{monzo| -339 230 -11 }}
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -339 230 -11 }}
 
{{Mapping|legend=1| 1 2 11 | 0 -11 -230 }}
: mapping generators: ~2, ~10737418240/10460353203
 
[[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 }}


POTE generator: 45.2769
{{Optimal ET sequence|legend=1| 53, …, 3313, 3366, 3419, 3472, 3525, 3578, 3631, 3684, 7421, 11105, 25894, 36999 }}


Mapping: [{{val| 1 2 11 }}, {{val| 0 -11 -230 }}]
[[Badness]] (Sintel): 6.48


Wedgie: {{wedgie| 11 230 339 }}
== 53 & 4296 ==
[[Subgroup]]: 2.3.5


{{Val list|legend=1| 53, 3684, 11105 }}
[[Comma list]]: {{monzo| 393 -267 13 }}


Badness: 0.2760
{{Mapping|legend=1| 1 -7 -174 | 0 13 267 }}
: mapping generators: ~2, ~{{monzo| 61 -41 2 }}


== 53 & 4190 ==
[[Optimal tuning]]s:
Comma list: {{monzo| 393 -267 13 }}
* [[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 }}


Wedgie: {{wedgie| 13 267 393 }}
{{Optimal ET sequence|legend=1| 53, …, 3872, 3925, 3978, 4031, 4084, 4137, 4190, 4243, 4296, 34315, 38611, 42907, 47203, 51499, 55795, 60091, 64387, 68683 }}


{{Val list|legend=1| 53, 4190, 4296 }}
[[Badness]] (Sintel): 4.07


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