Undim family: Difference between revisions

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== Undim ==
== Undim ==
Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, ~[[256/243]]) give the interval class of 5. Undim equates the [[Pythagorean comma]] with a stack of four [[schisma]]s. This makes it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap| ''n'' {{=}} 4 }}.
Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, [[~]][[256/243]]) give the interval class of 5. Undim has a [[ploidacot]] of tetraploid monocot, and equates the [[Pythagorean comma]] with a stack of four [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap| ''n'' {{=}} 4 }}.  
 
Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.


The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  
The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6754{{c}}
* [[WE]]: ~1215/1024 = 299.9421{{c}}, ~3/2 = 702.4699{{c}}
: [[error map]]: {{val| 0.0000 +0.7204 +0.3092 }}
: [[error map]]: {{val| -0.231 +0.283 +0.122 }}
* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6054{{c}}
* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6252{{c}}
: error map: {{val| 0.0000 +0.6504 +0.6591 }}
: error map: {{val| 0.000 +0.670 +0.560 }}


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


[[Badness]] (Smith): 0.241703
[[Badness]] (Sintel): 5.67


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.  
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.  
Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious place for prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]].


== Septimal undim ==
== 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
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7948{{c}}
* [[WE]]: ~25/21 = 299.9451{{c}}, ~3/2 = 702.6076{{c}}
: [[error map]]: {{val| 0.0000 +0.8398 -0.2879 +0.4308 }}
: [[error map]]: {{val| -0.220 +0.433 -0.505 +0.458 }}
* [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7362{{c}}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7528{{c}}
: error map: {{val| 0.0000 +0.7812 +0.0051 +1.0754 }}
: error map: {{val| 0.000 +0.798 -0.078 +0.894 }}


{{Optimal ET sequence|legend=1| 140, 152, 292 }}
{{Optimal ET sequence|legend=1| 12, …, 128, 140, 152, 292 }}


[[Badness]] (Smith): 0.062754
[[Badness]] (Sintel): 1.59


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7433{{c}}
* WE: ~25/21 = 299.9389{{c}}, ~3/2 = 702.5455{{c}}
* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.6886{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7038{{c}}


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


Badness (Smith): 0.034837
Badness (Sintel): 1.15


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7792{{c}}
* WE: ~25/21 = 299.9476{{c}}, ~3/2 = 702.6135{{c}}
* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7363{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7480{{c}}


{{Optimal ET sequence|legend=0| 140, 152f, 292 }}
{{Optimal ET sequence|legend=0| 12f, 128ef, 140, 152f, 292 }}


Badness (Smith): 0.028172
Badness (Sintel): 1.16


== Unlit ==
== 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. 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 2199023255552/2179240250625
[[Comma list]]: 4375/4374, {{monzo| 41 -20 -4 }}


{{Mapping|legend=1| 4 0 41 -160 | 0 1 -5 27 }}
{{Mapping|legend=1| 4 0 41 -160 | 0 1 -5 27 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5556{{c}}
* [[WE]]: ~1215/1024 = 299.9374{{c}}, ~3/2 = 702.4299{{c}}
: [[error map]]: {{val| 0.0000 +0.6006 +0.9081 +0.1761 }}
: [[error map]]: {{val| -0.250 +0.225 +0.223 +0.034 }}
* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5764{{c}}
* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5716{{c}}
: error map: {{val| 0.0000 +0.6214 +0.8043 +0.7369 }}
: error map: {{val| 0.000 +0.617 +0.828 +0.607 }}


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


[[Badness]] (Smith): 0.268206
[[Badness]] (Sintel): 6.79


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5582{{c}}
* WE: ~1215/1024 = 299.9426{{c}}, ~3/2 = 702.4482{{c}}
* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5826{{c}}
* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5772{{c}}


{{Optimal ET sequence|legend=0| 152, 468, 620 }}
{{Optimal ET sequence|legend=0| 152, 468, 620 }}


Badness (Smith): 0.070215
Badness (Sintel): 2.32


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5562{{c}}
* WE: ~1215/1024 = 299.9452{{c}}, ~3/2 = 702.4459{{c}}
* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5741{{c}}
* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5703{{c}}


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


Badness (Smith): 0.058390
Badness (Sintel): 2.41


== Twilight ==
== 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. 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
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 8 0 82 -79 | 0 1 -5 8 }}
{{Mapping|legend=1| 8 0 82 -79 | 0 1 -5 8 }}
: mapping generators: ~7168/6561, ~3
: mapping generators: ~7168/6561, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.4765{{c}}
* [[WE]]: ~7168/6561 = 149.9598{{c}}, ~3/2 = 702.3209{{c}}
: [[error map]]: {{val| 0.0000 +0.5215 +1.3036 +0.9865 }}
: [[error map]]: {{val| -0.321 +0.045 +0.395 +0.343 }}
* [[POTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}
* [[CWE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5023{{c}}
: error map: {{val| 0.0000 +0.5540 +1.1415 +1.2457 }}
: error map: {{val| 0.000 +0.547 +1.175 +1.192 }}


{{Optimal ET sequence|legend=1| 152, 328, 480, 1592bccddd }}
{{Optimal ET sequence|legend=1| 152, 328, 480, 1592bccddd }}


Badness (Smith): 0.213094
Badness (Sintel): 5.39


=== 11-limit ===
=== 11-limit ===
Line 152: Line 155:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4692{{c}}
* WE: ~12/11 = 149.9593{{c}}, ~3/2 = 702.3182{{c}}
* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5022{{c}}


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


Badness (Smith): 0.048007
Badness (Sintel): 1.59


=== 13-limit ===
=== 13-limit ===
Line 167: Line 170:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4168{{c}}
* WE: ~12/11 = 149.9582{{c}}, ~3/2 = 702.2815{{c}}
* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4773{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4670{{c}}


{{Optimal ET sequence|legend=0| 152f, 328, 480f, 808cdeff }}
{{Optimal ET sequence|legend=0| 152f, 328, 480f, 808cdeff }}


Badness (Smith): 0.041365
Badness (Sintel): 1.71


== References ==
== References ==

Latest revision as of 13:24, 12 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The undim family of temperaments tempers out the undim comma, [41 -20 -4⟩.

Undim

Undim features a quarter-octave period, which acts as the ptolemaic augmented second (1215/1024). That and five perfect fourths (i.e. a minor second, ~256/243) give the interval class of 5. Undim has a ploidacot of tetraploid monocot, and equates the Pythagorean comma with a stack of four schismas, making it a member of the schismic–Pythagorean equivalence continuum with n = 4.

The name undim was given by Petr Pařízek in 2011 for it is some sort of opposite to diminished[1].

Subgroup: 2.3.5

Comma list: [41 -20 -4⟩

Mapping: [⟨4 0 41], ⟨0 1 -5]]

mapping generators: ~1215/1024, ~3

Optimal tunings:

  • WE: ~1215/1024 = 299.9421 ¢, ~3/2 = 702.4699 ¢
error map: ⟨-0.231 +0.283 +0.122]
  • CWE: ~1215/1024 = 300.0000 ¢, ~3/2 = 702.6252 ¢
error map: ⟨0.000 +0.670 +0.560]

Optimal ET sequence: 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc

Badness (Sintel): 5.67

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal undim (140 & 152) tempers out 5120/5103 (hemifamity). Unlit (152 & 316) does 4375/4374 (ragisma) instead. Twilight (152 & 176) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.

Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious place for prime 19 by observing that sharpening 1215/1024 by 1216/1215 results in 19/16, thus mapping 19/16 to 1\4.

Septimal undim

Septimal undim tempers out the dimcomp comma, mapping ~25/21 to the 1/4-octave period. It can be described as 12 & 140, and is the unique temperament that tempers together the syntonic and septimal commas and equates it with a stack of three marvel commas. 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, 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

Comma list: 5120/5103, 390625/388962

Mapping: [⟨4 0 41 81], ⟨0 1 -5 -11]]

Optimal tunings:

  • WE: ~25/21 = 299.9451 ¢, ~3/2 = 702.6076 ¢
error map: ⟨-0.220 +0.433 -0.505 +0.458]
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 702.7528 ¢
error map: ⟨0.000 +0.798 -0.078 +0.894]

Optimal ET sequence: 12, …, 128, 140, 152, 292

Badness (Sintel): 1.59

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨4 0 41 81 128], ⟨0 1 -5 -11 -18]]

Optimal tunings:

  • WE: ~25/21 = 299.9389 ¢, ~3/2 = 702.5455 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 702.7038 ¢

Optimal ET sequence: 12, 128e, 140, 152, 292, 444d, 596d

Badness (Sintel): 1.15

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨4 0 41 81 128 148], ⟨0 1 -5 -11 -18 -21]]

Optimal tunings:

  • WE: ~25/21 = 299.9476 ¢, ~3/2 = 702.6135 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 702.7480 ¢

Optimal ET sequence: 12f, 128ef, 140, 152f, 292

Badness (Sintel): 1.16

Unlit

Unlit tempers out 4375/4374, the ragisma, and may be described as 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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [41 -20 -4⟩

Mapping: [⟨4 0 41 -160], ⟨0 1 -5 27]]

Optimal tunings:

  • WE: ~1215/1024 = 299.9374 ¢, ~3/2 = 702.4299 ¢
error map: ⟨-0.250 +0.225 +0.223 +0.034]
  • CWE: ~1215/1024 = 300.0000 ¢, ~3/2 = 702.5716 ¢
error map: ⟨0.000 +0.617 +0.828 +0.607]

Optimal ET sequence: 152, 316, 468, 620, 1088bcd, 1708bccdd

Badness (Sintel): 6.79

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨4 0 41 -160 -113], ⟨0 1 -5 27 20]]

Optimal tunings:

  • WE: ~1215/1024 = 299.9426 ¢, ~3/2 = 702.4482 ¢
  • CWE: ~1215/1024 = 300.0000 ¢, ~3/2 = 702.5772 ¢

Optimal ET sequence: 152, 468, 620

Badness (Sintel): 2.32

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨4 0 41 -160 -113 -334], ⟨0 1 -5 27 20 55]]

Optimal tunings:

  • WE: ~1215/1024 = 299.9452 ¢, ~3/2 = 702.4459 ¢
  • CWE: ~1215/1024 = 300.0000 ¢, ~3/2 = 702.5703 ¢

Optimal ET sequence: 152f, 316, 468, 620f, 1088bcdf

Badness (Sintel): 2.41

Twilight

Twilight tempers out 6144/6125, the porwell comma, and may be described as 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 and unlit.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 31470387200/31381059609

Mapping: [⟨8 0 82 -79], ⟨0 1 -5 8]]

mapping generators: ~7168/6561, ~3

Optimal tunings:

  • WE: ~7168/6561 = 149.9598 ¢, ~3/2 = 702.3209 ¢
error map: ⟨-0.321 +0.045 +0.395 +0.343]
  • CWE: ~7168/6561 = 150.0000 ¢, ~3/2 = 702.5023 ¢
error map: ⟨0.000 +0.547 +1.175 +1.192]

Optimal ET sequence: 152, 328, 480, 1592bccddd

Badness (Sintel): 5.39

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨8 0 82 -79 15], ⟨0 1 -5 8 1]]

Optimal tunings:

  • WE: ~12/11 = 149.9593 ¢, ~3/2 = 702.3182 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 702.5022 ¢

Optimal ET sequence: 152, 328, 480, 1112bccddee, 1592bccdddeee

Badness (Sintel): 1.59

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨8 0 82 -79 15 -186], ⟨0 1 -5 8 1 17]]

Optimal tunings:

  • WE: ~12/11 = 149.9582 ¢, ~3/2 = 702.2815 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 702.4670 ¢

Optimal ET sequence: 152f, 328, 480f, 808cdeff

Badness (Sintel): 1.71

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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