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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.  


== Llywelyn aka shoe ==
== Overview by mapping of 5 ==
{{See also| Chromatic pairs #Shoe }}
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
{{See also| Llywelyn clan #Llywelyn aka shoe }}


[[Subgroup]]: 2.5.7
* For no-fives, see [[#No-threes-or-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.


[[Comma list]]: 4194304/4117715
== 2.5.7 temperaments ==


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
Temperaments discussed elsewhere include
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic 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]]
=== Frostburn ===
{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}


Mapping generators: 2, ~8/7
[[Subgroup]]: 2.5.7


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
[[Comma list]]: 78125/76832


[[Gencom]]: [2 8/7; 4194304/4117715]
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
: Sval mapping generators: ~2, ~28/25


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389


=== 2.5.7.11 subgroup ===
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
Subgroup: 2.5.7.11


Comma list: 176/175, 1310720/1294139
[[Badness]] (Sintel): 0.886


Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
==== 2.5.7.11 ====
Subgroup: 2.5.7.11


Gencom: [2 8/7; 176/175 1310720/1294139]
Comma list: 245/242, 625/616


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}


Optimal tuning (POTE): ~8/7 = 227.114
: Sval mapping generators: ~2, ~28/25


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745


=== 2.5.7.11.13 subgroup ===
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
Subgroup: 2.5.7.11.13


Comma list: 176/175, 640/637, 847/845
Badness (Sintel): 0.463


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


Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
[[Subgroup]]: 2.5.7
 
Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
 
Optimal tuning (POTE): ~8/7 = 227.108
 
{{Optimal ET sequence|legend=1| 16, 21, 37 }}
 
=== 2.5.7.11.13.17 subgroup ===
Subgroup: 2.5.7.11.13.17
 
Comma list: 176/175, 221/200, 640/637, 833/832
 
Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
 
Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
 
Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
 
Optimal tuning (POTE): ~8/7 = 227.242


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Comma list]]: 1071875/1048576


== Didacus ==
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
{{See also| Hemimean clan #Didacus }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}


[[Subgroup]]: 2.5.7
: [[gencom]]: [2 175/128; 1071875/1048576]


[[Comma list]]: 3136/3125
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236


[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}


[[Gencom]]: [2 28/25; 3136/3125]
[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
=== 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]].)


Optimal tuning ([[POTE]]): ~28/25 = 93.772
A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
 
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
 
== Rainy ==
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 106: Line 92:
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents


== Mercy ==
=== French decimal ===
{{See also| Quince clan #Mercy }}
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.


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]].
Subgroup: 2.5.7


[[Subgroup]]: 2.5.7
Comma basis: {{monzo|372 -159 -1}}


[[Comma list]]: 823543/819200
Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
Optimal tuning (CTE): ~5/4 = 386.360


[[Gencom]]: [2 2744/2560; 823543/819200]
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
[[Badness]] (Sintel): 148.6


Optimal tuning ([[POTE]]): ~343/320 = 116.291
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}


=== 2.5.7.13 ===
Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
Optimal tuning (CTE): ~5/4 = 386.361


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
{{Optimal ET sequence|legend=0|264, 733}}, ...


[[Gencom]]: [2 14/13; 343/338 640/637]
Badness (Sintel): 52.150


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


Optimal tuning ([[POTE]]): ~14/13 = 116.094
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


{{Optimal ET sequence|legend=1| 10, 21, 31}}
Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]


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


[[Comma list]]: 170/169, 224/221, 640/637
{{Optimal ET sequence|legend=0|1525, 1789}}, ...


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
Badness (Sintel): 10.518


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
=== Bastille ===
{{Main| Bastille }}


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


Optimal tuning ([[POTE]]): ~14/13 = 115.769
Subgroup: 2.5.7


{{Optimal ET sequence|legend=1| 10, 21, 31}}
Comma list: {{Monzo|1426 -596 -15}}


=== 2.5.7.13.17.19 ===
Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
[[Badness]] (Sintel): 7224.3


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
=== Augment ===
{{See also| Chromatic pairs #Augment }}


Optimal tuning ([[POTE]]): ~14/13 = 115.716
Augment is related to [[augmented]].  


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
[[Subgroup]]: 2.5.7.11


== Pakkanen (rank 3) ==
[[Comma list]]: 56/55, 128/125
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 625/616
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}


{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}


: mapping generators: ~2, ~5, ~11
: [[gencom]]: [5/4 8/7; 56/55 128/125]


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}


== Frostburn ==
[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents
{{See also| Magic family #Quadrimage }}


[[Subgroup]]: 2.5.7
=== 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.  


[[Comma list]]: 78125/76832
Ostara can also refer to a collection of temperaments which temper out 16807/16796.


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
[[Subgroup]]: 2.5.7.11


: Sval mapping generators: ~2, ~28/25
[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢


=== 2.5.7.11 ===
{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
[[Badness]] (Sintel): 11.731


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


: Sval mapping generators: ~2, ~28/25
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢


== Yer (rank 3) ==
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
Badness (Sintel): 3.415


[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


== Yamablu ==
Optimal tunings:
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]].
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢


[[Subgroup]]: 2.11.13.17.19
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


[[Comma list]]: 209/208, 2057/2048, 83521/83486
Badness (Sintel): 1.985


[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19


Optimal tuning ([[POTE]]): ~17/13 = 462.9606
Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]


{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
Optimal tunings:  
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢


== Ostara ==
{{Optimal ET sequence|legend=0| 93, 338, 431 }}
'''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.
Badness (Sintel): 1.285


[[Subgroup]]: 2.5.7.11
=== 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.


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
Subgroup: 2.5.7.11.19.29.31


[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
Optimal tuning (CTE): ~58/31 = 1084.628


=== 2.5.7.11.13 subgroup ===
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
Subgroup: 2.5.7.11.13


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}


Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.


Optimal tuning (POTE): ~1664/1225 = 529.003¢
[[Subgroup]]: 2.5.7.13


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
[[Comma list]]: [[640/637]], [[10985/10976]]


=== 2.5.7.11.13.17 subgroup ===
{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}
Subgroup: 2.5.7.11.13.17


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
: [[gencom]]: [2 16/13; 640/637 10985/10976]


Optimal tuning (POTE): ~1664/1225 = 529.003¢
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}


=== 2.5.7.11.13.17.19 subgroup ===
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents
Subgroup: 2.5.7.11.13.17.19


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
==== Silver ====
{{See also| Chromatic pairs #Silver }}


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.


Optimal tuning (POTE): ~19/14 = 529.003¢
[[Subgroup]]: 2.5.7.13.17


== Pure onzonic ==
[[Comma list]]: [[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
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}


Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]


Optimal tuning (POTE): ~11/8 = 551.370
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711


{{Optimal ET sequence|legend=1|37, 1789}}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}


== Tricesimoprimal miracloid ==
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
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
=== Pakkanen ===
[[Subgroup]]: 2.5.7.11


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
[[Comma list]]: 625/616


Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}


Optimal tuning (CTE): ~58/31 = 1084.628
: mapping generators: ~2, ~5, ~11


{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653


== French decimal ==
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
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
[[Badness]] (Sintel): 0.573


Comma basis: {{monzo|372 -159 -1}}
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
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]].


Optimal tuning (CTE): ~5/4 = 386.360
[[Subgroup]]: 2.5.7.11


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
[[Comma list]]: 5021863/5000000


=== 2.5.7.11 subgroup ===
{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
Subgroup: 2.5.7.11


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
: mapping generators: ~2, ~5, ~100/77


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


Optimal tuning (CTE): ~5/4 = 386.361
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}


{{Optimal ET sequence|legend=1|264, 733}}, ...
[[Badness]] (Sintel): 1.862


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


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
Comma list: 847/845, 1001/1000


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}


Optimal tuning (CTE): ~5/4 = 386.361
Optimal tunings:
* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}


== Mabon ==
Badness (Sintel): 0.179
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.


Subgroup: 2.9.7
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


Comma basis: 44957696/43046721
Comma list: 170/169, 221/220, 847/845


Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}


Optimal tuning (CTE): ~729/448 = 870.792
Optimal tunings:  
* WE: ~2 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢


{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}


=== 2.9.7.11 subgroup ===
Badness (Sintel): 0.438
Subgroup: 2.9.7.11


Comma basis: 896/891, 1331/1296
== Higher 2.5 temperaments ==


Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


Optimal tuning (CTE): ~16/11 = 870.966
=== 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]].


{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
[[Subgroup]]: 2.5.11


== Bastille ==
[[Comma list]]: 1331/1280
{{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
[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]


Comma list: {{Monzo|1426 -596 -15}}
[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].  


== Shipwreck ==
[[Subgroup]]: 2.5.11


[[Subgroup]]: 2.7.53
[[Comma list]]: [[15625/15488]]


[[Comma list]]: 1048576/1042139
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}


[[Gencom]]: [2 64/53; 1048576/1042139]
{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}


[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
: [[gencom]]: [125/88 5/4; 15625/15488]


[[POTE generator]]: ~64/53 = 323.034
[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}


== Movila ==
[[Tp tuning #T2 tuning|RMS error]]: 0.3997
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]].


=== Insect ===
[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11


[[Comma list]]: 1331/1280
[[Comma list]]: 33275/32768
 
{{Mapping|legend=2|1 0 5|0 3 -2}}
 
: Mapping generators, ~2, ~[[55/32]]
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
 
{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}


[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
Sephiroth is the no-7 restriction of [[muggles]].  


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
[[Subgroup]]: 2.5.11.13.17


== Mavericks ==
[[Comma list]]: 65/64, 170/169, 221/220


[[Subgroup]]: 2.13.19
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}


[[Comma list]]: 47525504/47045881
{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}


[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
: [[gencom]]: [2 5/4; 65/64 170/169 221/220]


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}


== Vengeance ==
[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents
''Main article: [[vengeance]]''<br><br>


Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
=== Trader ===
[[Subgroup]]: 2.5.13


[[Subgroup]]: 2.5.17
[[Comma list]]: [[26/25]]


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


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079


{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}


== Superquintal ==
=== Superquintal ===
[[Subgroup]]: 2.5.13
[[Subgroup]]: 2.5.13


Line 449: Line 456:
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}


== Insect ==
== No-threes-or-fives subgroup temperaments ==
[[Subgroup]]: 2.5.11
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Berylic → [[4th-octave temperaments #Berylic|4th-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]]
* 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 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]]
 
{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}


[[Comma list]]: 33275/32768
{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}


{{Mapping|legend=2|1 0 5|0 3 -2}}
: [[gencom]]: [2, 16/13; 28672/28561]


: Mapping generators, ~2, ~[[55/32]]
[[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))


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}


{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents


== Supraminor ==
=== Ultrakleismic ===
[[Subgroup]]: 2.7.17
[[Subgroup]]: 2.7.17


Line 475: Line 556:
{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}
{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}


== Countersupraminor ==
=== Counterultrakleismic ===
[[Subgroup]]: 2.7.17
[[Subgroup]]: 2.7.17


Line 488: 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}}


== Trader ==
=== Shipwreck ===
[[Subgroup]]: 2.5.13
 
[[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 }}
 
: [[gencom]]: [2 143/128; 265837/262144]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents
 
=== 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]].
 
[[Subgroup]]: 2.11.13.17.19
 
[[Comma list]]: 209/208, 2057/2048, 83521/83486
 
[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
 
Optimal tuning ([[POTE]]): ~17/13 = 462.9606
 
{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
 
=== Mavericks ===
 
[[Subgroup]]: 2.13.19
 
[[Comma list]]: 47525504/47045881
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
 
{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}


[[Comma list]]: [[26/25]]
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19


{{Mapping|legend=2|1 2 3|0 1 2}}
[[Comma list]]: 209/208, 2057/2048


: Mapping generators, ~2, ~[[5/4]]
[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638


{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
{{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]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"