Aberschismic temperaments: Difference between revisions

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This is a collection of [[rank-2 temperament|rank-2]] ~~[[regular temperament|~~temperaments]] [[tempering out]] the [[~~hemifamity comma~~]] ({{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]].

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.

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* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]

* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]

* [[Leapday]] (+686/675) → [[Sengic temperaments #Leapday|Sengic temperaments]]

* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]

* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]

* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]

* ''[[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]]

* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]

* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]

* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]

* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]

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* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity 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]]

* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]

Considered below are septiquarter, kwai, ketchup, undecental~~, leapday~~, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].

Considered below are septiquarter, kwai, ketchup, undecental, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].

== Septiquarter ==

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[[Badness]] (Sintel): 2.39

~~== Leapday ==~~

~~{{Main| Leapday }}~~

~~: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''~~

Leapday tempers out [[686/675]], the senga, in addition to the hemifamity comma, 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 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 hemifamity comma tempered out.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 686/675, 5120/5103~~

~~{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}~~

~~: mapping generators: ~2, ~3~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}

~~: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}

~~: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}~~

~~{{Optimal ET sequence|legend=1| 17c, 29, 46 }}~~

~~[[Badness]] (Sintel): 2.43~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 121/120, 441/440, 686/675~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}

~~{{Optimal ET sequence|legend=0| 17c, 29, 46 }}~~

~~Badness (Sintel): 1.28~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 91/90, 121/120, 169/168, 352/351~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}

~~{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}~~

~~Badness (Sintel): 1.02~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 91/90, 121/120, 136/135, 154/153, 169/168~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}

~~{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}~~

~~Badness (Sintel): 0.910~~

~~=== 2.3.5.7.11.13.17.23 subgroup ===~~

~~Subgroup: 2.3.5.7.11.13.17.23~~

~~Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}

~~{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}~~

~~Badness (Sintel): 0.872~~

== Mystery ==

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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''

Jorgensen tempers out the [[linus comma]] in addition to the ~~hemifamity comma~~, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.

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>.

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== References ==

[[Category:Aberschismic temperaments| ]]

[[Category:Temperament collections]]

[[Category:~~Hemifamity temperaments| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

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 {{Technical data page}}
-This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[hemifamity comma]] ({{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]].
+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.
@@ Line 9: / Line 9: @@
 * [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
 * [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
+* [[Leapday]] (+686/675) → [[Sengic temperaments #Leapday|Sengic temperaments]]
 * [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
 * [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
 * [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
 * ''[[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]]
-* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
-* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
 * ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
 * ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
@@ Line 22: / Line 22: @@
 * [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity 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]]
 * ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]
-Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
+Considered below are septiquarter, kwai, ketchup, undecental, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
 == Septiquarter ==
@@ Line 304: / Line 306: @@
 [[Badness]] (Sintel): 2.39
-== Leapday ==
-{{Main| Leapday }}
-: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
-Leapday tempers out [[686/675]], the senga, in addition to the hemifamity comma, 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 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 hemifamity comma tempered out.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 686/675, 5120/5103
-{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
-: mapping generators: ~2, ~3
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
-: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
-: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
-{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
-[[Badness]] (Sintel): 2.43
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 441/440, 686/675
-Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
-Optimal tunings:
-* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
-{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
-Badness (Sintel): 1.28
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 121/120, 169/168, 352/351
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
-Optimal tunings:
-* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}
-{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}
-Badness (Sintel): 1.02
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
-Optimal tunings:
-* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
-{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
-Badness (Sintel): 0.910
-=== 2.3.5.7.11.13.17.23 subgroup ===
-Subgroup: 2.3.5.7.11.13.17.23
-Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
-Optimal tunings:
-* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}
-{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
-Badness (Sintel): 0.872
 == Mystery ==
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 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
-Jorgensen tempers out the [[linus comma]] in addition to the hemifamity comma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.
+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>.
@@ Line 797: / Line 712: @@
 == References ==
+[[Category:Aberschismic temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Hemifamity temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]