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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.  


Line 5: Line 6:


* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
* 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.
* 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.
* Didacus is with a ~28/25 generator, two of which give the ~5/4.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Ostara, movila and vengeance are with variety expressed generator, three of which give the ~5/2.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Insect is with a ~55/32 generator, three of which give the ~5/1.
* Frostburn has a ~28/25 generator, four of which give the ~8/5.
* Frostburn is with a ~28/25 generator, four of which give the ~8/5.
Others have a more complex mapping of 5.
Others are more far.
 
== 2.5.7 temperaments ==


Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* Jubilic → [[Jubilismic clan #Jubilic]]
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
 
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
== Llywelyn aka shoe ==
* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
{{See also| Chromatic pairs #Shoe }}
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
{{See also| Llywelyn clan #Llywelyn aka shoe }}
* Sidewalk ([[823543/800000]]) → [[2023/2000#Sidewalk]]
=== Frostburn ===
{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}


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


[[Comma list]]: 4194304/4117715
[[Comma list]]: 78125/76832


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}


Mapping generators: 2, ~8/7
: Sval mapping generators: ~2, ~28/25


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389


[[Gencom]]: [2 8/7; 4194304/4117715]
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
[[Badness]] (Sintel): 0.886


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
==== 2.5.7.11 ====
Subgroup: 2.5.7.11


=== 2.5.7.11 subgroup ===
Comma list: 245/242, 625/616
Subgroup: 2.5.7.11


Comma list: 176/175, 1310720/1294139
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}


Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
: Sval mapping generators: ~2, ~28/25


Gencom: [2 8/7; 176/175 1310720/1294139]
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


Optimal tuning (POTE): ~8/7 = 227.114
Badness (Sintel): 0.463


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
=== Mabilic ===
{{See also| Chromatic pairs #Mabilic }}{{Main|Mabilic and trismegistus}}Given below is the no-three version 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.


=== 2.5.7.11.13 subgroup ===
[[Subgroup]]: 2.5.7
Subgroup: 2.5.7.11.13


Comma list: 176/175, 640/637, 847/845
[[Comma list]]: 1071875/1048576


Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}


Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}


Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
: [[gencom]]: [2 175/128; 1071875/1048576]


Optimal tuning (POTE): ~8/7 = 227.108
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}


=== 2.5.7.11.13.17 subgroup ===
[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents
Subgroup: 2.5.7.11.13.17


Comma list: 176/175, 221/200, 640/637, 833/832
=== 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]].)


Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.


Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
[[Subgroup]]: 2.5.7


Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
[[Comma list]]: [[2100875/2097152]]


Optimal tuning (POTE): ~8/7 = 227.242
[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Gencom]]: [2 256/245; 2100875/2097152]


== Didacus ==
[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
{{See also| Hemimean clan #Didacus }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
Optimal tuning ([[POTE]]): ~256/245 = 77.205


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}


[[Comma list]]: 3136/3125
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents


[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]
=== French decimal ===
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.


[[Gencom]]: [2 28/25; 3136/3125]
Subgroup: 2.5.7


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
Comma basis: {{monzo|372 -159 -1}}


Optimal tuning ([[POTE]]): ~28/25 = 93.772
Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
Optimal tuning (CTE): ~5/4 = 386.360


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...


== Rainy ==
[[Badness]] (Sintel): 148.6
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


[[Subgroup]]: 2.5.7
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


[[Comma list]]: [[2100875/2097152]]
Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]


[[Gencom]]: [2 256/245; 2100875/2097152]
Optimal tuning (CTE): ~5/4 = 386.361


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
{{Optimal ET sequence|legend=0|264, 733}}, ...


Optimal tuning ([[POTE]]): ~256/245 = 77.205
Badness (Sintel): 52.150


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


== Mercy ==
Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
{{See also| Quince clan #Mercy }}


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]].
Optimal tuning (CTE): ~5/4 = 386.361


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=0|1525, 1789}}, ...


[[Comma list]]: 823543/819200
Badness (Sintel): 10.518


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
=== Bastille ===
{{Main| Bastille }}


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


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
Subgroup: 2.5.7


Optimal tuning ([[POTE]]): ~343/320 = 116.291
Comma list: {{Monzo|1426 -596 -15}}


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]


=== 2.5.7.13 ===
Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
[[Badness]] (Sintel): 7224.3


[[Gencom]]: [2 14/13; 343/338 640/637]
=== Augment ===
{{See also| Chromatic pairs #Augment }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
Augment is related to [[augmented]].


Optimal tuning ([[POTE]]): ~14/13 = 116.094
[[Subgroup]]: 2.5.7.11


{{Optimal ET sequence|legend=1| 10, 21, 31}}
[[Comma list]]: 56/55, 128/125


=== 2.5.7.13.17 ===
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
: [[gencom]]: [5/4 8/7; 56/55 128/125]


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}


Optimal tuning ([[POTE]]): ~14/13 = 115.769
[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents


{{Optimal ET sequence|legend=1| 10, 21, 31}}
=== 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.


=== 2.5.7.13.17.19 ===
Ostara can also refer to a collection of temperaments which temper out 16807/16796.
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
[[Subgroup]]: 2.5.7.11


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢


Optimal tuning ([[POTE]]): ~14/13 = 115.716
{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
[[Badness]] (Sintel): 11.731


== Pakkanen (rank 3) ==
==== 2.5.7.11.13 subgroup ====
[[Subgroup]]: 2.5.7.11
Subgroup: 2.5.7.11.13


[[Comma list]]: 625/616
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]


: mapping generators: ~2, ~5, ~11
Optimal tunings:  
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
Badness (Sintel): 3.415


== Frostburn ==
==== 2.5.7.11.13.17 subgroup ====
{{See also| Magic family #Quadrimage }}
Subgroup: 2.5.7.11.13.17


[[Subgroup]]: 2.5.7
Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]


[[Comma list]]: 78125/76832
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢


: Sval mapping generators: ~2, ~28/25
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
Badness (Sintel): 1.985


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19


=== 2.5.7.11 ===
Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢


: Sval mapping generators: ~2, ~28/25
{{Optimal ET sequence|legend=0| 93, 338, 431 }}


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Badness (Sintel): 1.285


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
=== Tricesimoprimal miracloid ===
{{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.


== Ostara ==
Subgroup: 2.5.7.11.19.29.31
'''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 can also refer to a collection of temperaments which temper out 16807/16796.
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688


[[Subgroup]]: 2.5.7.11
Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
Optimal tuning (CTE): ~58/31 = 1084.628


[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.


=== 2.5.7.11.13 subgroup ===
[[Subgroup]]: 2.5.7.13
Subgroup: 2.5.7.11.13


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
[[Comma list]]: [[640/637]], [[10985/10976]]


Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}


Optimal tuning (POTE): ~1664/1225 = 529.003¢
{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
: [[gencom]]: [2 16/13; 640/637 10985/10976]


=== 2.5.7.11.13.17 subgroup ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002
Subgroup: 2.5.7.11.13.17


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents


Optimal tuning (POTE): ~1664/1225 = 529.003¢
==== Silver ====
{{See also| Chromatic pairs #Silver }}


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.


=== 2.5.7.11.13.17.19 subgroup ===
[[Subgroup]]: 2.5.7.13.17
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]]: [[170/169]], [[640/637]], [[5525/5488]]


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}


Optimal tuning (POTE): ~19/14 = 529.003¢
{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


== Pure onzonic ==
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]
The 2.5.11.13 subgroup primarily contains temperaments developed for 1789edo, since it tempers out the jacobin comma 6656/6655, for which 2.5.11.13 is the subgroup, and the year 1789 is hallmark for the French revolution.


Subgroup: 2.5.11.13
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711


Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}


Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents


Optimal tuning (POTE): ~11/8 = 551.370
=== Pakkanen ===
[[Subgroup]]: 2.5.7.11


{{Optimal ET sequence|legend=1|37, 1789}}
[[Comma list]]: 625/616


== Tricesimoprimal miracloid ==
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
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
: mapping generators: ~2, ~5, ~11


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653


Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}


Optimal tuning (CTE): ~58/31 = 1084.628
[[Badness]] (Sintel): 0.573


{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


== French decimal ==
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]].
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
[[Subgroup]]: 2.5.7.11


Comma basis: {{monzo|372 -159 -1}}
[[Comma list]]: 5021863/5000000


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}


Optimal tuning (CTE): ~5/4 = 386.360
: mapping generators: ~2, ~5, ~100/77


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.080¢, ~5 = 2786.820¢, ~100/77 = 454.618¢
* [[CWE]]: ~2 = 1200.000¢, ~5 = 2786.740¢, ~100/77 = 454.590¢


=== 2.5.7.11 subgroup ===
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
Subgroup: 2.5.7.11


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
[[Badness]] (Sintel): 1.862


Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


Optimal tuning (CTE): ~5/4 = 386.361
Comma list: 847/845, 1001/1000


{{Optimal ET sequence|legend=1|264, 733}}, ...
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}


=== 2.5.7.11.13 subgroup ===
Optimal tunings:
Subgroup: 2.5.7.11.13
* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
Badness (Sintel): 0.179


Optimal tuning (CTE): ~5/4 = 386.361
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
Comma list: 170/169, 221/220, 847/845


== Bastille ==
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
{{Main|Bastille}}
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]].


Subgroup: 2.5.7
Optimal tunings:  
* WE: ~2 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢


Comma list: {{Monzo|1426 -596 -15}}
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
Badness (Sintel): 0.438


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
== Higher 2.5 temperaments ==


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


== Movila ==
=== Movila ===
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]].
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]].


Line 359: Line 375:
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


== Vengeance ==
=== Wizz ===
''Main article: [[vengeance]]''<br><br>
{{See also| Chromatic pairs #Wizz }}


Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].  


[[Subgroup]]: 2.5.17
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 78608/78125
 
{{Mapping|legend=2|1 1 1|0 3 7}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
[[Comma list]]: [[15625/15488]]


{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}


== Superquintal ==
{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
[[Subgroup]]: 2.5.13


[[Comma list]]: 64000000/62748517
: [[gencom]]: [125/88 5/4; 15625/15488]


{{Mapping|legend=2|1 5 6|0 -7 -6}}
[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768


: Mapping generators, ~2, ~13/10
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
[[Tp tuning #T2 tuning|RMS error]]: 0.3997


{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
=== Insect ===
 
== Insect ==
[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11


Line 400: Line 409:
{{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}}


== Trader ==
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
 
Sephiroth is the no-7 restriction of [[muggles]].
 
[[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=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}
 
: [[gencom]]: [2 5/4; 65/64 170/169 221/220]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236
 
{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents
 
=== Trader ===
[[Subgroup]]: 2.5.13
[[Subgroup]]: 2.5.13


Line 412: Line 442:


{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
=== Superquintal ===
[[Subgroup]]: 2.5.13
[[Comma list]]: 64000000/62748517
{{Mapping|legend=2|1 5 6|0 -7 -6}}
: Mapping generators, ~2, ~13/10
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}


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


=== Shipwreck ===
=== 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 2 -3 | 0 1 8 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~7/4 = 968.913
 
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}
 
Badness (Sintel): 0.031
 
=== Score ===
{{See also| Chromatic pairs #Score }}
 
[[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 }}
 
: [[gencom]]: [2 11/8; 343/338 847/832]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 540.099
 
{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents
 
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
 
Bossier can be described as the 3 &amp; 17 in the 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=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
 
: [[gencom]]: [2 14/11; 1573/1568 15488/15379]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/11 = 421.309
 
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
 
=== Voltage ===
Voltage is the 3 &amp; 7 temperament in the 2.7.13 subgroup.
 
[[Subgroup]]: 2.7.13
 
[[Comma list]]: [[28672/28561]]


[[Subgroup]]: 2.7.53
{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}


[[Comma list]]: 1048576/1042139
{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}


[[Gencom]]: [2 64/53; 1048576/1042139]
: [[gencom]]: [2, 16/13; 28672/28561]


[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
[[Optimal tuning]]:  
* [[POTE]]: ~2 = 1\1, ~16/13 = 357.677
* [[TOP tuning|POTT]]: ~2 = 1\1, ~16/13 = 357.794 (1200 - 300 log<sub>2</sub>(7))


[[POTE generator]]: ~64/53 = 323.034
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents


=== Ultrakleismic ===
=== Ultrakleismic ===
Line 461: Line 569:
{{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}}


=== Yer (rank 3) ===
=== Shipwreck ===
[[Subgroup]]: 2.11.13.17.19
 
[[Subgroup]]: 2.7.53
 
[[Comma list]]: 1048576/1042139
 
[[Gencom]]: [2 64/53; 1048576/1042139]
 
[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
 
[[POTE generator]]: ~64/53 = 323.034
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
 
=== 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.
 
[[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 }}
 
: [[gencom]]: [2 13/11; 1352/1331]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/11 = 279.318
 
{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents
 
=== 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]]
 
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
 
{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
 
: [[gencom]]: [13/11 11/8; 29282/28561]
 
[[Optimal tuning]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660
 
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents
 
=== 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 }}


[[Comma list]]: 209/208, 2057/2048
: [[gencom]]: [2 143/128; 265837/262144]


[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents


=== Yamablu ===
=== Yamablu ===
Line 498: Line 671:


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19
[[Comma list]]: 209/208, 2057/2048
[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
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