Aberschismic temperaments: Difference between revisions

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== Undecental ==

Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple diminished seventh (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas.

Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple diminished seventh (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting choice is the argent fifth, 2<sup>(2 - sqrt (2))</sup>.

[[Subgroup]]: 2.3.5.7

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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''

Leapday tempers out {{monzo|31 -21 1}} ~~(trisayo)~~ in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 & 46 temperament, which tempers out the hemifamity and [[686/675]] (senga). The alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46 & 121) tempers out the same 5-limit comma as the leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity tempered out.

Leapday tempers out the leapday comma, {{monzo| 31 -21 1 }}, in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 & 46 temperament, which tempers out the hemifamity and [[686/675]] (senga). The alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46 & 121) tempers out the same 5-limit comma as the leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity tempered out.

[[Subgroup]]: 2.3.5.7

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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mystery]].''

Mystery has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. [[145edo]] or [[232edo]] are good candidates for tunings.

[[Subgroup]]: 2.3.5.7

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Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, ~~847~~/~~845~~, 1331/1323

Comma list: 325/324, 352/351, 385/384, 1331/1323

Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]

Optimal tuning (POTE): ~99/70 = 1\2, ~64/63 = 25.697

Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 25.697

Optimal GPV sequence: {{Val list| 46, 94, 140 }}

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Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]

Optimal tuning (POTE): ~17/12 = 1\2, ~64/63 = 25.701

Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.701

Optimal GPV sequence: {{Val list| 46, 94, 140 }}

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Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]

Optimal tuning (POTE): ~17/12 = 1\2, ~64/63= 25.660

Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.660

Optimal GPV sequence: {{Val list| 46, 94, 140h, 234eh }}

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Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]

Optimal tuning (POTE): ~17/12 = 1\2, ~64/63 = 25.661

Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.661

Optimal GPV sequence: {{Val list| 46, 94, 140h, 234ehi }}

Badness: 0.014033

== Artoneutral ==

Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the 87 & 94 temperament. [[181edo]] is a recommendable tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 3828125/3779136

[[Mapping]]: [{{val| 1 8 18 -20 }}, {{val| 0 -9 -22 32 }}]

: mapping generators: ~2, ~105/64

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~105/64 = 855.2452

[[Badness]]: 0.157

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 4000/3993

Mapping: [{{val| 1 8 18 -20 17 }}, {{val| 0 -9 -22 32 -19 }}]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2397

Optimal GPV sequence: {{Val list| 87, 181 }}

Badness: 0.0459

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1575/1573

Mapping: [{{val| 1 8 18 -20 17 -2 }}, {{val| 0 -9 -22 32 -19 8 }}]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2369

Optimal GPV sequence: {{Val list| 87, 181 }}

Badness: 0.0263

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 385/384, 595/594

Mapping: [{{val| 1 8 18 -20 17 -2 44 }}, {{val| 0 -9 -22 32 -19 8 -56 }}]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2495

Optimal GPV sequence: {{Val list| 87, 94, 181 }}

Badness: 0.0227

[[Category:Temperament collections]]

[[Category:Hemifamity temperaments| ]]

[[Category:Rank 2]]

@@ Line 40: / Line 40: @@
 == Undecental ==
-Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple diminished seventh (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas.
+Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple diminished seventh (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting choice is the argent fifth, 2<sup>(2 - sqrt (2))</sup>.
 [[Subgroup]]: 2.3.5.7
@@ Line 60: / Line 60: @@
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''
-Leapday tempers out {{monzo|31 -21 1}} (trisayo) in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 &amp; 46 temperament, which tempers out the hemifamity and [[686/675]] (senga). The alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46 &amp; 121) tempers out the same 5-limit comma as the leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity tempered out.
+Leapday tempers out the leapday comma, {{monzo| 31 -21 1 }}, in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 &amp; 46 temperament, which tempers out the hemifamity and [[686/675]] (senga). The alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46 &amp; 121) tempers out the same 5-limit comma as the leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity tempered out.
 [[Subgroup]]: 2.3.5.7
@@ Line 170: / Line 170: @@
 {{Main| Mystery }}
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Mystery]].''
+Mystery has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. [[145edo]] or [[232edo]] are good candidates for tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 391: / Line 393: @@
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 352/351, 847/845, 1331/1323
+Comma list: 325/324, 352/351, 385/384, 1331/1323
 Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]
-Optimal tuning (POTE): ~99/70 = 1\2, ~64/63 = 25.697
+Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 25.697
 Optimal GPV sequence: {{Val list| 46, 94, 140 }}
@@ Line 408: / Line 410: @@
 Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]
-Optimal tuning (POTE): ~17/12 = 1\2, ~64/63 = 25.701
+Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.701
 Optimal GPV sequence: {{Val list| 46, 94, 140 }}
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 Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]
-Optimal tuning (POTE): ~17/12 = 1\2, ~64/63= 25.660
+Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.660
 Optimal GPV sequence: {{Val list| 46, 94, 140h, 234eh }}
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 Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]
-Optimal tuning (POTE): ~17/12 = 1\2, ~64/63 = 25.661
+Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.661
 Optimal GPV sequence: {{Val list| 46, 94, 140h, 234ehi }}
 Badness: 0.014033
+== Artoneutral ==
+Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the 87 & 94 temperament. [[181edo]] is a recommendable tuning.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 5120/5103, 3828125/3779136
+[[Mapping]]: [{{val| 1 8 18 -20 }}, {{val| 0 -9 -22 32 }}]
+: mapping generators: ~2, ~105/64
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~105/64 = 855.2452
+{{Val list|legend=1| 87, 94, 181 }}
+[[Badness]]: 0.157
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 385/384, 2200/2187, 4000/3993
+Mapping: [{{val| 1 8 18 -20 17 }}, {{val| 0 -9 -22 32 -19 }}]
+Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2397
+Optimal GPV sequence: {{Val list| 87, 181 }}
+Badness: 0.0459
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 325/324, 352/351, 385/384, 1575/1573
+Mapping: [{{val| 1 8 18 -20 17 -2 }}, {{val| 0 -9 -22 32 -19 8 }}]
+Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2369
+Optimal GPV sequence: {{Val list| 87, 181 }}
+Badness: 0.0263
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
+Mapping: [{{val| 1 8 18 -20 17 -2 44 }}, {{val| 0 -9 -22 32 -19 8 -56 }}]
+Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2495
+Optimal GPV sequence: {{Val list| 87, 94, 181 }}
+Badness: 0.0227
 [[Category:Temperament collections]]
 [[Category:Hemifamity temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]