Septischismic clan: Difference between revisions
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m FloraC moved page Garischismic clan to Septischismic clan: Rename (see talk at Septischismic family) |
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{{Technical data page}} | {{Technical data page}} | ||
The ''' | The '''septischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[septischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33 554 432 / 33 480 783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]], thus equating the two. | ||
== Gary == | == Gary == | ||
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==== Subgroup extensions ==== | ==== Subgroup extensions ==== | ||
Gary can be naturally extended into the no-5's 11-limit with good accuracy by equating (64/63)<sup>2</sup> with 33/32 | Gary can be naturally extended into the no-5's 11-limit with good accuracy by equating (64/63)<sup>2</sup> with 33/32 at the cost of doubling the complexity, acting as a temperament of [[olympic clan #Olympian|olympian]] which accurately represents the entire [[2.3.7.11 subgroup]] into the chain of fifths. The major third additionally is extremely close to [[19/15]], and is a natural interpretation in extensions with 5 and 19. | ||
=== 2.3.7.11 subgroup === | === 2.3.7.11 subgroup === | ||
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Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}} | * WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}} | ||
: error map: {{val| -0.037 0.216 -0.139 -0.173 }} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}} | ||
: error map: {{val| -0.000 0.274 -0.032 -0.051 }} | |||
{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }} | {{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }} | ||
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: ''For the 5-limit version, see [[Schismic–countercommatic equivalence continuum #Cotoneum (5-limit)]].'' | : ''For the 5-limit version, see [[Schismic–countercommatic equivalence continuum #Cotoneum (5-limit)]].'' | ||
Cotoneum tempers out 10976/10935 ([[hemimage comma]]) | Cotoneum tempers out 10976/10935 ([[hemimage comma]]) and 823543/819200 ([[quince comma]]) in addition to the septischisma. This temperament can be described as {{nowrap| 41 & 217 }}, and is supported by [[176edo|176-]], [[217edo|217-]], and [[258edo]]. 5/4 is found -49 generators away. In terms of chain-of-fifths notation, this is a sextuple-diminished octave, or a perfect fourth minus four generic commas. | ||
However, cotoneum can be notated like [[cassaschismic]], where 5/4 is conceptualized as an aberschisma-up comma-down major third (C–^↓E), but with the extra equivalence that the generic aberschisma is identical to the [[41-comma]]. In other words, we have C–^↑↑E = C–↓↓E. | However, cotoneum can be notated like [[cassaschismic]], where 5/4 is conceptualized as an aberschisma-up comma-down major third (C–^↓E), but with the extra equivalence that the generic aberschisma is identical to the [[41-comma]]. In other words, we have C–^↑↑E = C–↓↓E. | ||
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Like [[#Cotoneum|cotoneum]], newt can be notated in the same way as [[cassaschismic]], but with half-sharps and half-flats and the extra equivalence that two comma steps and an aberschisma step make a half-apotome step. In other words, C–^↑↑E = C–v↓↓E = C–Ed. | Like [[#Cotoneum|cotoneum]], newt can be notated in the same way as [[cassaschismic]], but with half-sharps and half-flats and the extra equivalence that two comma steps and an aberschisma step make a half-apotome step. In other words, C–^↑↑E = C–v↓↓E = C–Ed. | ||
Newt continues to be significant as an [[11-limit]] temperament, where it tempers out the lehmerisma ([[3025/3024]]). This extends into a very strong [[13-limit]] temperament and eventually a very strong no-17 [[19-limit]] temperament | Newt continues to be significant as an [[11-limit]] temperament, where it tempers out the lehmerisma ([[3025/3024]]). This extends into a very strong [[13-limit]] temperament and eventually a very strong no-17 [[19-limit]] temperament. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] are more accurate. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Badness (Sintel): 0.571 | Badness (Sintel): 0.571 | ||
=== 2.3.5.7.11.13.19 subgroup | === 2.3.5.7.11.13.19 subgroup === | ||
Subgroup: 2.3.5.7.11.13.19 | Subgroup: 2.3.5.7.11.13.19 | ||
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== Gariwizmic == | == Gariwizmic == | ||
Gariwizmic tempers out the [[wizma]] and the | Gariwizmic tempers out the [[wizma]] and the septischisma, and may be described as the {{nowrap| 94 & 176 }} temperament. It assumes a [[semioctave]] period and a [[3/2|perfect fifth]] generator that is slightly sharp of just. It finds [[5/4]] 39 fifths away, shifted by a semioctave. It extends extremely well to the 2.3.5.7.11.13.19 subgroup. Notable tunings not appearing in the optimal ET sequence include [[364edo]] and [[634edo]]. | ||
Gariwizmic was named by [[Eufalesio]] in 2026 as a concatenation of ''gary'' and ''wizmic''. | Gariwizmic was named by [[Eufalesio]] in 2026 as a concatenation of ''gary'' and ''wizmic''. | ||
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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Satin]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Satin]].'' | ||
Satin tempers out the [[rainy comma]] and the [[ | Satin tempers out the [[rainy comma]] and the [[cathartic comma]] in addition to the septischisma, and may be described as the {{nowrap| 94 & 217 }} temperament. It uses [[~]][[11/10]] as a generator, three of which gives a [[4/3|perfect fourth]], tempering out [[4000/3993]] in the 11-limit and onwards. Its [[ploidacot]] is omega-tricot. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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=== 2.3.5.7.11.13.19 subgroup === | === 2.3.5.7.11.13.19 subgroup === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.19 | ||
Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625 | Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625 | ||
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== World calendar == | == World calendar == | ||
World calendar tempers out the [[dimcomp comma]] and the | World calendar tempers out the [[dimcomp comma]] and the septischisma, and can be described as the {{nowrap| 12 & 364 }} temperament. The name derives from a {{w|World Calendar|certain calendar layout}} by the same name. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Badness (Sintel): 1.22 | Badness (Sintel): 1.22 | ||
[[Category:Septischismic clan| ]] <!-- main article --> | |||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||