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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 → [[Hemimean clan #Didacus]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Llywelyn a.k.a. shoe → [[Llywelynsmic clan #Llywelyn a.k.a. shoe]]
* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
* [[The Jacobins]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
** Genojacobin
 
** Onzonic
=== Frostburn ===
** Estates general
{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}
** Sextilimeans
 
** Pure bastille
[[Subgroup]]: 2.5.7
** Double bastille
 
** Acrosextilififths
[[Comma list]]: 78125/76832
** Declaration of rights
** Eternal revolutionary


== 2.5.7 temperaments ==
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
=== 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 generators: ~2, ~28/25


A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


[[Comma list]]: [[2100875/2097152]]
[[Badness]] (Sintel): 0.886


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
==== 2.5.7.11 ====
Subgroup: 2.5.7.11


[[Gencom]]: [2 256/245; 2100875/2097152]
Comma list: 245/242, 625/616


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


Optimal tuning ([[POTE]]): ~256/245 = 77.205
: Sval mapping generators: ~2, ~28/25


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


=== Bastille ===
Badness (Sintel): 0.463
{{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]].
=== 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 &amp; 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.


Subgroup: 2.5.7
[[Subgroup]]: 2.5.7


Comma list: {{Monzo|1426 -596 -15}}
[[Comma list]]: 1071875/1048576


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
: [[gencom]]: [2 175/128; 1071875/1048576]


=== Frostburn ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236
{{See also| Magic family #Quadrimage }}


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}


[[Comma list]]: 78125/76832
[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
=== 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 generators: ~2, ~28/25
A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
[[Subgroup]]: 2.5.7


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
[[Comma list]]: [[2100875/2097152]]


==== 2.5.7.11 ====
[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
[[Gencom]]: [2 256/245; 2100875/2097152]


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


: Sval mapping generators: ~2, ~28/25
Optimal tuning ([[POTE]]): ~256/245 = 77.205


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents


=== French decimal ===
=== French decimal ===
Line 99: Line 99:
Comma basis: {{monzo|372 -159 -1}}
Comma basis: {{monzo|372 -159 -1}}


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


Optimal tuning (CTE): ~5/4 = 386.360
Optimal tuning (CTE): ~5/4 = 386.360


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
[[Badness]] (Sintel): 148.6


==== 2.5.7.11 subgroup ====
==== 2.5.7.11 subgroup ====
Line 114: Line 116:
Optimal tuning (CTE): ~5/4 = 386.361
Optimal tuning (CTE): ~5/4 = 386.361


{{Optimal ET sequence|legend=1|264, 733}}, ...
{{Optimal ET sequence|legend=0|264, 733}}, ...
 
Badness (Sintel): 52.150


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
Line 125: Line 129:
Optimal tuning (CTE): ~5/4 = 386.361
Optimal tuning (CTE): ~5/4 = 386.361


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
{{Optimal ET sequence|legend=0|1525, 1789}}, ...
 
Badness (Sintel): 10.518
 
=== Bastille ===
{{Main| Bastille }}
 
Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an [[wikipedia:Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].
 
Subgroup: 2.5.7
 
Comma list: {{Monzo|1426 -596 -15}}
 
Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
 
Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
 
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
 
[[Badness]] (Sintel): 7224.3
 
=== Augment ===
{{See also| Chromatic pairs #Augment }}
 
Augment is related to [[augmented]].
 
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 56/55, 128/125
 
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
 
{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}
 
: [[gencom]]: [5/4 8/7; 56/55 128/125]
 
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275
 
{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents


=== Ostara ===
=== Ostara ===
Line 138: 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 149: 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 160: 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 ===
Line 187: Line 249:
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...


=== Mercy ===
=== Huntington ===
{{See also| Quince clan #Mercy }}
{{See also| Chromatic pairs #Huntington }}


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


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


[[Comma list]]: 823543/819200
[[Comma list]]: [[640/637]], [[10985/10976]]


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


[[Gencom]]: [2 2744/2560; 823543/819200]
{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
: [[gencom]]: [2 16/13; 640/637 10985/10976]


Optimal tuning ([[POTE]]): ~343/320 = 116.291
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}


==== 2.5.7.13 ====
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
==== Silver ====
{{See also| Chromatic pairs #Silver }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.


[[Gencom]]: [2 14/13; 343/338 640/637]
[[Subgroup]]: 2.5.7.13.17


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]


Optimal tuning ([[POTE]]): ~14/13 = 116.094
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}


{{Optimal ET sequence|legend=1| 10, 21, 31}}
{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


==== 2.5.7.13.17 ====
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
=== Pakkanen ===
[[Subgroup]]: 2.5.7.11


Optimal tuning ([[POTE]]): ~14/13 = 115.769
[[Comma list]]: 625/616


{{Optimal ET sequence|legend=1| 10, 21, 31}}
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}


==== 2.5.7.13.17.19 ====
: mapping generators: ~2, ~5, ~11
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
[[Badness]] (Sintel): 0.573


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


Optimal tuning ([[POTE]]): ~14/13 = 115.716
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 ET sequence|legend=1| 10, 21, 31, 52f }}
[[Subgroup]]: 2.5.7.11


=== Pakkanen (rank 3) ===
[[Comma list]]: 5021863/5000000
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 625/616
{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}


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


: mapping generators: ~2, ~5, ~11
[[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]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
[[Badness]] (Sintel): 1.862


== Higher 2.5 temperaments ==
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


=== Pure onzonic ===
Comma list: 847/845, 1001/1000
Named for the French word for eleven, ''onze'', since the generator is 11/8. Initially defined for 2.5.11.13, but it can be extended.


Pure onzonic is the temperament that was initially referred to as "jacobin" before it was pointed out that the same name would be reserved for the rank-5 temperamnet tempering out 6656/6655 alone (see above). 
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}


[[Subgroup]]: 2.5.11.13
Optimal tunings:  
* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢


[[Comma list]]: 6656/6655, {{monzo| -119 -46 15 47 }}
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}


[[Sval]] [[mapping]]: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
Badness (Sintel): 0.179


[[Optimal tuning]] ([[CTE]]): ~11/8 = 551.370
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


{{Optimal ET sequence|legend=1| 37, 1789 }}
Comma list: 170/169, 221/220, 847/845


==== Septimal onzonic ====
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
Septimal onzonic in between the 2.5.11.13 subgroup adds the mapping for 7.


Subgroup: 2.5.7.11.13
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, 200126927/200000000, 41322093568/41259765625
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}


Sval mapping: [{{val| 1 74 114 3 74 }}, {{val| 0 -156 -242 1 -153 }}]
Badness (Sintel): 0.438


Optimal tuning (CTE): ~11/8 = 551.369
== Higher 2.5 temperaments ==


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


=== Movila ===
=== Movila ===
Line 306: 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 320: 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 345: 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}}


=== Vengeance ===
=== Vengeance ===
Line 363: Line 473:
== 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]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic]]
* 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)]]
* 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 410: Line 598:
{{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 447: Line 686:


{{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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