Xenharmonic Wiki

No-threes subgroup temperaments: Difference between revisions

← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,046 edits
No edit summary
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,582 edits
 
(69 intermediate revisions by 9 users not shown)
Line 1: Line 1:
{{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.  


== Overview by mapping of 5 ==
== Overview by mapping of 5 ==
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* Trader has 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.


* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
== Temperaments with a 2.5.7 gene ==
* 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.
* Didacus is with a ~28/25 generator, two of which give the ~5/4.
* Ostara, movila and vengeance are with variety expressed generator, three of which give the ~5/2.
* Insect is with a ~55/32 generator, three of which give the ~5/1.
* Frostburn is with a ~28/25 generator, four of which give the ~8/5.
Others are more far.
 
Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* Jubilic → [[Jubilismic clan #Jubilic]]
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* 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]]


== 2.5.7 temperaments ==
=== Rainy ===
In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), 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 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.


=== Llywelyn aka shoe ===
A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.
{{See also| Chromatic pairs #Shoe }}
{{See also| Llywelyn clan #Llywelyn aka shoe }}


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


[[Comma list]]: 4194304/4117715
[[Comma list]]: [[2100875/2097152]]
 
{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
 
{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~256/245
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
: error map: {{val| 0.000 -0.267 -0.454 }}
 
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}
 
[[Badness]] (Sintel): 0.156
 
=== Augment ===
{{See also| Chromatic pairs #Augment }}
 
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.
 
[[Subgroup]]: 2.5.7


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
[[Comma list]]: 128/125


Mapping generators: 2, ~8/7
{{Mapping|legend=2| 3 7 0 | 0 0 1 }}


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7


[[Gencom]]: [2 8/7; 4194304/4117715]
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}
: [[error map]]: {{val| -2.962 +6.776 -0.040 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 974.3418{{c}}
: error map: {{val| 0.000 +13.686 +5.516 }}


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
[[Badness]] (Sintel): 0.296


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


Comma list: 176/175, 1310720/1294139
Comma list: 56/55, 128/125
 
Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}
 
Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}
 
Optimal tunings:
* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}


Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}


Gencom: [2 8/7; 176/175 1310720/1294139]
Badness (Sintel): 0.196


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
=== Frostburn ===
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].


Optimal tuning (POTE): ~8/7 = 227.114
[[Subgroup]]: 2.5.7


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Comma list]]: 78125/76832


==== 2.5.7.11.13 subgroup ====
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
Subgroup: 2.5.7.11.13
: mapping generators: ~2, ~28/25


Comma list: 176/175, 640/637, 847/845
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}
: [[error map]]: {{val| +0.346 -2.630 +2.189 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 204.2027{{c}}
: error map: {{val| 0.000 -3.125 +1.755 }}


Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
[[Badness]] (Sintel): 0.886


Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


Optimal tuning (POTE): ~8/7 = 227.108
Comma list: 245/242, 625/616


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25


==== 2.5.7.11.13.17 subgroup ====
Optimal tunings:
Subgroup: 2.5.7.11.13.17
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}


Comma list: 176/175, 221/200, 640/637, 833/832
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
Badness (Sintel): 0.463


Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
=== Mabilic ===
{{Main| Mabilic and trismegistus }}
{{See also| Chromatic pairs #Mabilic }}


Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
Mabilic is the no-3 [[restriction]] 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.


Optimal tuning (POTE): ~8/7 = 227.242
[[Subgroup]]: 2.5.7


{{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 }}
: mapping generators: ~2, ~175/128


[[Subgroup]]: 2.5.7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}
: [[error map]]: {{val| +1.254 -1.698 -1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}
: error map: {{val| 0.000 -4.701 -4.846 }}


[[Comma list]]: 3136/3125
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}


[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]
[[Badness]] (Sintel): 1.70


[[Gencom]]: [2 28/25; 3136/3125]
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.


Optimal tuning ([[POTE]]): ~28/25 = 93.772
[[Subgroup]]: 2.5.7


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
[[Comma list]]: 40960000/40353607


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}


=== Rainy ===
{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
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]].)
: mapping generators: ~2, ~80/49


A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
: error map: {{val| 0.000 -0.091 +2.828 }}


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}


[[Comma list]]: [[2100875/2097152]]
Badness (Sintel): 1.87


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13


[[Gencom]]: [2 256/245; 2100875/2097152]
Comma list: 640/637, 10985/10976


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}


Optimal tuning ([[POTE]]): ~256/245 = 77.205
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
Optimal tunings:
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}


=== Mercy ===
Badness (Sintel): 0.319
{{See also| Quince clan #Mercy }}


Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
==== Silver ====
{{See also| Chromatic pairs #Silver }}


[[Subgroup]]: 2.5.7
Silver can be described as the 10 & 37 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]
Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}


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


Optimal tuning ([[POTE]]): ~343/320 = 116.291
Optimal tunings:  
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}


==== 2.5.7.13 ====
Badness (Sintel): 0.504
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
=== Ostara ===
Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.


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


[[Gencom]]: [2 14/13; 343/338 640/637]
Comma list: 8589934592/8544921875, 30691800524/30517578125


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773


Optimal tuning ([[POTE]]): ~14/13 = 116.094
Optimal tunings:  
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}


{{Optimal ET sequence|legend=1| 10, 21, 31}}
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


==== 2.5.7.13.17 ====
Badness (Sintel): 11.7
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
Optimal tunings:  
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}


Optimal tuning ([[POTE]]): ~14/13 = 115.769
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


{{Optimal ET sequence|legend=1| 10, 21, 31}}
Badness (Sintel): 3.42


==== 2.5.7.13.17.19 ====
==== 2.5.7.11.13.17 subgroup ====
[[Subgroup]]: 2.5.7.13.17.19
Subgroup: 2.5.7.11.13.17


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
Optimal tunings:  
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


Optimal tuning ([[POTE]]): ~14/13 = 115.716
Badness (Sintel): 1.99


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19


=== Frostburn ===
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
{{See also| Magic family #Quadrimage }}


[[Subgroup]]: 2.5.7
Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}


[[Comma list]]: 78125/76832
Optimal tunings:  
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


: Sval mapping generators: ~2, ~28/25
Badness (Sintel): 1.29


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
=== Bastille ===
{{Main| Bastille }}


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
Bastille is 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]].


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


Comma list: 245/242, 625/616
[[Comma list]]: {{monzo| 1426 -596 -15 }}


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


: Sval mapping generators: ~2, ~28/25
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
: error map: {{val| 0.000 +0.041 +0.018 }}


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


=== Ostara ===
=== No-threes naiad (rank-3) ===
'''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.
{{Todo|inline=1| review | comment = Devise the permanent name for this temperament; determine its status with respect to its extensions linked below. }}
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


Ostara can also refer to a collection of temperaments which temper out 16807/16796.
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
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
[[Comma list]]: 5021863/5000000


[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{c}}
: [[error map]]: {{val| +0.080 +0.507 -1.318 -0.643 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.7404{{c}}, ~77/50 = 745.4102{{c}}
: error map: {{val| 0.000 +0.427 -0.445 -0.808 }}


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}
 
[[Badness]] (Sintel): 1.86


==== 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 list: 1001/1000, 34420736/34328125, 5670699008/5661858125
Comma list: 847/845, 1001/1000
 
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}


Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
Optimal tunings:  
* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}


Optimal tuning (POTE): ~1664/1225 = 529.003¢
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
Badness (Sintel): 0.179


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


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
Comma list: 170/169, 221/220, 847/845
 
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}
 
Optimal tunings:
* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}
 
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}
 
Badness (Sintel): 0.438
 
== Temperaments with a higher 2.5.''p'' gene ==
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
 
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}
 
Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[astrology]] and [[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 }}
: mapping generators: ~125/88, ~5/4


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
[[Optimal tuning]]s:
* [[WE]]: ~125/88 = 600.1831{{c}}, ~5/4 = 383.8848{{c}}
: [[error map]]: {{val| +0.366 -1.697 +1.252 }}
* [[CWE]]: ~125/88 = 600.0000{{c}}, ~5/4 = 383.9977{{c}}
: error map: {{val| 0.000 -2.316 +0.675 }}
 
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}
 
[[Badness]] (Sintel): 0.266
 
=== Insect ===
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 33275/32768
 
{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}
: mapping generators, ~2, ~55/32
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1238{{c}}, ~55/32 = 928.5003{{c}}
: [[error map]]: {{val| +1.124 -0.813 -2.700 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~55/32 = 927.7384{{c}}
: error map: {{val| 0.000 -3.099 -6.975 }}
 
{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}
 
[[Badness]] (Sintel): 0.564
 
=== 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]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 1331/1280
 
{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.0339{{c}}, ~11/8 = 528.4296{{c}}
: [[error map]]: {{val| +3.034 +2.009 -13.787 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 528.1575{{c}}
: error map: {{val| 0.000 -1.841 -23.160 }}
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
 
[[Badness]] (Sintel): 0.718
 
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
 
Sephiroth is the common [[restriction]] of [[muggles]] and [[submerged]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 34375/32768
 
{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}


Optimal tuning (POTE): ~1664/1225 = 529.003¢
{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3290{{c}}, ~5/4 = 373.6097{{c}}
: [[error map]]: {{val| +3.329 -6.046 -2.722 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 372.1586{{c}}
: error map: {{val| 0.000 -14.155 -12.111 }}


==== 2.5.7.11.13.17.19 subgroup ====
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
Subgroup: 2.5.7.11.13.17.19


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
[[Badness]] (Sintel): 1.85


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13


Optimal tuning (POTE): ~19/14 = 529.003¢
Comma list: 65/64, 6875/6656


=== Tricesimoprimal miracloid ===
Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
Described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.


Subgroup: 2.5.7.11.19.29.31
Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
Optimal tunings:  
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}


Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}


Optimal tuning (CTE): ~58/31 = 1084.628
Badness (Sintel): 0.410


{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
==== 2.5.11.13.17 subgroup ====
Subgroup: 2.5.11.13.17


=== French decimal ===
Comma list: 65/64, 170/169, 221/220
Conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.


Subgroup: 2.5.7
Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}


Comma basis: {{monzo|372 -159 -1}}
Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
Optimal tunings:  
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}


Optimal tuning (CTE): ~5/4 = 386.360
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
Badness (Sintel): 0.299


==== 2.5.7.11 subgroup ====
=== Trader ===
Subgroup: 2.5.7.11
[[Subgroup]]: 2.5.13


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
[[Comma list]]: 26/25


Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
{{Mapping|legend=2| 1 2 3 | 0 1 2 }}
: mapping generators, ~2, ~5/4


Optimal tuning (CTE): ~5/4 = 386.361
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.0216{{c}}, ~5/4 = 410.2152{{c}}
: [[error map]]: {{val| -1.978 +19.945 -26.033 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 408.9029{{c}}
: error map: {{val| 0.000 +22.589 -22.722 }}


{{Optimal ET sequence|legend=1|264, 733}}, ...
{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}


==== 2.5.7.11.13 subgroup ====
[[Badness]] (Sintel): 0.138
Subgroup: 2.5.7.11.13


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
=== Superquintal ===
[[Subgroup]]: 2.5.13


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
[[Comma list]]: 64000000/62748517


Optimal tuning (CTE): ~5/4 = 386.361
{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~20/13


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5925{{c}}, ~20/13 = 740.6286{{c}}
: [[error map]]: {{val| -0.408 -1.098 +3.244 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 740.8058{{c}}
: error map: {{val| 0.000 -0.673 +4.307 }}


=== Bastille ===
{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}
{{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
[[Badness]] (Sintel): 1.93


Comma list: {{Monzo|1426 -596 -15}}
== No-threes no-fives subgroup temperaments ==
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* 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.


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
=== Amaranthine ===
{{See also| No-fives subgroup temperaments #Chrysanthemum }}


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
Amaranthine is the high-accuracy 2.7.11-subgroup strong [[restriction]] of [[Gamelismic clan #11-limit 3|undecimal mothra]].


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
[[Subgroup]]: 2.7.11


=== Pakkanen (rank 3) ===
[[Comma list]]: 5767168/5764801
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 625/616
{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7


{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9846{{c}}, ~7/4 = 968.9078{{c}}
: [[error map]]: {{val| -0.015 +0.051 -0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 968.9174{{c}}
: error map: {{val| 0.000 +0.091 +0.021 }}


: mapping generators: ~2, ~5, ~11
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
Badness (Sintel): 0.0309


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].


== Higher 2.5 temperaments ==
[[Subgroup]]: 2.7.11


=== Pure onzonic ===
[[Comma list]]: 1372/1331
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| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7


Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
[[Optimal tuning]]s:  
* [[WE]]: ~14/11 = 399.8041{{c}}, ~7/4 = 963.1666{{c}}
: [[error map]]: {{val| -0.588 -6.835 +10.281 }}
* [[CWE]]: ~14/11 = 400.0000{{c}}, ~4/4 = 962.7466{{c}}
: error map: {{val| 0.000 -6.079 +11.429 }}


Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}


Optimal tuning (POTE): ~11/8 = 551.370
[[Badness]] (Sintel): 0.475


{{Optimal ET sequence|legend=1|37, 1789}}
=== Score ===
{{See also| Chromatic pairs #Score }}


=== Movila ===
Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.  
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]].


[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.7.11.13


[[Comma list]]: 1331/1280
[[Comma list]]: 343/338, 847/832


[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
{{Mapping|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}


=== Insect ===
{{Optimal ET sequence|legend=1| 9, 11, 20 }}
[[Subgroup]]: 2.5.11


[[Comma list]]: 33275/32768
[[Badness]] (Sintel): 0.368


{{Mapping|legend=2|1 0 5|0 3 -2}}
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}


: Mapping generators, ~2, ~[[55/32]]
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
[[Subgroup]]: 2.7.11


{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
[[Comma list]]: 214358881/210827008


=== Superquintal ===
{{Mapping|legend=2| 1 0 1 | 0 8 7 }}
[[Subgroup]]: 2.5.13


[[Comma list]]: 64000000/62748517
{{Mapping|legend=3| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11


{{Mapping|legend=2|1 5 6|0 -7 -6}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1886{{c}}, ~14/11 = 421.2661{{c}}
: [[error map]]: {{val| +0.189 +1.303 -2.266 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2365{{c}}
: error map: {{val| 0.000 +1.066 -2.662 }}


: Mapping generators, ~2, ~13/10
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
[[Badness]] (Sintel): 1.73


{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
==== 2.7.11.13 subgroup ====
Subgroup: 2.7.11.13


=== Trader ===
Comma list: 1573/1568, 15488/15379
[[Subgroup]]: 2.5.13


[[Comma list]]: [[26/25]]
Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}


{{Mapping|legend=2|1 2 3|0 1 2}}
Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}


: Mapping generators, ~2, ~[[5/4]]
Optimal tunings:  
* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}


{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
Badness (Sintel): 0.307


=== Vengeance ===
=== Voltage ===
''Main article: [[vengeance]]''<br><br>
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 7<sup>1/4</sup> of 842.2065{{c}}, which is also the CTC (constrained Tenney–Chebyshevian) tuning.


Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
[[Subgroup]]: 2.7.13


[[Subgroup]]: 2.5.17
[[Comma list]]: 28672/28561


[[Comma list]]: 78608/78125
{{Mapping|legend=2| 1 0 3 | 0 4 1 }}


{{Mapping|legend=2|1 1 1|0 3 7}}
{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~13


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}
: [[error map]]: {{val| -0.217 -0.143 +0.991 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/8 = 842.2568{{c}}
: error map: {{val| 0.000 +0.201 +1.729 }}


{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}


== No-threes-or-fives subgroup temperaments ==
[[Badness]] (Sintel): 0.115
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone]]
* Berylic → [[4th-octave temperaments #Berylic]]
* [[21st-octave temperaments #21-23-commatic]]
* [[31st-octave temperaments #31-17/13-commatic]]
* [[37th-octave temperaments #37-11-commatic (rank-1)]]
* etc.


=== Ultrakleismic ===
=== Ultrakleismic ===
Line 433: Line 646:
[[Comma list]]: 4913/4802
[[Comma list]]: 4913/4802


{{Mapping|legend=2|1 2 3|0 3 4}}
{{Mapping|legend=2| 1 2 3 | 0 3 4 }}
: mapping generators, ~2, ~17/14


: Mapping generators, ~2, ~[[17/14]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1379{{c}}, ~17/14 = 324.3440{{c}}
: [[error map]]: {{val| +0.138 +4.482 -7.166 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 324.3738{{c}}
: error map: {{val| 0.000 +4.295 -7.460 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446
{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}


{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}
[[Badness]] (Sintel): 0.460


=== Counterultrakleismic ===
=== Counterultrakleismic ===
Line 446: Line 664:
[[Comma list]]: 2024782584832/2015993900449
[[Comma list]]: 2024782584832/2015993900449


{{Mapping|legend=2|1 0 1|0 10 11}}
{{Mapping|legend=2| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14


: Mapping generators, ~2, ~[[17/14]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9723{{c}}, ~17/14 = 336.8586{{c}}
: [[error map]]: {{val| -0.028 -0.240 +0.462 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 336.8621{{c}}
: error map: {{val| 0.000 -0.205 +0.528 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858
{{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}}
[[Badness]] (Sintel): 0.860


=== Shipwreck ===
=== Shipwreck ===
[[Subgroup]]: 2.7.53
[[Subgroup]]: 2.7.53


[[Comma list]]: 1048576/1042139
[[Comma list]]: 1048576/1042139


[[Gencom]]: [2 64/53; 1048576/1042139]
{{Mapping|legend=2| 1 0 6 | 0 3 -1 }}]
: mapping generators, ~2, ~64/53
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6967{{c}}, ~64/53 = 323.1839{{c}}
: [[error map]]: {{val| -0.303 +0.119 +1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/53 = 323.1959{{c}}
: error map: {{val| 0.000 +0.762 +3.300 }}
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
 
[[Badness]] (Sintel): 0.224
 
=== Lovecraft ===
{{See also | Chromatic pairs #Lovecraft }}
 
Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, tempers out [[1352/1331]], and 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 }}
: mapping generators, ~2, ~13/11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5223{{c}}, ~13/11 = 279.2064{{c}}
: [[error map]]: {{val| -0.478 +5.662 -4.341 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/11 = 278.9918{{c}}
: error map: {{val| 0.000 +6.666 -3.552 }}
 
{{Optimal ET sequence|legend=1| 4, 9, 13, 30, 43, 73, 116e }}
 
[[Badness]] (Sintel): 0.175
 
=== 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 }}
: mapping generators, ~2, ~143/128
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8795{{c}}, ~143/128 = 182.5017{{c}}
: [[error map]]: {{val| +0.880 -1.174 -2.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~143/128 = 182.4386{{c}}
: error map: {{val| 0.000 -4.002 -5.405 }}
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}
 
[[Badness]] (Sintel): 0.451


[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
=== Blackbirds ===
{{See also | Chromatic pairs #Blackbirds }}


[[POTE generator]]: ~64/53 = 323.034
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.  


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 29282/28561


=== Yer (rank 3) ===
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
: mapping generators, ~13/11, ~11


[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~13/11 = 299.9728{{c}}, ~11/8 = 546.6107{{c}}
: [[error map]]: {{val| -0.109 -5.033 +5.730 }}
* [[CWE]]: ~13/11 = 300.0000{{c}}, ~11/8 = 546.4664{{c}}
: error map: {{val| 0.000 -4.852 +5.939 }}


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
[[Badness]] (Sintel): 0.668


=== Yamablu ===
=== 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]].
Yamablu, with a generator of ~26/17, 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). It extends the 2.11.13-subgroup temperament tempering out 556573090931/549755813888. The [[Kite's Genchain mode numbering|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].  


[[Subgroup]]: 2.11.13.17.19
[[Subgroup]]: 2.11.13.17.19
Line 486: Line 773:
[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 209/208, 2057/2048, 83521/83486


[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}
 
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}
 
[[Badness]] (Sintel): 0.386


Optimal tuning ([[POTE]]): ~17/13 = 462.9606
=== Arsenic ===
Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.  


{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
[[Subgroup]]: 2.11.13.17.19


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432
 
{{Mapping|legend=2| 33 0 8 249 26 | 0 1 1 -1 1 }}
: mapping generators: ~247/242, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}
* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}
 
{{Optimal ET sequence|legend=1| 33, 165, 198, 231, 495h }}
 
[[Badness]] (Sintel): 2.30
 
=== Berylic ===
Berylic tempers out the [[berylisma]] in the 2.11.37 subgroup, representing the fact that [[44/37]] is a {{w|continued fraction}} convergent to 2<sup>1/4</sup> the fourth root of 2. Beryllic is an example of a temperament which has an astronomically low [[badness]], being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by edo systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within an obscure subgroup, 2.11.37.
 
If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.
 
[[Subgroup]]: 2.11.37
 
[[Comma list]]: 1874161/1874048
 
{{Mapping|legend=2| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~44/37 = 300.0003{{c}}, ~11/8 = 551.3211{{c}}
: [[error map]]: {{val| +0.001 +0.007 -0.017 }}
* [[CWE]]: ~44/37 = 300.0000{{c}}, ~11/8 = 551.3237{{c}}
: error map: {{val| 0.000 +0.006 -0.020 }}
 
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}
 
[[Badness]] (Sintel): 0.00188


=== Mavericks ===
=== Mavericks ===
[[Subgroup]]: 2.13.19
[[Subgroup]]: 2.13.19


[[Comma list]]: 47525504/47045881
[[Comma list]]: 47525504/47045881


[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8817{{c}}, ~26/19 = 539.9150{{c}}
: [[error map]]: {{val| -0.118 -1.156 +1.825 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/19 = 539.9280{{c}}
: error map: {{val| 0.000 -0.960 +2.127 }}
 
{{Optimal ET sequence|legend=1| 9, 11, 20 }}
 
[[Badness]] (Sintel): 0.559
 
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19
 
[[Comma list]]: 209/208, 2057/2048
 
{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4447{{c}}, ~11/8 = 548.4929{{c}}, ~13/8 = 841.3613{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 548.2193{{c}}, ~13/8 = 841.4707{{c}}
 
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
[[Badness]] (Sintel): 0.106


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"