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Septischismic clan: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]], thus equating the two.  
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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[[Comma list]]: 33554432/33480783
[[Comma list]]: 33554432/33480783


{{Mapping|legend=2| 1 0 25 | 0 1 -14 }}
{{Mapping|legend=3| 1 0 25 | 0 1 -14 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


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* ''[[Quintagar]]'' (+3136/3125) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Quintagar]]'' (+3136/3125) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Paramity]]'' (+65625/65536) → [[Amity family #Paramity|Amity family]]
* ''[[Paramity]]'' (+65625/65536) → [[Amity family #Paramity|Amity family]]
* ''[[Garistearn]]'' (+118098/117649) → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]
* ''[[Garistearn]]'' (+118098/117649) → [[94th-octave temperaments #Garistearn|94th-octave temperaments]]


Considered below are cotoneum, newt, gariwizmic, satin, sextile, and world calendar.  
Considered below are [[#Cotoneum|cotoneum]], [[#Newt|newt]], [[#Gariwizmic|gariwizmic]], [[#Satin|satin]], [[#Sextile|sextile]], [[#World calendar|world calendar]], [[#Heptacot|heptacot]], and [[#Garitritonic|garitritonic]].  


==== 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 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 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.  
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 ===
Line 48: Line 48:
Comma list: 19712/19683, 41503/41472
Comma list: 19712/19683, 41503/41472


Subgroup-val mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}
{{Mapping|legend=2| 1 0 25 -33 | 0 1 -14 23 }}


Optimal tunings:  
Optimal tunings:  
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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]]), and 823543/819200 ([[quince comma]]) in addition to the garischisma. 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.   
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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Comma list: 441/440, 10976/10935, 16384/16335
Comma list: 441/440, 10976/10935, 16384/16335


Mapping: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }}
{{Mapping|legend=0| 1 0 80 25 -33 | 0 1 -49 -14 23 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 364/363, 441/440, 3584/3575, 10976/10935
Comma list: 364/363, 441/440, 3584/3575, 10976/10935


Mapping: {{mapping| 1 0 80 25 -33 -93 | 0 1 -49 -14 23 61 }}
{{Mapping|legend=0| 1 0 80 25 -33 -93 | 0 1 -49 -14 23 61 }}


Optimal tunings:  
Optimal tunings:  
Line 135: Line 135:
Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262


Mapping: {{mapping| 1 0 80 25 -33 -93 -137 | 0 1 -49 -14 23 61 89 }}
{{Mapping|legend=0| 1 0 80 25 -33 -93 -137 | 0 1 -49 -14 23 61 89 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728


Mapping: {{mapping| 1 0 80 25 -33 -93 -137 74 | 0 1 -49 -14 23 61 89 -44 }}
{{Mapping|legend=0| 1 0 80 25 -33 -93 -137 74 | 0 1 -49 -14 23 61 89 -44 }}


Optimal tunings:  
Optimal tunings:  
Line 199: Line 199:
Comma list: 2401/2400, 3025/3024, 19712/19683
Comma list: 2401/2400, 3025/3024, 19712/19683


Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
{{Mapping|legend=0| 1 1 19 11 -10 | 0 2 -57 -28 46 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095


Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
{{Mapping|legend=0| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}


Optimal tunings:  
Optimal tunings:  
Line 229: Line 229:
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400


Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
{{Mapping|legend=0| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}


Optimal tunings:  
Optimal tunings:  
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== Gariwizmic ==
== Gariwizmic ==
Gariwizmic tempers out the [[wizma]] and the garischisma, 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 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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Comma list: 9801/9800, 19712/19683, 41503/41472
Comma list: 9801/9800, 19712/19683, 41503/41472


Mapping: {{mapping| 2 0 -119 50 -66 | 0 1 39 -14 23 }}
{{Mapping|legend=0| 2 0 -119 50 -66 | 0 1 39 -14 23 }}


Optimal tunings:
Optimal tunings:
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Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683
Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683


Mapping: {{mapping| 2 0 -119 50 -66 93 | 0 1 39 -14 23 -27 }}
{{Mapping|legend=0| 2 0 -119 50 -66 93 | 0 1 39 -14 23 -27 }}


Optimal tunings:
Optimal tunings:
Line 300: Line 300:
Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079
Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079


Mapping: {{mapping| 2 0 -119 50 -66 93 -131 | 0 1 39 -14 23 -27 44 }}
{{Mapping|legend=0| 2 0 -119 50 -66 93 -131 | 0 1 39 -14 23 -27 44 }}


Optimal tunings:
Optimal tunings:
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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 [[canousma]] in addition to the garischisma, 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.  
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
Line 339: Line 339:
Comma list: 4000/3993, 19712/19683, 41503/41472
Comma list: 4000/3993, 19712/19683, 41503/41472


Mapping: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}
{{Mapping|legend=0| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}


Optimal tunings:  
Optimal tunings:  
Line 354: Line 354:
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689


Mapping: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}
{{Mapping|legend=0| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}


Optimal tunings:  
Optimal tunings:  
Line 369: Line 369:
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095


Mapping: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}
{{Mapping|legend=0| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}


Optimal tunings:  
Optimal tunings:  
Line 384: Line 384:
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573


Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}
{{Mapping|legend=0| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}


Optimal tunings:  
Optimal tunings:  
Line 399: Line 399:
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155


Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 16 | 0 -3 -70 42 -69 34 -50 -85 -83 }}
{{Mapping|legend=0| 1 2 12 -3 13 -1 11 16 16 | 0 -3 -70 42 -69 34 -50 -85 -83 }}


Optimal tunings:  
Optimal tunings:  
Line 436: Line 436:
Comma list: 5632/5625, 9801/9800, 151263/151250
Comma list: 5632/5625, 9801/9800, 151263/151250


Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}
{{Mapping|legend=0| 6 0 71 150 230 | 0 1 -6 -14 -22 }}


Optimal tunings:  
Optimal tunings:  
Line 451: Line 451:
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647


Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}
{{Mapping|legend=0| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}


Optimal tunings:  
Optimal tunings:  
Line 466: Line 466:
Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625
Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625


Mapping: {{mapping| 6 0 71 150 230 279 35 | 0 1 -6 -14 -22 -27 -1 }}
{{Mapping|legend=0| 6 0 71 150 230 279 35 | 0 1 -6 -14 -22 -27 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 477: Line 477:


== World calendar ==
== World calendar ==
World calendar tempers out the [[dimcomp comma]] and the garischisma, 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.  
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 ― because a possible scale is [31 30 30 31 30 30 31 30 30 31 30 30] in 364edo, which is the proposed layout for month lengths in the namesake calendar from January to December.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 501: Line 501:
Comma list: 2025/2023, 24576/24565, 390625/388962
Comma list: 2025/2023, 24576/24565, 390625/388962


Subgroup-val mapping: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}
{{Mapping|legend=2| 4 1 -44 86 3 | 0 2 -13 -28 5 }}


Optimal tunings:  
Optimal tunings:  
Line 516: Line 516:
Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000
Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000


Subgroup-val mapping: {{mapping| 4 1 -44 86 3 25 | 0 2 -13 -28 5 -3 }}
{{Mapping|legend=2| 4 1 -44 86 3 25 | 0 2 -13 -28 5 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 555: Line 555:
Comma list: 19712/19683, 41503/41472, 703125/702464
Comma list: 19712/19683, 41503/41472, 703125/702464


Mapping: {{mapping| 1 1 6 11 -10 | 0 7 -44 -98 161 }}
{{Mapping|legend=0| 1 1 6 11 -10 | 0 7 -44 -98 161 }}
: mapping generators: ~2, ~1323/1250
: mapping generators: ~2, ~1323/1250


Line 571: Line 571:
Comma list: 2080/2079, 4096/4095, 19712/19683, 31250/31213
Comma list: 2080/2079, 4096/4095, 19712/19683, 31250/31213


Mapping: {{mapping| 1 1 6 11 -10 -7 | 0 7 -44 -98 161 128 }}
{{Mapping|legend=0| 1 1 6 11 -10 -7 | 0 7 -44 -98 161 128 }}
: mapping generators: ~2, ~1323/1250
: mapping generators: ~2, ~1323/1250


Line 587: Line 587:
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 31250/31213
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 31250/31213


Mapping: {{mapping| 1 1 6 11 -10 -7 5 | 0 7 -44 -98 161 128 -9 }}
{{Mapping|legend=0| 1 1 6 11 -10 -7 5 | 0 7 -44 -98 161 128 -9 }}
: mapping generators: ~2, ~1323/1250
: mapping generators: ~2, ~1323/1250


Line 627: Line 627:
Comma list: 19712/19683, 41503/41472, 1953125/1948617
Comma list: 19712/19683, 41503/41472, 1953125/1948617


Mapping: {{mapping| 1 -3 -15 67 -102 | 0 9 34 -126 207 }}
{{Mapping|legend=0| 1 -3 -15 67 -102 | 0 9 34 -126 207 }}


Optimal tunings:  
Optimal tunings:  
Line 642: Line 642:
Comma list: 2080/2079, 4096/4095, 19712/19683, 78125/78078
Comma list: 2080/2079, 4096/4095, 19712/19683, 78125/78078


Mapping: {{mapping| 1 -3 -15 67 -102 -34 | 0 9 34 -126 207 74 }}
{{Mapping|legend=0| 1 -3 -15 67 -102 -34 | 0 9 34 -126 207 74 }}


Optimal tunings:  
Optimal tunings:  
Line 657: Line 657:
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 59375/59319
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 59375/59319


Mapping: {{mapping| 1 -3 -15 67 -102 -34 -36 | 0 9 34 -126 207 74 79 }}
{{Mapping|legend=0| 1 -3 -15 67 -102 -34 -36 | 0 9 34 -126 207 74 79 }}


Optimal tunings:  
Optimal tunings:  
Line 667: Line 667:
Badness (Sintel): 1.22
Badness (Sintel): 1.22


[[Category:Septischismic clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Garischismic clan| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Septischismic_clan"