Aberschismic temperaments: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
This is a collection of [[rank-2 temperament|rank-2]] | This is a collection of [[rank-2 temperament|rank-2]] '''aberschismic temperaments''', which [[tempering out|temper out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]]. | ||
Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1. | Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1. | ||
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* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]] | * [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]] | ||
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]] | * ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]] | ||
* | * [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]] | ||
* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]] | |||
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]] | * [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]] | ||
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]] | * ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]] | ||
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]] | * ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]] | ||
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* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]] | * [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]] | ||
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]] | * ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]] | ||
* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]] | |||
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]] | |||
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]] | * ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]] | ||
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]] | * ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]] | ||
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== Ketchup == | == Ketchup == | ||
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its ploidacot is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament. | Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 289/288, 325/324, 352/351, 385/384, | Comma list: 289/288, 325/324, 352/351, 385/384, 442/441 | ||
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }} | Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }} | ||
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Badness (Sintel): 0.845 | Badness (Sintel): 0.845 | ||
=== | === 2.3.5.7.11.13.17.23 subgroup === | ||
Subgroup: 2.3.5.7.11.13.17. | Subgroup: 2.3.5.7.11.13.17.23 | ||
Comma list: | Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390 | ||
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 | Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~17/12 = 600. | * WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}} | ||
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25. | * CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}} | ||
{{Optimal ET sequence|legend=0| 46, 94, | {{Optimal ET sequence|legend=0| 46, 94, 140 }} | ||
Badness (Sintel): 0.772 | |||
Badness (Sintel): | |||
== Undecental == | == Undecental == | ||
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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].'' | ||
Leapday tempers out [[686/675]], the senga, in addition to the | Leapday tempers out [[686/675]], the senga, in addition to the aberschisma, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this. | ||
Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however. | Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however. | ||
Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell ([[6144/6125]]) rather than the | Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell comma ([[6144/6125]]) rather than the aberschisma tempered out. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Badness (Sintel): 0.910 | Badness (Sintel): 0.910 | ||
=== | === 2.3.5.7.11.13.17.23 subgroup === | ||
Subgroup: 2.3.5.7.11.13.17. | Subgroup: 2.3.5.7.11.13.17.23 | ||
Comma list: 91/90, 121/120 | Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168 | ||
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704. | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}} | ||
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, | {{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }} | ||
Badness (Sintel): | Badness (Sintel): 0.872 | ||
== Mystery == | == Mystery == | ||
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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].'' | ||
Jorgensen tempers out the [[linus comma]] in addition to the | Jorgensen tempers out the [[linus comma]] in addition to the aberschisma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot. | ||
It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>. | It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>. | ||
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[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category: | [[Category:Aberschismic temperaments| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||