Undim family: Difference between revisions

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* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6252{{c}}

: error map: {{val| 0.000 +0.670 +0.560 }}

~~<!-- * [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6754{{c}}~~

~~: [[error map]]: {{val| 0.0000 +0.7204 +0.3092 }}~~

* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6054{{c}}

~~: error map: {{val| 0.0000 +0.6504 +0.6591 }} -->~~

{{Optimal ET sequence|legend=1| 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}

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== Septimal undim ==

Septimal undim tempers out the [[dimcomp comma]], mapping [[~]][[25/21]] to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that ~~equates a~~ syntonic~septimal ~~comma~~ with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In ~~some~~ 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.

Septimal undim tempers out the [[dimcomp comma]], mapping [[~]][[25/21]] to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that tempers together the syntonic and septimal commas and equates it with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7528{{c}}

: error map: {{val| 0.000 +0.798 -0.078 +0.894 }}

~~<!-- * [[CTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7948{{c}}~~

~~: [[error map]]: {{val| 0.0000 +0.8398 -0.2879 +0.4308 }}~~

* [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7362{{c}}

~~: error map: {{val| 0.0000 +0.7812 +0.0051 +1.0754 }} -->~~

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Comma list: 1375/1372, 5120/5103, 5632/5625

~~Mapping:~~ {{~~mapping~~| 4 0 41 81 128 | 0 1 -5 -11 -18 }}

Optimal tunings:

* WE: ~25/21 = 299.9389{{c}}, ~3/2 = 702.5455{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7038{{c}}

~~<!-- * CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7433{{c}}~~

* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.6886{{c}} -->

{{Optimal ET sequence|legend=0| 12, 128e, 140, 152, 292, 444d, 596d }}

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Comma list: 352/351, 625/624, 847/845, 1375/1372

~~Mapping:~~ {{~~mapping~~| 4 0 41 81 128 148 | 0 1 -5 -11 -18 -21 }}

Optimal tunings:

* WE: ~25/21 = 299.9476{{c}}, ~3/2 = 702.6135{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7480{{c}}

~~<!-- * CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7792{{c}}~~

* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7363{{c}} -->

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== Unlit ==

Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity.

Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity. It was named by [[Flora Canou]] in 2021 for it is some sort of opposite of [[#Septimal undim|septimal undim]].

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5716{{c}}

: error map: {{val| 0.000 +0.617 +0.828 +0.607 }}

~~<!-- * [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5556{{c}}~~

~~: [[error map]]: {{val| 0.0000 +0.6006 +0.9081 +0.1761 }}~~

* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5764{{c}}

~~: error map: {{val| 0.0000 +0.6214 +0.8043 +0.7369 }} -->~~

{{Optimal ET sequence|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }}

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Comma list: 3025/3024, 4375/4374, 5767168/5740875

~~Mapping:~~ {{~~mapping~~| 4 0 41 -160 -113 | 0 1 -5 27 20 }}

Optimal tunings:

* WE: ~1215/1024 = 299.9426{{c}}, ~3/2 = 702.4482{{c}}

* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5772{{c}}

~~<!-- * CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5582{{c}}~~

* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5826{{c}} -->

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Comma list: 1716/1715, 2080/2079, 3025/3024, 1835008/1828125

~~Mapping:~~ {{~~mapping~~| 4 0 41 -160 -113 -334 | 0 1 -5 27 20 55 }}

Optimal tunings:

* WE: ~1215/1024 = 299.9452{{c}}, ~3/2 = 702.4459{{c}}

* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5703{{c}}

~~<!-- * CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5562{{c}}~~

* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5741{{c}} -->

{{Optimal ET sequence|legend=0| 152f, 316, 468, 620f, 1088bcdf }}

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== Twilight ==

Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot.

Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot. It was named by [[Flora Canou]] in 2021 for it splits the period in halves to reconcile the different mappings of [[#Septimal undim|septimal undim]] and [[#Unlit|unlit]].

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5023{{c}}

: error map: {{val| 0.000 +0.547 +1.175 +1.192 }}

~~<!-- * [[CTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.4765{{c}}~~

~~: [[error map]]: {{val| 0.0000 +0.5215 +1.3036 +0.9865 }}~~

* [[POTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}

~~: error map: {{val| 0.0000 +0.5540 +1.1415 +1.2457 }} -->~~

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Comma list: 6144/6125, 9801/9800, 19712/19683

~~Mapping:~~ {{~~mapping~~| 8 0 82 -79 15 | 0 1 -5 8 1 }}

Optimal tunings:

* WE: ~12/11 = 149.9593{{c}}, ~3/2 = 702.3182{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5022{{c}}

~~<!-- * CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4692{{c}}~~

* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5090{{c}} -->

{{Optimal ET sequence|legend=0| 152, 328, 480, 1112bccddee, 1592bccdddeee }}

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Comma list: 1716/1715, 2080/2079, 3584/3575, 14641/14625

~~Mapping:~~ {{~~mapping~~| 8 0 82 -79 15 -186 | 0 1 -5 8 1 17 }}

Optimal tunings:

* WE: ~12/11 = 149.9582{{c}}, ~3/2 = 702.2815{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4670{{c}}

~~<!-- * CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4168{{c}}~~

* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4773{{c}} -->

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== References ==

[[Category:Undim family| ]]

[[Category:Temperament families]]

[[Category:~~Undim family| ]] <!~~-~~- main article -->~~

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

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

@@ Line 19: / Line 19: @@
 * [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6252{{c}}
 : error map: {{val| 0.000 +0.670 +0.560 }}
-<!-- * [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6754{{c}}
-: [[error map]]: {{val| 0.0000 +0.7204 +0.3092 }}
-* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6054{{c}}
-: error map: {{val| 0.0000 +0.6504 +0.6591 }} -->
 {{Optimal ET sequence|legend=1| 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}
@@ Line 34: / Line 30: @@
 == Septimal undim ==
-Septimal undim tempers out the [[dimcomp comma]], mapping [[~]][[25/21]] to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.
+Septimal undim tempers out the [[dimcomp comma]], mapping [[~]][[25/21]] to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that tempers together the syntonic and septimal commas and equates it with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 47: / Line 43: @@
 * [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7528{{c}}
 : error map: {{val| 0.000 +0.798 -0.078 +0.894 }}
-<!-- * [[CTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7948{{c}}
-: [[error map]]: {{val| 0.0000 +0.8398 -0.2879 +0.4308 }}
-* [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7362{{c}}
-: error map: {{val| 0.0000 +0.7812 +0.0051 +1.0754 }} -->
 {{Optimal ET sequence|legend=1| 12, …, 128, 140, 152, 292 }}
@@ Line 61: / Line 53: @@
 Comma list: 1375/1372, 5120/5103, 5632/5625
-Mapping: {{mapping| 4 0 41 81 128 | 0 1 -5 -11 -18 }}
+{{Mapping|legend=0| 4 0 41 81 128 | 0 1 -5 -11 -18 }}
 Optimal tunings:
 * WE: ~25/21 = 299.9389{{c}}, ~3/2 = 702.5455{{c}}
 * CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7038{{c}}
-<!-- * CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7433{{c}}
-* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.6886{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 128e, 140, 152, 292, 444d, 596d }}
@@ Line 78: / Line 68: @@
 Comma list: 352/351, 625/624, 847/845, 1375/1372
-Mapping: {{mapping| 4 0 41 81 128 148 | 0 1 -5 -11 -18 -21 }}
+{{Mapping|legend=0| 4 0 41 81 128 148 | 0 1 -5 -11 -18 -21 }}
 Optimal tunings:
 * WE: ~25/21 = 299.9476{{c}}, ~3/2 = 702.6135{{c}}
 * CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7480{{c}}
-<!-- * CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7792{{c}}
-* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7363{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 128ef, 140, 152f, 292 }}
@@ Line 91: / Line 79: @@
 == Unlit ==
-Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity.
+Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity. It was named by [[Flora Canou]] in 2021 for it is some sort of opposite of [[#Septimal undim|septimal undim]].
 [[Subgroup]]: 2.3.5.7
@@ Line 104: / Line 92: @@
 * [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5716{{c}}
 : error map: {{val| 0.000 +0.617 +0.828 +0.607 }}
-<!-- * [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5556{{c}}
-: [[error map]]: {{val| 0.0000 +0.6006 +0.9081 +0.1761 }}
-* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5764{{c}}
-: error map: {{val| 0.0000 +0.6214 +0.8043 +0.7369 }} -->
 {{Optimal ET sequence|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }}
@@ Line 118: / Line 102: @@
 Comma list: 3025/3024, 4375/4374, 5767168/5740875
-Mapping: {{mapping| 4 0 41 -160 -113 | 0 1 -5 27 20 }}
+{{Mapping|legend=0| 4 0 41 -160 -113 | 0 1 -5 27 20 }}
 Optimal tunings:
 * WE: ~1215/1024 = 299.9426{{c}}, ~3/2 = 702.4482{{c}}
 * CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5772{{c}}
-<!-- * CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5582{{c}}
-* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5826{{c}} -->
 {{Optimal ET sequence|legend=0| 152, 468, 620 }}
@@ Line 135: / Line 117: @@
 Comma list: 1716/1715, 2080/2079, 3025/3024, 1835008/1828125
-Mapping: {{mapping| 4 0 41 -160 -113 -334 | 0 1 -5 27 20 55 }}
+{{Mapping|legend=0| 4 0 41 -160 -113 -334 | 0 1 -5 27 20 55 }}
 Optimal tunings:
 * WE: ~1215/1024 = 299.9452{{c}}, ~3/2 = 702.4459{{c}}
 * CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5703{{c}}
-<!-- * CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5562{{c}}
-* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5741{{c}} -->
 {{Optimal ET sequence|legend=0| 152f, 316, 468, 620f, 1088bcdf }}
@@ Line 148: / Line 128: @@
 == Twilight ==
-Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot.
+Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot. It was named by [[Flora Canou]] in 2021 for it splits the period in halves to reconcile the different mappings of [[#Septimal undim|septimal undim]] and [[#Unlit|unlit]].
 [[Subgroup]]: 2.3.5.7
@@ Line 162: / Line 142: @@
 * [[CWE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5023{{c}}
 : error map: {{val| 0.000 +0.547 +1.175 +1.192 }}
-<!-- * [[CTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.4765{{c}}
-: [[error map]]: {{val| 0.0000 +0.5215 +1.3036 +0.9865 }}
-* [[POTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}
-: error map: {{val| 0.0000 +0.5540 +1.1415 +1.2457 }} -->
 {{Optimal ET sequence|legend=1| 152, 328, 480, 1592bccddd }}
@@ Line 176: / Line 152: @@
 Comma list: 6144/6125, 9801/9800, 19712/19683
-Mapping: {{mapping| 8 0 82 -79 15 | 0 1 -5 8 1 }}
+{{Mapping|legend=0| 8 0 82 -79 15 | 0 1 -5 8 1 }}
 Optimal tunings:
 * WE: ~12/11 = 149.9593{{c}}, ~3/2 = 702.3182{{c}}
 * CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5022{{c}}
-<!-- * CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4692{{c}}
-* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5090{{c}} -->
 {{Optimal ET sequence|legend=0| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
@@ Line 193: / Line 167: @@
 Comma list: 1716/1715, 2080/2079, 3584/3575, 14641/14625
-Mapping: {{mapping| 8 0 82 -79 15 -186 | 0 1 -5 8 1 17 }}
+{{Mapping|legend=0| 8 0 82 -79 15 -186 | 0 1 -5 8 1 17 }}
 Optimal tunings:
 * WE: ~12/11 = 149.9582{{c}}, ~3/2 = 702.2815{{c}}
 * CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4670{{c}}
-<!-- * CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4168{{c}}
-* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4773{{c}} -->
 {{Optimal ET sequence|legend=0| 152f, 328, 480f, 808cdeff }}
@@ Line 207: / Line 179: @@
 == References ==
+[[Category:Undim family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Undim family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]

Undim family: Difference between revisions

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