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No-threes subgroup temperaments: Difference between revisions

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{{Technical data page}}
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.  
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.  


== Overview by mapping of 5 ==
== Overview by mapping of 5 ==
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]].
* 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.
* 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.
* Frostburn has a ~28/25 generator, four of which give the ~8/5.
Others have a more complex mapping of 5.


* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
== Temperaments with a 2.5.7 gene ==
* French decimal and trader are with ~2/1 period and ~5/4 generator. There is one-to-one correspondence between 2.5 subgroup and mapped intervals.
* Didacus is with a ~28/25 generator, two of which give the ~5/4.
* Ostara, movila and vengeance are with variety expressed generator, three of which give the ~5/2.
* Insect is with a ~55/32 generator, three of which give the ~5/1.
* Frostburn is with a ~28/25 generator, four of which give the ~8/5.
Others are more far.
 
== 2.5.7 temperaments ==
 
Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic]]
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe]]
* 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]]


=== Rainy ===
=== Rainy ===
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is 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 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)
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.  


A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
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
Line 28: Line 28:
[[Comma list]]: [[2100875/2097152]]
[[Comma list]]: [[2100875/2097152]]


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
 
{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~256/245
 
[[Optimal tuning]]s:
* [[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 }}
 
[[Badness]] (Sintel): 0.156
 
=== Augment ===
{{See also| Chromatic pairs #Augment }}
 
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 128/125


[[Gencom]]: [2 256/245; 2100875/2097152]
{{Mapping|legend=2| 3 7 0 | 0 0 1 }}


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


Optimal tuning ([[POTE]]): ~256/245 = 77.205
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}
: [[error map]]: {{val| -2.962 +6.776 -0.040 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 974.3418{{c}}
: error map: {{val| 0.000 +13.686 +5.516 }}


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
[[Badness]] (Sintel): 0.296


=== Bastille ===
==== 2.5.7.11 subgroup ====
{{Main| Bastille }}
Subgroup: 2.5.7.11


Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Pure Bastille|pure bastille]].
Comma list: 56/55, 128/125


Subgroup: 2.5.7
Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}


Comma list: {{Monzo|1426 -596 -15}}
Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
Optimal tunings:  
* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
Badness (Sintel): 0.196


=== Frostburn ===
=== Frostburn ===
{{See also| Magic family #Quadrimage }}
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 63: Line 92:


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


: Sval 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 tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
[[Badness]] (Sintel): 0.886


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


Comma list: 245/242, 625/616
Comma list: 245/242, 625/616


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25
 
Optimal tunings:
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}
 
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
 
Badness (Sintel): 0.463
 
=== Mabilic ===
{{Main| Mabilic and trismegistus }}
{{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.
 
[[Subgroup]]: 2.5.7


: Sval mapping generators: ~2, ~28/25
[[Comma list]]: 1071875/1048576


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128


=== French decimal ===
[[Optimal tuning]]s:
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.
* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}
: [[error map]]: {{val| +1.254 -1.698 -1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}
: error map: {{val| 0.000 -4.701 -4.846 }}
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}
 
[[Badness]] (Sintel): 1.70
 
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}
 
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 40960000/40353607
 
{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}
 
{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~80/49
 
[[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


Subgroup: 2.5.7
Comma list: 640/637, 10985/10976


Comma basis: {{monzo|372 -159 -1}}
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


Optimal tuning (CTE): ~5/4 = 386.360
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=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}


==== 2.5.7.11 subgroup ====
Badness (Sintel): 0.319
Subgroup: 2.5.7.11


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
==== Silver ====
{{See also| Chromatic pairs #Silver }}


Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.


Optimal tuning (CTE): ~5/4 = 386.361
Subgroup: 2.5.7.13.17


{{Optimal ET sequence|legend=1|264, 733}}, ...
Comma list: 170/169, 640/637, 5525/5488


==== 2.5.7.11.13 subgroup ====
Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
Subgroup: 2.5.7.11.13


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
Optimal tunings:  
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


Optimal tuning (CTE): ~5/4 = 386.361
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
Badness (Sintel): 0.504


=== Ostara ===
=== Ostara ===
'''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.  
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.  


Ostara can also refer to a collection of temperaments which temper out 16807/16796.
Subgroup: 2.5.7.11


[[Subgroup]]: 2.5.7.11
Comma list: 8589934592/8544921875, 30691800524/30517578125


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773


[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
Optimal tunings:  
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
Badness (Sintel): 11.7


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
Line 138: Line 231:
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}


Optimal tuning (POTE): ~1664/1225 = 529.003¢
Optimal tunings:  
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
 
Badness (Sintel): 3.42


==== 2.5.7.11.13.17 subgroup ====
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11.13.17


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
 
Optimal tunings:
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}


Optimal tuning (POTE): ~1664/1225 = 529.003¢
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
Badness (Sintel): 1.99


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


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
 
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
 
Optimal tunings:
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}
 
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.29
 
=== French decimal ===
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
 
[[Comma list]]: {{monzo| 372 -159 -1 }}
 
{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
: mapping generators: ~2, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
* [[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 }}
 
[[Badness]] (Sintel): 148
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
 
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
 
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
Optimal tunings:  
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


Optimal tuning (POTE): ~19/14 = 529.003¢
{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}


=== Tricesimoprimal miracloid ===
Badness (Sintel): 52.2
{{See also|Tricesimoprimal miracloid/Eliora's approach|l1=Eliora's approach to tricesimoprimal miracloid}}
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.


Subgroup: 2.5.7.11.19.29.31
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}


Optimal tuning (CTE): ~58/31 = 1084.628
Optimal tunings:  
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}


=== Mercy ===
Badness (Sintel): 10.5
{{See also| Quince clan #Mercy }}
 
=== Bastille ===
{{Main| Bastille }}


Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
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
[[Subgroup]]: 2.5.7


[[Comma list]]: 823543/819200
[[Comma list]]: {{monzo| 1426 -596 -15 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


[[Gencom]]: [2 2744/2560; 823543/819200]
[[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 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


Optimal tuning ([[POTE]]): ~343/320 = 116.291
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
=== 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 }}


==== 2.5.7.13 ====
This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands [[Subgroup temperaments #Tridec|tridec]] and [[Subgroup temperaments #Naiadec|naiadec]].
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
[[Subgroup]]: 2.5.7.11


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
[[Comma list]]: 5021863/5000000


[[Gencom]]: [2 14/13; 343/338 640/637]
{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{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 tuning ([[POTE]]): ~14/13 = 116.094
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}


{{Optimal ET sequence|legend=1| 10, 21, 31}}
[[Badness]] (Sintel): 1.86


==== 2.5.7.13.17 ====
==== 2.5.7.11.13 subgroup ====
[[Subgroup]]: 2.5.7.13.17
Subgroup: 2.5.7.11.13


[[Comma list]]: 170/169, 224/221, 640/637
Comma list: 847/845, 1001/1000


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
Optimal tunings:  
* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}


Optimal tuning ([[POTE]]): ~14/13 = 115.769
Badness (Sintel): 0.179


{{Optimal ET sequence|legend=1| 10, 21, 31}}
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


==== 2.5.7.13.17.19 ====
Comma list: 170/169, 221/220, 847/845
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
Optimal tunings:  
* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
Badness (Sintel): 0.438


Optimal tuning ([[POTE]]): ~14/13 = 115.716
== Temperaments with a higher 2.5.''p'' gene ==
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}


=== Pakkanen (rank 3) ===
Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[astrology]] and [[wizard]].
[[Subgroup]]: 2.5.7.11
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 15625/15488
 
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
 
{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
: mapping generators: ~125/88, ~5/4
 
[[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, 394c }}


[[Comma list]]: 625/616
[[Badness]] (Sintel): 0.266


{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
=== Insect ===
[[Subgroup]]: 2.5.11


: mapping generators: ~2, ~5, ~11
[[Comma list]]: 33275/32768


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}
: mapping generators, ~2, ~55/32


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
[[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 }}


== Higher 2.5 temperaments ==
{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}


Temperaments discussed elsewhere include:
[[Badness]] (Sintel): 0.564
* The Jacobins (superfamily tempering out [[6656/6655]]) → [[The Jacobins]]


=== Movila ===
=== Movila ===
Line 267: Line 449:
[[Comma list]]: 1331/1280
[[Comma list]]: 1331/1280


[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
[[Optimal tuning]]s:
* [[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 ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


=== Insect ===
[[Badness]] (Sintel): 0.718
 
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
 
Sephiroth is the no-7 [[restriction]] of [[muggles]].
 
[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11


[[Comma list]]: 33275/32768
[[Comma list]]: 34375/32768
 
{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
 
{{Mapping|legend=3| 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
 
Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
 
Gencom mapping: {{mapping| 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
 
Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
 
Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}
 
Badness (Sintel): 0.299
 
=== Trader ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 26/25


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


: Mapping generators, ~2, ~[[55/32]]
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}


{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
[[Badness]] (Sintel): 0.138


=== Superquintal ===
=== Superquintal ===
Line 291: Line 543:
[[Comma list]]: 64000000/62748517
[[Comma list]]: 64000000/62748517


{{Mapping|legend=2|1 5 6|0 -7 -6}}
{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~20/13
 
[[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 }}
 
[[Badness]] (Sintel): 1.93
 
== No-threes no-fives subgroup temperaments ==
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* 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]]
* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)|37th-octave temperaments]]
* etc.
 
=== Amaranthine ===
{{See also| No-fives subgroup temperaments #Chrysanthemum }}
 
Amaranthine is the high-accuracy 2.7.11-subgroup strong [[restriction]] of [[Gamelismic clan #11-limit 3|undecimal mothra]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 5767168/5764801
 
{{Mapping|legend=2| 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 }}


: Mapping generators, ~2, ~13/10
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
Badness (Sintel): 0.0309


{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].


=== Trader ===
[[Subgroup]]: 2.7.11
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 1372/1331
 
{{Mapping|legend=2| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}
 
[[Badness]] (Sintel): 0.475
 
=== Score ===
{{See also| Chromatic pairs #Score }}
 
Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.
 
[[Subgroup]]: 2.7.11.13
 
[[Comma list]]: 343/338, 847/832
 
{{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 generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[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| 9, 11, 20 }}
 
[[Badness]] (Sintel): 0.368
 
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
 
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
 
{{Mapping|legend=2| 1 0 1 | 0 8 7 }}
 
{{Mapping|legend=3| 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


[[Comma list]]: [[26/25]]
Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}


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


: Mapping generators, ~2, ~[[5/4]]
Optimal tunings:  
* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}


{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
Badness (Sintel): 0.307


=== Vengeance ===
=== Voltage ===
{{Main| Vengeance }}
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.


Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
[[Subgroup]]: 2.7.13


[[Subgroup]]: 2.5.17
[[Comma list]]: 28672/28561


[[Comma list]]: 78608/78125
{{Mapping|legend=2| 1 0 3 | 0 4 1 }}


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}
: [[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|7g, 9, 25, 34, 93, 127, 288, 415}}
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}


== No-threes-or-fives subgroup temperaments ==
[[Badness]] (Sintel): 0.115
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone]]
* Berylic → [[4th-octave temperaments #Berylic]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic]]
* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)]]
* etc.


=== Ultrakleismic ===
=== Ultrakleismic ===
Line 341: Line 696:
[[Comma list]]: 4913/4802
[[Comma list]]: 4913/4802


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


: Mapping generators, ~2, ~[[17/14]]
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446
{{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 354: Line 714:
[[Comma list]]: 2024782584832/2015993900449
[[Comma list]]: 2024782584832/2015993900449


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


: Mapping generators, ~2, ~[[17/14]]
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858
{{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 ===
[[Subgroup]]: 2.7.53
[[Subgroup]]: 2.7.53


[[Comma list]]: 1048576/1042139
[[Comma list]]: 1048576/1042139


[[Gencom]]: [2 64/53; 1048576/1042139]
{{Mapping|legend=2| 1 0 6 | 0 3 -1 }}]
: mapping generators, ~2, ~64/53


[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
[[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 }}


[[POTE generator]]: ~64/53 = 323.034
{{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 ===
{{See also | Chromatic pairs #Lovecraft }}
 
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
 
[[Comma list]]: 1352/1331
 
{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
 
{{Mapping|legend=3| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
: mapping generators, ~2, ~13/11
 
[[Optimal tuning]]s:
* [[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 ET sequence|legend=1| 4, 9, 13, 30, 43, 73, 116e }}
 
[[Badness]] (Sintel): 0.175
 
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 265837/262144
 
{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
 
{{Mapping|legend=3| 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 ===
{{See also | Chromatic pairs #Blackbirds }}
 
Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 29282/28561


=== Yer (rank 3) ===
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
: mapping generators, ~13/11, ~11


[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
[[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 tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
[[Badness]] (Sintel): 0.668


=== Yamablu ===
=== Yamablu ===
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]].
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
[[Subgroup]]: 2.11.13.17.19
Line 394: Line 823:
[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 209/208, 2057/2048, 83521/83486


[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}
 
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}
 
[[Badness]] (Sintel): 0.386
 
=== Arsenic ===
Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.
 
[[Subgroup]]: 2.11.13.17.19
 
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432
 
{{Mapping|legend=2| 33 0 8 249 26 | 0 1 1 -1 1 }}
: mapping generators: ~247/242, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}
* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}
 
{{Optimal ET sequence|legend=1| 33, 165, 198, 231, 495h }}
 
[[Badness]] (Sintel): 2.30


Optimal tuning ([[POTE]]): ~17/13 = 462.9606
=== 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.


{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
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.


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Subgroup]]: 2.11.37
 
[[Comma list]]: 1874161/1874048
 
{{Mapping|legend=2| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11
 
[[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| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}
 
[[Badness]] (Sintel): 0.00188


=== Mavericks ===
=== Mavericks ===
[[Subgroup]]: 2.13.19
[[Subgroup]]: 2.13.19


[[Comma list]]: 47525504/47045881
[[Comma list]]: 47525504/47045881


[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19
 
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
{{Optimal ET sequence|legend=1| 9, 11, 20 }}
 
[[Badness]] (Sintel): 0.559
 
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19
 
[[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 generators: ~2, ~11, ~13
 
[[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| 7fh, 9, 11, 20 }}
[[Badness]] (Sintel): 0.106


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"