Schismic–countercommatic equivalence continuum: Difference between revisions
m FloraC moved page Schismic-counterpyth equivalence continuum to Schismic-countercommatic equivalence continuum: Name change following pythagorean -> compton |
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The '''schismic- | 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 microtemperaments. | ||
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 tempers both commas and thus tempers 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 (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 tempers both commas and thus tempers 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. | ||
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-Pythagorean equivalence continuum]]. | ||
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| [[ | | [[Countercomp family #Countercomp|Countercomp]] | ||
| | | | ||
| {{monzo|65 -41}} | | {{monzo|65 -41}} | ||
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== Rodan (5-limit) == | == Rodan (5-limit) == | ||
{{See also|Gamelismic clan #Rodan}} | {{See also| Gamelismic clan #Rodan }} | ||
[[Comma]]: 131072000/129140163 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 131072000/129140163 | |||
[[Mapping]]: [{{val| 1 1 -1 }}, {{val| 0 3 17 }}] | [[Mapping]]: [{{val| 1 1 -1 }}, {{val| 0 3 17 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~729/640 = 234.528 | ||
{{Val list|legend=1| 5, 31c, 36c, 41, 46, 87, 220, 307 }} | {{Val list|legend=1| 5, 31c, 36c, 41, 46, 87, 220, 307 }} | ||
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{{See also|Breedsmic temperaments #Hemififths}} | {{See also|Breedsmic temperaments #Hemififths}} | ||
[[Comma]]: 858993459200/847288609443 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 858993459200/847288609443 | |||
[[Mapping]]: [{{val| 1 1 -5 }}, {{val| 0 2 25 }}] | [[Mapping]]: [{{val| 1 1 -5 }}, {{val| 0 2 25 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476 | ||
{{Val list|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }} | {{Val list|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }} | ||
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== Kwai (5-limit) == | == Kwai (5-limit) == | ||
{{See also|Mirkwai clan #Kwai}} | {{See also| Mirkwai clan #Kwai }} | ||
[[Subgroup]]: 2.3.5 | |||
[[Comma]]: {{ | [[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523 | ||
[[Mapping]]: [{{val| 1 2 16 }}, {{val| 0 -1 -33 }}] | [[Mapping]]: [{{val| 1 2 16 }}, {{val| 0 -1 -33 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630 | ||
{{Val list|legend=1| 41, 111, 152 }} | {{Val list|legend=1| 41, 111, 152 }} | ||
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[[Badness]]: 0.636715 | [[Badness]]: 0.636715 | ||
== | == Countercomp == | ||
:''See also: [[ | :''See also: [[Countercomp family]] and [[41-comma]]'' | ||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: {{monzo|65 -41}} | [[Comma list]]: {{monzo| 65 -41 }} | ||
[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}] | [[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668 | ||
{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }} | {{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }} | ||
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== Cotoneum (5-limit) == | == Cotoneum (5-limit) == | ||
{{See also|Hemimage temperaments #Cotoneum}} | {{See also| Hemimage temperaments #Cotoneum }} | ||
[[Comma]]: {{ | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: {{monzo| 80 -49 -1 }} | |||
[[Mapping]]: [{{val| 1 2 -18 }}, {{val| 0 -1 49 }}] | [[Mapping]]: [{{val| 1 2 -18 }}, {{val| 0 -1 49 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315 | ||
{{Val list|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b }} | {{Val list|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b }} | ||
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== Newt (5-limit) == | == Newt (5-limit) == | ||
{{See also|Breedsmic temperaments #Newt}} | {{See also| Breedsmic temperaments #Newt }} | ||
[[Subgroup]]: 2.3.5 | |||
[[Comma]]: {{ | [[Comma list]]: {{monzo| 95 -57 -2 }} | ||
[[Mapping]]: [{{val| 1 1 19 }}, {{val| 0 2 -57 }}] | [[Mapping]]: [{{val| 1 1 19 }}, {{val| 0 2 -57 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~{{monzo| 47 -28 -1 }} = 351.114 | ||
{{Val list|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }} | {{Val list|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }} | ||