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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.
 
Temperaments discussed elsewhere include
* Jubilic → [[Jubilismic clan #Jubilic]]


== 2.5.7 temperaments ==
== 2.5.7 temperaments ==


=== Llywelyn aka shoe ===
Temperaments discussed elsewhere include
{{See also| Chromatic pairs #Shoe }}
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
{{See also| Llywelyn clan #Llywelyn aka shoe }}
* 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 }}


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


Comma list: 176/175, 1310720/1294139
Comma list: 245/242, 625/616


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


Gencom: [2 8/7; 176/175 1310720/1294139]
: Sval mapping generators: ~2, ~28/25


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745


Optimal tuning (POTE): ~8/7 = 227.114
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
Badness (Sintel): 0.463


==== 2.5.7.11.13 subgroup ====
=== Mabilic ===
Subgroup: 2.5.7.11.13
{{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.
 
Comma list: 176/175, 640/637, 847/845
 
Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
 
Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
 
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 }}
 
=== Didacus ===
{{See also| Hemimean clan #Didacus }}
 
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]


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


[[Comma list]]: 3136/3125
[[Comma list]]: 1071875/1048576


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


[[Gencom]]: [2 28/25; 3136/3125]
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
: [[gencom]]: [2 175/128; 1071875/1048576]


Optimal tuning ([[POTE]]): ~28/25 = 93.772
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents


=== Rainy ===
=== Rainy ===
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]]. Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> = [[128/125]] (the 2.3.5 diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the [[cloudy comma]]), which has a similar size and role to the 2.3.5 diesis so might be considered the 2.3.7 diesis, in that if you temper it, it imparts significant error on the tuning of 8/7 (so that it is now ~8.8{{cent}} sharp), similar to how tempering 128/125 imparts significant error on the tuning of 5/4 (so that it is ~13.7{{cent}} sharp). Combined with the small size of these "dieses", it implies that the locations of 5/4 and 8/7 are in some sense awkward to conceptualize due to the small but not insignificant size of the remnant with the octave, so it is might be favorable to equate 128/125 with the cloudy comma to create a general-purpose diesis. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)
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]].)


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.
Line 124: 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
 
=== Frostburn ===
{{See also| Magic family #Quadrimage }}
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 78125/76832
 
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}


: Sval mapping generators: ~2, ~28/25
[[Comma list]]: 56/55, 128/125


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


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}


==== 2.5.7.11 ====
: [[gencom]]: [5/4 8/7; 56/55 128/125]
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}


: Sval mapping generators: ~2, ~28/25
[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents
 
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
 
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}


=== Ostara ===
=== Ostara ===
Line 227: Line 182:
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢  
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
[[Badness]] (Sintel): 11.731


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
Line 238: Line 197:
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]


Optimal tuning (POTE): ~1664/1225 = 529.003¢  
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
 
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
Badness (Sintel): 3.415


==== 2.5.7.11.13.17 subgroup ====
==== 2.5.7.11.13.17 subgroup ====
Line 249: Line 212:
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


Optimal tuning (POTE): ~1664/1225 = 529.003¢
Optimal tunings:  
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
 
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
Badness (Sintel): 1.985


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


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


Optimal tuning (POTE): ~19/14 = 529.003¢
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢
 
{{Optimal ET sequence|legend=0| 93, 338, 431 }}
 
Badness (Sintel): 1.285


=== Tricesimoprimal miracloid ===
=== 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.
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.


Line 275: Line 249:
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...


=== French decimal ===
=== Huntington ===
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.
{{See also| Chromatic pairs #Huntington }}


Subgroup: 2.5.7
Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.


Comma basis: {{monzo|372 -159 -1}}
[[Subgroup]]: 2.5.7.13


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
[[Comma list]]: [[640/637]], [[10985/10976]]


Optimal tuning (CTE): ~5/4 = 386.360
{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}


==== 2.5.7.11 subgroup ====
: [[gencom]]: [2 16/13; 640/637 10985/10976]
Subgroup: 2.5.7.11


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002


Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}


Optimal tuning (CTE): ~5/4 = 386.361
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents


{{Optimal ET sequence|legend=1|264, 733}}, ...
==== Silver ====
{{See also| Chromatic pairs #Silver }}


==== 2.5.7.11.13 subgroup ====
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.  
Subgroup: 2.5.7.11.13


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
[[Subgroup]]: 2.5.7.13.17


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]


Optimal tuning (CTE): ~5/4 = 386.361
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}
 
{{Optimal ET sequence|legend=1|1525, 1789}}, ...
 
=== Bastille ===
{{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|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


Comma list: {{Monzo|1426 -596 -15}}
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents


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


Line 337: Line 304:
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}


== Higher 2.5 temperaments ==
[[Badness]] (Sintel): 0.573
 
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}
 
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]]: 2.5.7.11
 
[[Comma list]]: 5021863/5000000
 
{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
 
: mapping generators: ~2, ~5, ~100/77
 
[[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 ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
 
[[Badness]] (Sintel): 1.862
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma list: 847/845, 1001/1000
 
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}
 
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=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
 
Badness (Sintel): 0.179
 
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
 
Comma list: 170/169, 221/220, 847/845


=== Pure onzonic ===
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
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 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: 6656/6655, {{monzo| -119 -46 15 47 }}
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}


Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
Badness (Sintel): 0.438


Optimal tuning (POTE): ~11/8 = 551.370
== Higher 2.5 temperaments ==


{{Optimal ET sequence|legend=1|37, 1789}}
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


=== Movila ===
=== Movila ===
Line 364: Line 374:


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}
Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].
[[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 }}
: [[gencom]]: [125/88 5/4; 15625/15488]
[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3997


=== Insect ===
=== Insect ===
Line 378: 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}}


=== Superquintal ===
=== Sephiroth ===
[[Subgroup]]: 2.5.13
{{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 }}


[[Comma list]]: 64000000/62748517
{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}


{{Mapping|legend=2|1 5 6|0 -7 -6}}
: [[gencom]]: [2 5/4; 65/64 170/169 221/220]


: Mapping generators, ~2, ~13/10
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}


{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents


=== Trader ===
=== Trader ===
Line 404: Line 443:
{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}


=== Vengeance ===
=== Superquintal ===
''Main article: [[vengeance]]''<br><br>
[[Subgroup]]: 2.5.13


Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
[[Comma list]]: 64000000/62748517


[[Subgroup]]: 2.5.17
{{Mapping|legend=2|1 5 6|0 -7 -6}}


[[Comma list]]: 78608/78125
: Mapping generators, ~2, ~13/10


{{Mapping|legend=2|1 1 1|0 3 7}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
 
{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}


== No-threes-or-fives subgroup temperaments ==
== No-threes-or-fives subgroup temperaments ==
Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone]]
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Berylic → [[4th-octave temperaments #Berylic]]
* Berylic → [[4th-octave temperaments #Berylic|4th-octave temperaments]]
* [[21st-octave temperaments #21-23-commatic]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* [[31st-octave temperaments #31-17/13-commatic]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
* [[37th-octave temperaments #37-11-commatic (rank-1)]]
* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)|37th-octave temperaments]]
* etc.
* 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 }}
{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
: [[gencom]]: [2, 16/13; 28672/28561]
[[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 ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents


=== Ultrakleismic ===
=== Ultrakleismic ===
Line 468: Line 583:
{{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 }}


=== Yer (rank 3) ===
=== Lovecraft ===
[[Subgroup]]: 2.11.13.17.19
{{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 505: 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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