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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]].)


A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
: Sval mapping generators: ~2, ~28/25


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


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


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
[[Badness]] (Sintel): 0.886


[[Gencom]]: [2 256/245; 2100875/2097152]
==== 2.5.7.11 ====
Subgroup: 2.5.7.11


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
Comma list: 245/242, 625/616


Optimal tuning ([[POTE]]): ~256/245 = 77.205
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
: Sval mapping generators: ~2, ~28/25


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745


=== Bastille ===
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
{{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]].
Badness (Sintel): 0.463


Subgroup: 2.5.7
=== 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.


Comma list: {{Monzo|1426 -596 -15}}
[[Subgroup]]: 2.5.7


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
[[Comma list]]: 1071875/1048576


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}


=== Frostburn ===
: [[gencom]]: [2 175/128; 1071875/1048576]
{{See also| Magic family #Quadrimage }}


[[Subgroup]]: 2.5.7
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236


[[Comma list]]: 78125/76832
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents


: Sval mapping generators: ~2, ~28/25
=== 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]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
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, 29, 35, 41, 47 }}
[[Subgroup]]: 2.5.7


==== 2.5.7.11 ====
[[Comma list]]: [[2100875/2097152]]
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
[[Gencom]]: [2 256/245; 2100875/2097152]


: Sval mapping generators: ~2, ~28/25
[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Optimal tuning ([[POTE]]): ~256/245 = 77.205


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
 
[[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=1| 93, 245e, 338, 431, 1386c }}
{{Optimal ET sequence|legend=0| 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 }}
 
Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.
 
[[Subgroup]]: 2.5.7.13
 
[[Comma list]]: [[640/637]], [[10985/10976]]
 
{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}
 
{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}
 
: [[gencom]]: [2 16/13; 640/637 10985/10976]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002
 
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents


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]].
==== Silver ====
{{See also| Chromatic pairs #Silver }}


[[Subgroup]]: 2.5.7
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.


[[Comma list]]: 823543/819200
[[Subgroup]]: 2.5.7.13.17


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


[[Gencom]]: [2 2744/2560; 823543/819200]
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}


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


Optimal tuning ([[POTE]]): ~343/320 = 116.291
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711


==== 2.5.7.13 ====
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents


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


[[Gencom]]: [2 14/13; 343/338 640/637]
[[Comma list]]: 625/616


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


Optimal tuning ([[POTE]]): ~14/13 = 116.094
: mapping generators: ~2, ~5, ~11


{{Optimal ET sequence|legend=1| 10, 21, 31}}
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653


==== 2.5.7.13.17 ====
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
[[Badness]] (Sintel): 0.573


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
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]].


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


Optimal tuning ([[POTE]]): ~14/13 = 115.769
[[Comma list]]: 5021863/5000000


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


==== 2.5.7.13.17.19 ====
: mapping generators: ~2, ~5, ~100/77
[[Subgroup]]: 2.5.7.13.17.19


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


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
[[Badness]] (Sintel): 1.862


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


Optimal tuning ([[POTE]]): ~14/13 = 115.716
Comma list: 847/845, 1001/1000


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}


=== Pakkanen (rank 3) ===
Optimal tunings:
[[Subgroup]]: 2.5.7.11
* 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]]: 625/616
Comma list: 170/169, 221/220, 847/845


{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}


: mapping generators: ~2, ~5, ~11
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 tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
Badness (Sintel): 0.438


== Higher 2.5 temperaments ==
== Higher 2.5 temperaments ==
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


=== Movila ===
=== Movila ===
Line 278: 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 292: 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 317: 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 335: 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 382: 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 419: 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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