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Schismic–Mercator equivalence continuum: Difference between revisions

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The '''syntonic-chromatic equivalence continuum''' is a continuum of temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]].
{{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
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]]
| (22 digits)
| {{Monzo| -36 11 8 }}
|-
| 9
| [[Miscellaneous 5-limit temperaments #Untriton|Untriton]]
| (32 digits)
| {{Monzo| -51 19 9 }}
|-
|-
| …
| …
Line 63: 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.
*3684 & 11105 (''n'' = 11/6 = 1.8{{overline|3}})
 
{| class="wikitable center-1"
|+ 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 }}]]
|}
 
{| 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 ==
== 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 }}
[[Badness]] (Sintel): 2.14
== Demibuzzard (5-limit) ==
: ''For extensions, see [[Buzzardsmic clan #Demibuzzard]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: 69198046875/68719476736
{{Mapping|legend=1| 1 -4 10 | 0 8 -11 }}
: mapping generators: ~2, ~16384/10125
[[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| 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]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 33 -34 9 }}
{{Mapping|legend=1| 1 -3 -15 | 0 9 34 }}
: mapping generators: ~2, ~20480000/14348907
[[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| 51c, 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b }}
[[Badness]] (Sintel): 6.00


Comma: {{monzo|-69 45 -1}}
== 53 & 3684 ==
[[Subgroup]]: 2.3.5


Map: [<1 2 21|, <0 -1 -45|]
[[Comma list]]: {{monzo| -339 230 -11 }}


Wedgie: <<1 45 69||
{{Mapping|legend=1| 1 2 11 | 0 -11 -230 }}
: mapping generators: ~2, ~10737418240/10460353203


POTE generator: ~3/2 = 701.9175
[[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 }}


EDOs: {{EDOs|53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}}
{{Optimal ET sequence|legend=1| 53, …, 3313, 3366, 3419, 3472, 3525, 3578, 3631, 3684, 7421, 11105, 25894, 36999 }}


Badness: 0.09123
[[Badness]] (Sintel): 6.48


== 3684 & 11105 ==
== 53 & 4296 ==
Comma: {{monzo|-339 230 -11}}
[[Subgroup]]: 2.3.5


Map: {{val|1 2 11}}, {{val|0 -11 -230}}
[[Comma list]]: {{monzo| 393 -267 13 }}


Wedgie: {{wedgie|11 230 339}}
{{Mapping|legend=1| 1 -7 -174 | 0 13 267 }}
: mapping generators: ~2, ~{{monzo| 61 -41 2 }}


POTE generator: 45.2769
[[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 }}


EDOs: {{Val list|53, 3684, 11105}}
{{Optimal ET sequence|legend=1| 53, …, 3872, 3925, 3978, 4031, 4084, 4137, 4190, 4243, 4296, 34315, 38611, 42907, 47203, 51499, 55795, 60091, 64387, 68683 }}


== Unnamed temperament ==
[[Badness]] (Sintel): 4.07
Comma: {{monzo|393 -267 13}}


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