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No-threes subgroup temperaments: Difference between revisions

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Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Trader has a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Insect has a ~55/32 generator, three of which give the ~5/1.
Line 15: Line 15:
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]


=== Frostburn ===
=== Augment ===
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].  
{{See also| Chromatic pairs #Augment }}
 
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.  


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 78125/76832
[[Comma list]]: 128/125
 
{{Mapping|legend=3| 3 7 0 | 0 0 1 }}


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
{{Mapping|legend=5| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~2, ~28/25
: mapping generators: ~5/4, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}
* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}
: [[error map]]: {{val| +0.346 -2.630 +2.189 }}
: [[error map]]: {{val| -2.962 +6.776 -0.040 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 204.2027{{c}}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 974.3418{{c}}
: error map: {{val| 0.000 -3.125 +1.755 }}
: error map: {{val| 0.000 +13.686 +5.516 }}


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}


[[Badness]] (Sintel): 0.886
[[Badness]] (Sintel): 0.296


==== 2.5.7.11 subgroup ====
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
Comma list: 56/55, 128/125
 
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}


Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
{{Mapping|legend=4| 3 0 7 0 2 | 0 0 0 1 1 }}
: mapping generators: ~2, ~28/25


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}


{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}


Badness (Sintel): 0.463
Badness (Sintel): 0.196


=== Mabilic ===
=== Mabilic ===
Line 59: Line 62:
{{See also| Chromatic pairs #Mabilic }}
{{See also| Chromatic pairs #Mabilic }}


Mabilic is the no-3 [[restriction]] of [[mavila family #Armodue|armodue]], [[mabila family #Semabila|semabila]], and [[magic family #Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
Mabilic is the no-3 [[restriction]] of [[mavila family #Armodue|armodue]] (7 & 16), [[magic family #Trismegistus|trismegistus]] (16 & 25), and [[mabila family #Semabila|semabila]] (25 & 34). It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma. [[Weak extension]]s of mabilic include [[archytas clan #Sixix|sixix]] (7 & 25), [[vishnu family #Vishnean|vishnean]] (16 & 34), [[marvel temperaments #Septimin|septimin]] (9 & 41) and [[tetracot family #Monkey|monkey]] (7 & 34).  


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 65: Line 68:
[[Comma list]]: 1071875/1048576
[[Comma list]]: 1071875/1048576


{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
{{Mapping|legend=3| 1 1 5 | 0 3 -5 }}


{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
{{Mapping|legend=5| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128
: mapping generators: ~2, ~175/128


Line 80: Line 83:
[[Badness]] (Sintel): 1.70
[[Badness]] (Sintel): 1.70


=== Rainy ===
=== Ostara ===
In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.
Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.
 
Subgroup: 2.5.7.11
 
Comma list: 8589934592/8544921875, 30691800524/30517578125
 
{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773


A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.
Optimal tunings:
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


[[Comma list]]: [[2100875/2097152]]
Badness (Sintel): 11.7


{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
: mapping generators: ~2, ~256/245


[[Optimal tuning]]s:
{{Mapping|legend=2| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
: error map: {{val| 0.000 -0.267 -0.454 }}


{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}
Optimal tunings:
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}


[[Badness]] (Sintel): 0.156
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


=== French decimal ===
Badness (Sintel): 3.42
French decimal is conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.


[[Subgroup]]: 2.5.7
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


[[Comma list]]: {{monzo| 372 -159 -1 }}
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
{{Mapping|legend=2| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
: mapping generators: ~2, ~5


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
: error map: {{val| 0.000 +0.046 +0.019 }}


{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


[[Badness]] (Sintel): 148
Badness (Sintel): 1.99


==== 2.5.7.11 subgroup ====
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11.13.17.19


Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}
{{Mapping|legend=2| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}


{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


Badness (Sintel): 52.2
Badness (Sintel): 1.29


==== 2.5.7.11.13 subgroup ====
=== Frostburn ===
Subgroup: 2.5.7.11.13
{{Redirect|Frostburn|the software developer|Lumi Pakkanen}}
 
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 78125/76832
 
{{Mapping|legend=3| 1 3 4 | 0 -4 -7 }}
: mapping generators: ~2, ~28/25
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}
: [[error map]]: {{val| +0.346 -2.630 +2.189 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 204.2027{{c}}
: error map: {{val| 0.000 -3.125 +1.755 }}
 
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
 
[[Badness]] (Sintel): 0.886
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
Comma list: 245/242, 625/616


Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}


{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


Badness (Sintel): 10.5
Badness (Sintel): 0.463


=== Bastille ===
=== Rainy ===
{{Main| Bastille }}
In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.


Bastille is described as the 2.5.7-subgroup 1407 & 1789 temperament, and named after an [[Wikipedia: Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins #Double bastille|double bastille]].
A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: {{monzo| 1426 -596 -15 }}
[[Comma list]]: [[2100875/2097152]]
 
{{Mapping|legend=3| 1 2 3 | 0 5 -3 }}


{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
{{Mapping|legend=5| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}
: mapping generators: ~2, ~256/245


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
: error map: {{val| 0.000 +0.041 +0.018 }}
: error map: {{val| 0.000 -0.267 -0.454 }}


{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}


[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
[[Badness]] (Sintel): 0.156


=== Augment ===
=== Llywelyn a.k.a. shoe ===
{{See also| Chromatic pairs #Augment }}
Llywelyn a.k.a. shoe is the common [[restriction]] of [[gamelismic clan #Gorgo|gorgo]] (16 & 21), [[breedsmic temperaments #Gorgik|gorgik]] (21 & 37), and [[gariboh clan #Semaja|semaja]] (16 & 37). It can be described as 16 & 21, and tempers out the [[llywelynsma]]. Its generator is a flattened [[~]][[8/7]] whole tone, seven of which give the interval class of [[5/1|5]]. Its [[ploidacot]] is alpha-heptaseph. [[Weak extension]]s to the full 7-limit include [[marvel temperaments #Misneb|misneb]] (21 & 32), which splits 8/7 into two for [[15/14]]~[[16/15]] each, [[archytas clan #Ultrapyth|ultrapyth]] (32 & 37), which splits [[16/7]] into two for a [[3/2|perfect fifth]], and [[buzzardsmic clan #Septimal buzzard|buzzard]] (53 & 58), which splits 16/7 into three, each for [[21/16]].


Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.
; Scales
* [[Shoe5]], [[Shoe6]], [[Shoe11]], [[Shoe16]]


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 128/125
[[Comma list]]: 4194304/4117715


{{Mapping|legend=2| 3 7 0 | 0 0 1 }}
{{Mapping|legend=3| 1 1 3 | 0 7 -1 }}


{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
{{Mapping|legend=5| 1 0 1 3 | 0 0 7 -1 }}
: mapping generators: ~5/4, ~7
: mapping generators: ~2, ~8/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}
* [[WE]]: ~2 = 1199.1974{{c}}, ~8/7 = 226.7586{{c}}
: [[error map]]: {{val| -2.962 +6.776 -0.040 }}
: [[error map]]: {{val| -0.803 +0.194 +2.008 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 974.3418{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.8388{{c}}
: error map: {{val| 0.000 +13.686 +5.516 }}
: error map: {{val| 0.000 +1.558 +4.335 }}


{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37, 164d }}


[[Badness]] (Sintel): 0.296
[[Badness]] (Sintel): 2.06


==== 2.3.7.11 subgroup ====
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


Comma list: 56/55, 128/125
Comma list: 176/175, 1310720/1294139


Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}
{{Mapping|legend=2| 1 1 3 1 | 0 7 -1 13 }}


Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}
{{Mapping|legend=4| 1 0 1 3 1 | 0 0 7 -1 13 }}
: mapping generators: ~5/4, ~7


Optimal tunings:  
Optimal tunings:  
* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* WE: ~2 = 1199.1784{{c}}, ~8/7 = 226.9582{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0438{{c}}


{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}


Badness (Sintel): 0.196
Badness (Sintel): 0.651


=== Ostara ===
==== 2.5.7.11.13 subgroup ====
Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.
Subgroup: 2.5.7.11.13


Ostara can also refer to a collection of temperaments which temper out 16807/16796.{{clarify}}
Comma list: 176/175, 640/637, 847/845


[[Subgroup]]: 2.5.7.11
{{Mapping|legend=2| 1 1 3 1 2 | 0 7 -1 13 9 }}


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
{{Mapping|legend=4| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}


{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
Optimal tunings:
* WE: ~2 = 1199.1643{{c}}, ~8/7 = 226.9495{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0315{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }}
* [[CTE]]: ~2 = 1200.000{{c}}, ~5120/3773 = 529.003{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5120/3773 = 529.004{{c}}


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
Badness (Sintel): 0.319


[[Badness]] (Sintel): 11.731
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


==== 2.5.7.11.13 subgroup ====
Comma list: 176/175, 221/220, 640/637, 847/845
Subgroup: 2.5.7.11.13


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
{{Mapping|legend=2| 1 1 3 1 2 2 | 0 7 -1 13 9 11 }}


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
{{Mapping|legend=4| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}
* WE: ~2 = 1199.3162{{c}}, ~8/7 = 227.1125{{c}}
* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.1774{{c}}


{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
{{Optimal ET sequence|legend=0| 16, 21, 37, 95 }}
 
Badness (Sintel): 0.458
 
=== Mercy ===
Named by {{u|KingHyperio}} in 2021, mercy is the common restriction of [[gamelismic clan #Miracle|miracle]] and [[porwell temperaments #Countermiracle|countermiracle]]. It tempers out the [[quince comma]]. In this temperament, two generators make [[~]][[8/7]], five make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph. [[Weak extension]]s of this temperament include [[starling temperaments #Cassablanca|cassablanca]], [[octagar temperaments #Slendi|slendi]], [[ragismic microtemperaments #Quincy|quincy]], [[septischismic clan #Cotoneum|cotoneum]], and [[varunismic temperaments #Octowerck|octowerck]].
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 823543/819200
 
{{Mapping|legend=3| 1 3 3 | 0 -7 -2 }}
 
{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}
: mapping generators: ~2, ~343/320
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2169{{c}}, ~343/320 = 116.3116{{c}}
: [[error map]]: {{val| +0.217 +0.156 -0.798 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
: error map: {{val| 0.000 -0.117 -1.341 }}
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
 
[[Badness]] (Sintel): 0.428
 
==== 2.5.7.13 subgroup ====
Presented here is a low-accuracy extension, which may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].


Badness (Sintel): 3.415
Subgroup: 2.5.7.13


==== 2.5.7.11.13.17 subgroup ====
Comma list: 343/338, 640/637
Subgroup: 2.5.7.11.13.17


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
{{Mapping|legend=2| 1 3 3 4 | 0 -7 -2 -3 }}


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
{{Mapping|legend=4| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}


{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
 
Badness (Sintel): 0.546


Badness (Sintel): 1.985
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17


==== 2.5.7.11.13.17.19 subgroup ====
Comma list: 170/169, 224/221, 343/338
Subgroup: 2.5.7.11.13.17.19


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
{{Mapping|legend=2| 1 3 3 4 4 | 0 -7 -2 -3 1 }}


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
{{Mapping|legend=4| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~19/14 = 529.006{{c}}
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.000{{c}}, ~19/14 = 529.005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}


{{Optimal ET sequence|legend=0| 93, 338, 431 }}
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}


Badness (Sintel): 1.285
Badness (Sintel): 0.451


=== Tricesimoprimal miracloid ===
==== 2.5.7.13.17.19 subgroup ====
{{See also| Tricesimoprimal miracloid/Eliora's approach }}
Subgroup: 2.5.7.13.17.19


Tricesimoprimal miracloid is described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.
Comma list: 170/169, 224/221, 343/338, 476/475


Subgroup: 2.5.7.11.19.29.31
{{Mapping|legend=2| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
{{Mapping|legend=4| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}


Subgroup-val mapping: {{mapping| 1 419 48 177 157 624 625 | 0 -461 -50 -192 -169 -685 -686 }}
Optimal tunings:  
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~58/31 = 1084.628{{c}}
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}


{{Optimal ET sequence|legend=1| 52, 1737, 1789, … }}
Badness (Sintel): 0.759


=== Huntington ===
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}
{{See also| Chromatic pairs #Huntington }}


Huntington may be described as the 10 & 27 temperament in the 2.5.7.13 subgroup.  
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  


[[Subgroup]]: 2.5.7.13
[[Subgroup]]: 2.5.7


[[Comma list]]: [[640/637]], [[10985/10976]]
[[Comma list]]: 40960000/40353607


{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}
{{Mapping|legend=3| 1 -4 0 | 0 9 4 }}


{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}
{{Mapping|legend=5| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~16/13
: mapping generators: ~2, ~80/49


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/13 = 357.002{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
: error map: {{val| 0.000 -0.091 +2.828 }}
 
{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}
 
Badness (Sintel): 1.87
 
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13
 
Comma list: 640/637, 10985/10976
 
{{Mapping|legend=2| 1 -4 0 3 | 0 9 4 1 }}


{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}
{{Mapping|legend=4| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents
Optimal tunings:
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}
 
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}
 
Badness (Sintel): 0.319


==== Silver ====
==== Silver ====
{{See also| Chromatic pairs #Silver }}
{{See also| Chromatic pairs #Silver }}


Silver can be described as the 10 & 27 temperament in the 2.5.7.13.17 subgroup.  
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.  
 
Subgroup: 2.5.7.13.17


[[Subgroup]]: 2.5.7.13.17
Comma list: 170/169, 640/637, 5525/5488


[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]
{{Mapping|legend=2| 1 -4 0 3 9 | 0 9 4 1 -7 }}


{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}
{{Mapping|legend=4| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
Optimal tunings:
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/8 = 842.711{{c}}
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}


{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}
Badness (Sintel): 0.504


[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
=== Bastille ===
{{Main| Bastille }}


=== Pakkanen ===
Bastille is described as the 2.5.7-subgroup 1407 & 1789 temperament, and named after an [[Wikipedia: Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins #Double bastille|double bastille]].
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 625/616
[[Subgroup]]: 2.5.7


{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
[[Comma list]]: {{monzo| 1426 -596 -15 }}
: mapping generators: ~2, ~5, ~11


[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.6544{{c}}, ~5/4 = 380.3004{{c}}, ~11/8 = 551.9653{{c}}
{{Mapping|legend=3| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
: error map: {{val| 0.000 +0.041 +0.018 }}


[[Badness]] (Sintel): 0.573
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}
 
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


=== No-threes naiad ===
=== No-threes naiad (rank-3) ===
{{Todo|inline=1| review | comment = Devise the permanent name for this temperament; determine its status with respect to its extensions linked below. }}
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


Line 359: Line 454:
[[Comma list]]: 5021863/5000000
[[Comma list]]: 5021863/5000000


{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
{{Mapping|legend=3| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~100/77
: mapping generators: ~2, ~5, ~77/50


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.080{{c}}, ~5 = 2786.820{{c}}, ~100/77 = 454.618{{c}}
* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5 = 2786.740{{c}}, ~100/77 = 454.590{{c}}
: [[error map]]: {{val| +0.080 +0.507 -1.318 -0.643 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.7404{{c}}, ~77/50 = 745.4102{{c}}
: error map: {{val| 0.000 +0.427 -0.445 -0.808 }}


{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}


[[Badness]] (Sintel): 1.862
[[Badness]] (Sintel): 1.86


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
Line 375: Line 472:
Comma list: 847/845, 1001/1000
Comma list: 847/845, 1001/1000


Subgroup-val mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}
{{Mapping|legend=2| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.034{{c}}, ~5 = 2786.678{{c}}, ~13/10 = 454.569{{c}}
* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5 = 2786.646{{c}}, ~13/10 = 454.557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}


{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}


Badness (Sintel): 0.179
Badness (Sintel): 0.179
Line 390: Line 487:
Comma list: 170/169, 221/220, 847/845
Comma list: 170/169, 221/220, 847/845


Subgroup-val mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
{{Mapping|legend=2| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.407{{c}}, ~5 = 2787.484{{c}}, ~13/10 = 455.036{{c}}
* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5 = 2787.107{{c}}, ~13/10 = 454.906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}


{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}


Badness (Sintel): 0.438
Badness (Sintel): 0.438
Line 404: Line 501:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


=== Movila ===
=== Sephiroth ===
This temperament has a structure very similar to [[mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].
{{See also| Chromatic pairs #Sephiroth }}
 
Sephiroth is the common [[restriction]] of [[magic family #Muggles|muggles]] and [[avicennmic temperaments #Submerged|submerged]].  


[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11


[[Comma list]]: 1331/1280
[[Comma list]]: 34375/32768
 
{{Mapping|legend=3| 1 0 15 | 0 1 -5 }}
 
{{Mapping|legend=5| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3290{{c}}, ~5/4 = 373.6097{{c}}
: [[error map]]: {{val| +3.329 -6.046 -2.722 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 372.1586{{c}}
: error map: {{val| 0.000 -14.155 -12.111 }}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
 
[[Badness]] (Sintel): 1.85
 
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13
 
Comma list: 65/64, 6875/6656
 
{{Mapping|legend=2| 1 0 15 6 | 0 1 -5 -1 }}
 
{{Mapping|legend=4| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}
 
Badness (Sintel): 0.410
 
==== 2.5.11.13.17 subgroup ====
Subgroup: 2.5.11.13.17
 
Comma list: 65/64, 170/169, 221/220
 
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
 
{{Mapping|legend=4| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}


{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
Optimal tunings:
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}


[[Optimal tuning]] (CTE): ~2 = 1200.000{{c}}, ~11/8 = 529.846{{c}}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
Badness (Sintel): 0.299


=== Wizz ===
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}
{{See also| Chromatic pairs #Wizz }}


Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].  
Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[jubilismic clan #Astrology|astrology]] and [[stearnsmic clan #Wizard|wizard]].  


[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11


[[Comma list]]: [[15625/15488]]
[[Comma list]]: 15625/15488


{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
{{Mapping|legend=3| 2 0 -7 | 0 1 3 }}


{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
{{Mapping|legend=5| 2 0 4 0 5 | 0 0 1 0 3 }}
: mapping generators: ~125/88, ~5/4
: mapping generators: ~125/88, ~5/4


[[Optimal tuning]] ([[POTE]]): ~125/88 = 600.000{{c}}, ~5/4 = 383.768{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~125/88 = 600.1831{{c}}, ~5/4 = 383.8848{{c}}
: [[error map]]: {{val| +0.366 -1.697 +1.252 }}
* [[CWE]]: ~125/88 = 600.0000{{c}}, ~5/4 = 383.9977{{c}}
: error map: {{val| 0.000 -2.316 +0.675 }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3997
[[Badness]] (Sintel): 0.266


=== Insect ===
=== Insect ===
Line 442: Line 588:
[[Comma list]]: 33275/32768
[[Comma list]]: 33275/32768


{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}
{{Mapping|legend=3| 1 0 5 | 0 3 -2 }}
: mapping generators, ~2, ~55/32
: mapping generators, ~2, ~55/32


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~55/32 = 928.032{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1238{{c}}, ~55/32 = 928.5003{{c}}
: [[error map]]: {{val| +1.124 -0.813 -2.700 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~55/32 = 927.7384{{c}}
: error map: {{val| 0.000 -3.099 -6.975 }}


{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}
{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}


=== Sephiroth ===
[[Badness]] (Sintel): 0.564
{{See also| Chromatic pairs #Sephiroth }}


Sephiroth is the no-7 [[restriction]] of [[muggles]].  
=== Movila ===
This temperament has a structure very similar to [[mavila family #Mavila|mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].


[[Subgroup]]: 2.5.11.13.17
[[Subgroup]]: 2.5.11


[[Comma list]]: 65/64, 170/169, 221/220
[[Comma list]]: 1331/1280


{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
{{Mapping|legend=3| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8


{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}
[[Optimal tuning]]s:
: mapping generators: ~2, ~5/4
* [[WE]]: ~2 = 1203.0339{{c}}, ~11/8 = 528.4296{{c}}
: [[error map]]: {{val| +3.034 +2.009 -13.787 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 528.1575{{c}}
: error map: {{val| 0.000 -1.841 -23.160 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~5/4 = 372.236{{c}}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}
[[Badness]] (Sintel): 0.718
 
[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents


=== Trader ===
=== Trader ===
[[Subgroup]]: 2.5.13
[[Subgroup]]: 2.5.13


[[Comma list]]: [[26/25]]
[[Comma list]]: 26/25


{{Mapping|legend=2| 1 2 3 | 0 1 2 }}
{{Mapping|legend=3| 1 0 -1 | 0 1 2 }}
: mapping generators, ~2, ~5/4
: mapping generators, ~2, ~5


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~5/4 = 407.079{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0216{{c}}, ~5/4 = 410.2152{{c}}
: [[error map]]: {{val| -1.978 +19.945 -26.033 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 408.9029{{c}}
: error map: {{val| 0.000 +22.589 -22.722 }}


{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}
{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}
[[Badness]] (Sintel): 0.138


=== Superquintal ===
=== Superquintal ===
Line 486: Line 644:
[[Comma list]]: 64000000/62748517
[[Comma list]]: 64000000/62748517


{{Mapping|legend=2| 1 5 6 | 0 -7 -6 }}
{{Mapping|legend=3| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~13/10
: mapping generators, ~2, ~20/13


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~13/10 = 459.281{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5925{{c}}, ~20/13 = 740.6286{{c}}
: [[error map]]: {{val| -0.408 -1.098 +3.244 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 740.8058{{c}}
: error map: {{val| 0.000 -0.673 +4.307 }}


{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}
{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}
[[Badness]] (Sintel): 1.93


== No-threes no-fives subgroup temperaments ==
== No-threes no-fives subgroup temperaments ==
Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Berylic → [[4th-octave temperaments #Berylic|4th-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
Line 511: Line 674:
[[Comma list]]: 5767168/5764801
[[Comma list]]: 5767168/5764801


{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}
{{Mapping|legend=3| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9846{{c}}, ~7/4 = 968.9078{{c}}
: [[error map]]: {{val| -0.015 +0.051 -0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 968.9174{{c}}
: error map: {{val| 0.000 +0.091 +0.021 }}
 
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}
 
Badness (Sintel): 0.0309
 
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup, equating [[14/11]] to a step of [[3edo]]. It is the no-3 [[restriction]] of [[#Augment|augment]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 1372/1331


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~7/4 = 968.913{{c}}
{{Mapping|legend=3| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7


{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}
[[Optimal tuning]]s:
* [[WE]]: ~14/11 = 399.8041{{c}}, ~7/4 = 963.1666{{c}}
: [[error map]]: {{val| -0.588 -6.835 +10.281 }}
* [[CWE]]: ~14/11 = 400.0000{{c}}, ~4/4 = 962.7466{{c}}
: error map: {{val| 0.000 -6.079 +11.429 }}


Badness (Sintel): 0.031
{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}


=== Score ===
[[Badness]] (Sintel): 0.475
 
=== Scaevola ===
Scaevola is the 2.7.11-subgroup temperament tempering out 14641/14336, with a generator of [[11/8]] and mapping +4 generators to [[7/4]]. Two strong extensions are [[#Score|score]] to 2.7.11.13 and [[subgroup temperaments #Joan|joan]] to 2.9.7.11.
 
Scaevola has a structure that is very similar to [[mavila family #Mavila|mavila]], but whereas mavila has a "fifth" of [[3/2]], "major third" of [[5/4]], and "minor third" of [[6/5]], scaevola has a "fifth" of [[16/11]], "major third" of [[14/11]], and "minor third" of [[8/7]], substituting mavila's 4:5:6 for 11:14:16. Essentially, Scaevola can be seen as a counterpart of mavila for fifths even flatter than [[9edo]], such as [[11edo]]'s fifth.
 
It was named by [[CompactStar]] in 2026, referring to ''{{w|Scaevola}}'' (the genus of flowering plants), which is meant to sound reminiscent of ''score'' and ''mavila'' (the score and joan temperaments were actually named prior to this temperament).
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 14641/14336
 
{{Mapping|legend=3| 1 1 3 | 0 4 1 }}
 
{{Mapping|legend=5| 1 0 0 1 3 | 0 0 0 4 1 }}
: mapping generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2479{{c}}, ~11/8 = 542.1162{{c}}
: [[error map]]: {{val| +1.248 +0.887 -5.458 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 541.9403{{c}}
: error map: {{val| 0.000 -1.065 -9.378 }}
 
{{Optimal ET sequence|legend=1| 2, 7d, 9, 11, 20, 31, 51, 82e, 113e, 195eee }}
 
[[Badness]] (Sintel): 0.578
 
==== Score ====
{{See also| Chromatic pairs #Score }}
{{See also| Chromatic pairs #Score }}


[[Subgroup]]: 2.7.11.13
Score is an extension of scaevola that maps [[13/8]] to +6 generators.


[[Comma list]]: 343/338, 847/832
Subgroup: 2.7.11.13
 
Comma list: 343/338, 847/832


{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}


{{Mapping|legend=3| 1 0 0 1 3 1| 0 0 0 4 1 6 }}
{{Mapping|legend=4| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~11/8 = 540.099{{c}}
Optimal tunings:
* WE: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}


{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
{{Optimal ET sequence|legend=0| 9, 11, 20 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents
Badness (Sintel): 0.368


=== Bossier ===
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
{{See also| Chromatic pairs #Bossier }}


Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.  
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].  
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 214358881/210827008


[[Subgroup]]: 2.7.11.13
{{Mapping|legend=3| 1 0 1 | 0 8 7 }}


[[Comma list]]: [[1573/1568]], [[15488/15379]]
{{Mapping|legend=5| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1886{{c}}, ~14/11 = 421.2661{{c}}
: [[error map]]: {{val| +0.189 +1.303 -2.266 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2365{{c}}
: error map: {{val| 0.000 +1.066 -2.662 }}
 
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}
 
[[Badness]] (Sintel): 1.73
 
==== 2.7.11.13 subgroup ====
Subgroup: 2.7.11.13
 
Comma list: 1573/1568, 15488/15379


{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}


{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
{{Mapping|legend=4| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
: mapping generators: ~2, ~14/11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~14/11 = 421.309{{c}}
Optimal tunings:
* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}


{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}
{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
Badness (Sintel): 0.307


=== Voltage ===
=== Voltage ===
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.  
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 7<sup>1/4</sup> of 842.2065{{c}}, which is also the CTC (constrained Tenney–Chebyshevian) tuning.  


[[Subgroup]]: 2.7.13
[[Subgroup]]: 2.7.13


[[Comma list]]: [[28672/28561]]
[[Comma list]]: 28672/28561


{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}
{{Mapping|legend=3| 1 0 3 | 0 4 1 }}


{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
{{Mapping|legend=5| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~16/13
: mapping generators: ~2, ~13


[[Optimal tuning]]:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~16/13 = 357.677{{c}}
* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}
* [[TOP tuning|POTT]]: ~2 = 1200.000{{c}}, ~16/13 = 357.794{{c}} (1200 - 300 log<sub>2</sub>(7))
: [[error map]]: {{val| -0.217 -0.143 +0.991 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/8 = 842.2568{{c}}
: error map: {{val| 0.000 +0.201 +1.729 }}


{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents
[[Badness]] (Sintel): 0.115


=== Ultrakleismic ===
=== Ultrakleismic ===
Line 582: Line 822:
[[Comma list]]: 4913/4802
[[Comma list]]: 4913/4802


{{Mapping|legend=2| 1 2 3 | 0 3 4 }}
{{Mapping|legend=3| 1 2 3 | 0 3 4 }}
: mapping generators, ~2, ~[[17/14]]
: mapping generators, ~2, ~17/14


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 324.446{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1379{{c}}, ~17/14 = 324.3440{{c}}
: [[error map]]: {{val| +0.138 +4.482 -7.166 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 324.3738{{c}}
: error map: {{val| 0.000 +4.295 -7.460 }}


{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}
[[Badness]] (Sintel): 0.460


=== Counterultrakleismic ===
=== Counterultrakleismic ===
Line 594: Line 840:
[[Comma list]]: 2024782584832/2015993900449
[[Comma list]]: 2024782584832/2015993900449


{{Mapping|legend=2| 1 0 1 | 0 10 11 }}
{{Mapping|legend=3| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14
: mapping generators, ~2, ~17/14


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 336.858{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9723{{c}}, ~17/14 = 336.8586{{c}}
: [[error map]]: {{val| -0.028 -0.240 +0.462 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 336.8621{{c}}
: error map: {{val| 0.000 -0.205 +0.528 }}


{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}
{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}
[[Badness]] (Sintel): 0.860


=== Shipwreck ===
=== Shipwreck ===
Line 606: Line 858:
[[Comma list]]: 1048576/1042139
[[Comma list]]: 1048576/1042139


{{Mapping|legend=2| 1 0 6 | 0 3 -1 }}]
{{Mapping|legend=3| 1 0 6 | 0 3 -1 }}]
: mapping generators, ~2, ~64/53
: mapping generators, ~2, ~64/53


[[Optimal tuning]]s ([[POTE]]): ~2 = 1200.000{{c}}, ~64/53 = 323.034{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6967{{c}}, ~64/53 = 323.1839{{c}}
: [[error map]]: {{val| -0.303 +0.119 +1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/53 = 323.1959{{c}}
: error map: {{val| 0.000 +0.762 +3.300 }}


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
[[Badness]] (Sintel): 0.224


=== Lovecraft ===
=== Lovecraft ===
{{See also | Chromatic pairs #Lovecraft }}
{{See also | Chromatic pairs #Lovecraft }}


Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.  
Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, tempers out [[1352/1331]], and is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.  


[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13


[[Comma list]]: [[1352/1331]]
[[Comma list]]: 1352/1331
 
{{Mapping|legend=3| 1 1 0 | 0 2 3 }}


{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
{{Mapping|legend=5| 1 0 0 0 1 0 | 0 0 0 0 2 3 }}
: mapping generators: ~2, ~26/11


{{Mapping|legend=3| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
[[Optimal tuning]]s:
: mapping generators, ~2, ~13/11
* [[WE]]: ~2 = 1199.5223{{c}}, ~13/11 = 279.2064{{c}}
: [[error map]]: {{val| -0.478 +5.662 -4.341 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/11 = 278.9918{{c}}
: error map: {{val| 0.000 +6.666 -3.552 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/11 = 279.318{{c}}
{{Optimal ET sequence|legend=1| 4, 9, 13, 30, 43, 73, 116e }}


{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}
[[Badness]] (Sintel): 0.175


[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 265837/262144
 
{{Mapping|legend=3| 1 0 6 | 0 3 -2 }}
 
{{Mapping|legend=5| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
: mapping generators, ~2, ~143/128
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8795{{c}}, ~143/128 = 182.5017{{c}}
: [[error map]]: {{val| +0.880 -1.174 -2.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~143/128 = 182.4386{{c}}
: error map: {{val| 0.000 -4.002 -5.405 }}
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}
 
[[Badness]] (Sintel): 0.451


=== Blackbirds ===
=== Blackbirds ===
Line 640: Line 925:
[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13


[[Comma list]]: [[29282/28561]]
[[Comma list]]: 29282/28561


{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
{{Mapping|legend=3| 4 0 1 | 0 1 1 }}


{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
{{Mapping|legend=5| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
: mapping generators, ~13/11, ~11
: mapping generators, ~13/11, ~11


[[Optimal tuning]] ([[POTE]]): ~13/11 = 300.000{{c}}, ~11/8 = 546.660{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~13/11 = 299.9728{{c}}, ~11/8 = 546.6107{{c}}
: [[error map]]: {{val| -0.109 -5.033 +5.730 }}
* [[CWE]]: ~13/11 = 300.0000{{c}}, ~11/8 = 546.4664{{c}}
: error map: {{val| 0.000 -4.852 +5.939 }}
 
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}
 
[[Badness]] (Sintel): 0.668
 
=== Yamablu ===
Yamablu, with a generator of ~26/17, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). It extends the 2.11.13-subgroup temperament tempering out 556573090931/549755813888. The [[Kite's Genchain mode numbering|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].
 
[[Subgroup]]: 2.11.13.17.19
 
[[Comma list]]: 209/208, 2057/2048, 83521/83486
 
{{Mapping|legend=3| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17


{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}


=== Bluebirds ===
[[Badness]] (Sintel): 0.386
{{Distinguish| Bluebird }}
 
{{See also| Chromatic pairs #Bluebirds }}
=== Arsenic ===
Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.


[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: [[265837/262144]]
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432


{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
{{Mapping|legend=3| 33 0 8 249 26 | 0 1 1 -1 1 }}
: mapping generators: ~247/242, ~11


{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
[[Optimal tuning]]s:
: mapping generators, ~2, ~143/128
* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}
* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~143/128 = 182.368{{c}}
{{Optimal ET sequence|legend=1| 33, 165, 198, 231, 495h }}


{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}
[[Badness]] (Sintel): 2.30


[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents
=== Berylic ===
Berylic tempers out the [[berylisma]] in the 2.11.37 subgroup, representing the fact that [[44/37]] is a {{w|continued fraction}} convergent to 2<sup>1/4</sup> the fourth root of 2. Beryllic is an example of a temperament which has an astronomically low [[badness]], being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by edo systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within an obscure subgroup, 2.11.37.


=== Yamablu ===
If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].


[[Subgroup]]: 2.11.13.17.19
[[Subgroup]]: 2.11.37


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 1874161/1874048


{{Mapping|legend=2| 1 5 1 1 0 | 0 -4 7 8 11 }}
{{Mapping|legend=3| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~17/13 = 462.9606{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~44/37 = 300.0003{{c}}, ~11/8 = 551.3211{{c}}
: [[error map]]: {{val| +0.001 +0.007 -0.017 }}
* [[CWE]]: ~44/37 = 300.0000{{c}}, ~11/8 = 551.3237{{c}}
: error map: {{val| 0.000 +0.006 -0.020 }}


{{Optimal ET sequence|legend=1| 13, 44, 57, 70 }}
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Badness]] (Sintel): 0.00188


=== Mavericks ===
=== Mavericks ===
Line 692: Line 1,005:
[[Comma list]]: 47525504/47045881
[[Comma list]]: 47525504/47045881


{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
{{Mapping|legend=3| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~26/19 = 539.886{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8817{{c}}, ~26/19 = 539.9150{{c}}
: [[error map]]: {{val| -0.118 -1.156 +1.825 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/19 = 539.9280{{c}}
: error map: {{val| 0.000 -0.960 +2.127 }}


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
{{Optimal ET sequence|legend=1| 9, 11, 20 }}
 
[[Badness]] (Sintel): 0.559


=== Yer (rank 3) ===
=== Yer (rank 3) ===
Line 703: Line 1,023:
[[Comma list]]: 209/208, 2057/2048
[[Comma list]]: 209/208, 2057/2048


{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
{{Mapping|legend=3| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13


[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.4457{{c}}, ~11/8 = 548.4934{{c}}, ~16/13 = 358.638{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4447{{c}}, ~11/8 = 548.4929{{c}}, ~13/8 = 841.3613{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 548.2193{{c}}, ~13/8 = 841.4707{{c}}


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


[[Category:Temperament collections]]
[[Badness]] (Sintel): 0.106
[[Category:Subgroup temperaments]]
 
[[Category:Catalogs of subgroup temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"