No-threes subgroup temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (2/)
Delete tricesimoprimal miracloid. + todo for no-3 naiad
 
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* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
=== 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.
A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.
[[Subgroup]]: 2.5.7
[[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
[[Comma list]]: 128/125
{{Mapping|legend=2| 3 7 0 | 0 0 1 }}
{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7
[[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 ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}
[[Badness]] (Sintel): 0.296
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
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}}
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}
Badness (Sintel): 0.196


=== Frostburn ===
=== Frostburn ===
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[[Badness]] (Sintel): 1.70
[[Badness]] (Sintel): 1.70


=== Rainy ===
=== Huntington ===
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.
{{See also| Chromatic pairs #Huntington }}


A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  


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


[[Comma list]]: [[2100875/2097152]]
[[Comma list]]: 40960000/40353607


{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}


{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~256/245
: mapping generators: ~2, ~80/49


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
: error map: {{val| 0.000 -0.267 -0.454 }}
: error map: {{val| 0.000 -0.091 +2.828 }}


{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}
{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}


[[Badness]] (Sintel): 0.156
Badness (Sintel): 1.87


=== French decimal ===
==== 2.5.7.13 subgroup ====
French decimal is 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: 2.5.7.13
 
[[Subgroup]]: 2.5.7


[[Comma list]]: {{monzo| 372 -159 -1 }}
Comma list: 640/637, 10985/10976


{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
: mapping generators: ~2, ~5


[[Optimal tuning]]s:
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
: mapping generators: ~2, ~13/8
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
: error map: {{val| 0.000 +0.046 +0.019 }}
 
{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
 
[[Badness]] (Sintel): 148
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
 
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
 
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}


Badness (Sintel): 52.2
Badness (Sintel): 0.319


==== 2.5.7.11.13 subgroup ====
==== Silver ====
Subgroup: 2.5.7.11.13
{{See also| Chromatic pairs #Silver }}


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.


Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
Subgroup: 2.5.7.13.17


Optimal tunings:  
Comma list: 170/169, 640/637, 5525/5488
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}
Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}


Badness (Sintel): 10.5
Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
 
=== Bastille ===
{{Main| Bastille }}
 
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]].
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: {{monzo| 1426 -596 -15 }}
 
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}
 
[[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 ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}
 
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
 
=== 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
 
[[Comma list]]: 128/125
 
{{Mapping|legend=2| 3 7 0 | 0 0 1 }}
 
{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7
 
[[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 ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}
 
[[Badness]] (Sintel): 0.296
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
 
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 }}
: mapping generators: ~5/4, ~7


Optimal tunings:  
Optimal tunings:  
* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}


Badness (Sintel): 0.196
Badness (Sintel): 0.504


=== Ostara ===
=== Ostara ===
Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.  
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.  
 
Ostara can also refer to a collection of temperaments which temper out 16807/16796.{{clarify}}


Subgroup: 2.5.7.11
Subgroup: 2.5.7.11
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Comma list: 8589934592/8544921875, 30691800524/30517578125
Comma list: 8589934592/8544921875, 30691800524/30517578125


{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773
: mapping generators: ~2, ~5120/3773


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Badness (Sintel): 1.29
Badness (Sintel): 1.29


=== Tricesimoprimal miracloid ===
=== French decimal ===
{{See also| Tricesimoprimal miracloid/Eliora's approach }}
French decimal is 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.


Tricesimoprimal miracloid is 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


Subgroup: 2.5.7.11.19.29.31
[[Comma list]]: {{monzo| 372 -159 -1 }}


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
: mapping generators: ~2, ~5


Subgroup-val mapping: {{mapping| 1 -42 -2 -15 -12 -61 -61 | 0 461 50 192 169 685 686 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
: error map: {{val| 0.000 +0.046 +0.019 }}


Optimal tunings:
{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
* WE: ~2 = 1200.0079{{c}}, ~31/29 = 115.3723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/29 = 115.3716{{c}}


{{Optimal ET sequence|legend=0| 52, 1737, 1789 }}
[[Badness]] (Sintel): 148


Badness (Sintel): 4.51
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


=== Huntington ===
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
{{See also| Chromatic pairs #Huntington }}


Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}
 
Subgroup: 2.5.7.13
 
Comma list: [[640/637]], [[10985/10976]]
 
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
 
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}
{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}


Badness (Sintel): 0.319
Badness (Sintel): 52.2


==== Silver ====
==== 2.5.7.11.13 subgroup ====
{{See also| Chromatic pairs #Silver }}
Subgroup: 2.5.7.11.13


Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


Subgroup: 2.5.7.13.17
Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}


Comma list: [[170/169]], [[640/637]], [[5525/5488]]
Optimal tunings:  
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}


Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
Badness (Sintel): 10.5


Optimal tunings:
=== Bastille ===
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
{{Main| Bastille }}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


{{Optimal ET sequence|legend=1| 10, 27, 37, 47, 84, 131, 178g }}
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]].


Badness (Sintel): 0.504
[[Subgroup]]: 2.5.7


=== Pakkanen ===
[[Comma list]]: {{monzo| 1426 -596 -15 }}
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 625/616
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
 
: mapping generators: ~2, ~{{monzo| -380 159 4 }}
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
: mapping generators: ~2, ~5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.6499{{c}}, ~5/4 = 380.2990{{c}}, ~7/4 = 969.2326{{c}}
* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
: [[error map]]: {{val| +0.650 -4.715 +1.706 +2.595 }}
: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.6249{{c}}, ~7/4 = 969.6520{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
: error map: {{val| 0.000 -5.689 +0.826 +1.530 }}
: error map: {{val| 0.000 +0.041 +0.018 }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


[[Badness]] (Sintel): 0.573
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


=== No-threes naiad ===
=== No-threes naiad (rank-3) ===
{{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 }}
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


Line 418: Line 399:
Temperaments discussed elsewhere include:
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
=== 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 }}
[[Optimal tuning]] (CTE): ~2 = 1200.000{{c}}, ~11/8 = 529.846{{c}}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


=== Wizz ===
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}
{{See also| Chromatic pairs #Wizz }}


Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].  
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
[[Subgroup]]: 2.5.11


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


{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
Line 446: Line 414:
: mapping generators: ~125/88, ~5/4
: mapping generators: ~125/88, ~5/4


[[Optimal tuning]] ([[POTE]]): ~125/88 = 600.000{{c}}, ~5/4 = 383.768{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3997
[[Badness]] (Sintel): 0.266


=== Insect ===
=== Insect ===
Line 460: Line 432:
: mapping generators, ~2, ~55/32
: mapping generators, ~2, ~55/32


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~55/32 = 928.032{{c}}
[[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 }}
{{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 ===
=== Sephiroth ===
Line 469: Line 467:
Sephiroth is the no-7 [[restriction]] of [[muggles]].  
Sephiroth is the no-7 [[restriction]] of [[muggles]].  


[[Subgroup]]: 2.5.11.13.17
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 34375/32768
 
{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
 
{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5
 
[[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 }}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
 
[[Badness]] (Sintel): 1.85
 
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13
 
Comma list: 65/64, 6875/6656
 
Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
 
Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}
 
Badness (Sintel): 0.410


[[Comma list]]: 65/64, 170/169, 221/220
==== 2.5.11.13.17 subgroup ====
Subgroup: 2.5.11.13.17


{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
Comma list: 65/64, 170/169, 221/220


{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}
Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
: mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~5/4 = 372.236{{c}}
Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}


{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}


[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents
Badness (Sintel): 0.299


=== Trader ===
=== Trader ===
[[Subgroup]]: 2.5.13
[[Subgroup]]: 2.5.13


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


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~5/4 = 407.079{{c}}
[[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| 3, 20c, 23c, 26c }}
{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}
[[Badness]] (Sintel): 0.138


=== Superquintal ===
=== Superquintal ===
Line 501: Line 543:
[[Comma list]]: 64000000/62748517
[[Comma list]]: 64000000/62748517


{{Mapping|legend=2| 1 5 6 | 0 -7 -6 }}
{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~13/10
: mapping generators, ~2, ~20/13


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~13/10 = 459.281{{c}}
[[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 }}


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


== No-threes no-fives subgroup temperaments ==
== No-threes no-fives subgroup temperaments ==
Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Berylic → [[4th-octave temperaments #Berylic|4th-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
Line 526: Line 573:
[[Comma list]]: 5767168/5764801
[[Comma list]]: 5767168/5764801


{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}
{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}
 
Badness (Sintel): 0.0309
 
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 1372/1331
 
{{Mapping|legend=2| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~7/4 = 968.913{{c}}
[[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 }}


{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}
{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}


Badness (Sintel): 0.031
[[Badness]] (Sintel): 0.475


=== Score ===
=== Score ===
{{See also| Chromatic pairs #Score }}
{{See also| Chromatic pairs #Score }}
Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.


[[Subgroup]]: 2.7.11.13
[[Subgroup]]: 2.7.11.13
Line 543: Line 617:
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}
{{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 }}
{{Mapping|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~11/8 = 540.099{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}


{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
{{Optimal ET sequence|legend=1| 9, 11, 20 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents
[[Badness]] (Sintel): 0.368


=== Bossier ===
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
{{See also| Chromatic pairs #Bossier }}


Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.  
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].  


[[Subgroup]]: 2.7.11.13
[[Subgroup]]: 2.7.11


[[Comma list]]: [[1573/1568]], [[15488/15379]]
[[Comma list]]: 214358881/210827008


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


{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
{{Mapping|legend=3| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11
: mapping generators: ~2, ~14/11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~14/11 = 421.309{{c}}
[[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 }}
 
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}
 
[[Badness]] (Sintel): 1.73
 
==== 2.7.11.13 subgroup ====
Subgroup: 2.7.11.13
 
Comma list: 1573/1568, 15488/15379
 
Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}
 
Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}


{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}
{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
Badness (Sintel): 0.307


=== Voltage ===
=== Voltage ===
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.  
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.  


[[Subgroup]]: 2.7.13
[[Subgroup]]: 2.7.13


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


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


{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~16/13
: mapping generators: ~2, ~13


[[Optimal tuning]]:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~16/13 = 357.677{{c}}
* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}
* [[TOP tuning|POTT]]: ~2 = 1200.000{{c}}, ~16/13 = 357.794{{c}} (1200 - 300 log<sub>2</sub>(7))
: [[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| 3, 7, 10, 27, 37, 47, 57, 104 }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents
[[Badness]] (Sintel): 0.115


=== Ultrakleismic ===
=== Ultrakleismic ===
Line 598: Line 697:


{{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]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 324.446{{c}}
[[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 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 612: Line 717:
: mapping generators, ~2, ~17/14
: mapping generators, ~2, ~17/14


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 336.858{{c}}
[[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 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 ===
Line 624: Line 735:
: mapping generators, ~2, ~64/53
: mapping generators, ~2, ~64/53


[[Optimal tuning]]s ([[POTE]]): ~2 = 1200.000{{c}}, ~64/53 = 323.034{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
[[Badness]] (Sintel): 0.224


=== Lovecraft ===
=== Lovecraft ===
{{See also | Chromatic pairs #Lovecraft }}
{{See also | Chromatic pairs #Lovecraft }}


Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.  
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
[[Subgroup]]: 2.11.13


[[Comma list]]: [[1352/1331]]
[[Comma list]]: 1352/1331


{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
Line 642: Line 759:
: mapping generators, ~2, ~13/11
: mapping generators, ~2, ~13/11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/11 = 279.318{{c}}
[[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| 13, 30, 43, 73, 116 }}
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents
[[Badness]] (Sintel): 0.451


=== Blackbirds ===
=== Blackbirds ===
Line 655: Line 799:
[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13


[[Comma list]]: [[29282/28561]]
[[Comma list]]: 29282/28561


{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
Line 662: Line 806:
: mapping generators, ~13/11, ~11
: mapping generators, ~13/11, ~11


[[Optimal tuning]] ([[POTE]]): ~13/11 = 300.000{{c}}, ~11/8 = 546.660{{c}}
[[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 ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents
[[Badness]] (Sintel): 0.668


=== Bluebirds ===
=== Yamablu ===
{{Distinguish| Bluebird }}
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]].
{{See also| Chromatic pairs #Bluebirds }}


[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: [[265837/262144]]
[[Comma list]]: 209/208, 2057/2048, 83521/83486


{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17


{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
[[Optimal tuning]]s:
: mapping generators, ~2, ~143/128
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~143/128 = 182.368{{c}}
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}


{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}
[[Badness]] (Sintel): 0.386


[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents
=== 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.


=== Yamablu ===
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.
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].


[[Subgroup]]: 2.11.13.17.19
[[Subgroup]]: 2.11.37


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 1874161/1874048


{{Mapping|legend=2| 1 5 1 1 0 | 0 -4 7 8 11 }}
{{Mapping|legend=2| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~17/13 = 462.9606{{c}}
[[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| 13, 44, 57, 70 }}
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Badness]] (Sintel): 0.00188


=== Mavericks ===
=== Mavericks ===
Line 708: Line 862:


{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19
[[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 tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~26/19 = 539.886{{c}}
{{Optimal ET sequence|legend=1| 9, 11, 20 }}


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
[[Badness]] (Sintel): 0.559


=== Yer (rank 3) ===
=== Yer (rank 3) ===
Line 719: Line 880:


{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
{{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]] ([[TE tuning|TE]]): ~2 = 1200.4457{{c}}, ~11/8 = 548.4934{{c}}, ~16/13 = 358.638{{c}}
[[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| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
[[Badness]] (Sintel): 0.106


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]

Latest revision as of 16:20, 3 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of subgroup temperaments which omit the prime harmonic of 3.

Overview by mapping of 5

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.
  • French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
  • Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
  • Insect has a ~55/32 generator, three of which give the ~5/1.
  • Frostburn has a ~28/25 generator, four of which give the ~8/5.

Others have a more complex mapping of 5.

Temperaments with a 2.5.7 gene

Temperaments discussed elsewhere include

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)3 (128/125, the lesser diesis) with (2/1)/(8/7)5 (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 ¢ sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7 ¢). By tempering out their difference, stacked 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.

A highly notable tuning of rainy not shown here is 311edo, which is 140 + 171 so tuned between them.

Subgroup: 2.5.7

Comma list: 2100875/2097152

Subgroup-val mapping: [⟨1 2 3], ⟨0 5 -3]]

Gencom mapping: [⟨1 0 2 3], ⟨0 0 5 -3]]

mapping generators: ~2, ~256/245

Optimal tunings:

  • WE: ~2 = 1200.0939 ¢, ~256/245 = 77.2107 ¢
error map: ⟨+0.094 -0.072 -0.176]
  • CWE: ~2 = 1200.0000 ¢, ~256/245 = 77.2093 ¢
error map: ⟨0.000 -0.267 -0.454]

Optimal ET sequence: 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

Comma list: 128/125

Subgroup-val mapping: [⟨3 7 0], ⟨0 0 1]]

Gencom mapping: [⟨3 0 7 0], ⟨0 0 0 1]]

mapping generators: ~5/4, ~7

Optimal tunings:

  • WE: ~5/4 = 399.0128 ¢, ~7/4 = 974.7085 ¢
error map: ⟨-2.962 +6.776 -0.040]
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 974.3418 ¢
error map: ⟨0.000 +13.686 +5.516]

Optimal ET sequence: 3, 6, 15, 21, 27, 102ccd, 129ccd

Badness (Sintel): 0.296

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 56/55, 128/125

Subgroup-val mapping: [⟨3 7 0 2], ⟨0 0 1 1]]

Gencom mapping: [⟨3 0 7 0 2], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~5/4 = 398.9239 ¢, ~7/4 = 969.1106 ¢
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 968.4397 ¢

Optimal ET sequence: 3, 6, 15, 21

Badness (Sintel): 0.196

Frostburn

Frostburn is the common restriction of quadrimage and baldy.

Subgroup: 2.5.7

Comma list: 78125/76832

Subgroup-val mapping: [⟨1 3 4], ⟨0 -4 -7]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • WE: ~2 = 1200.3462 ¢, ~28/25 = 204.3386 ¢
error map: ⟨+0.346 -2.630 +2.189]
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 204.2027 ¢
error map: ⟨0.000 -3.125 +1.755]

Optimal ET sequence: 6, 29, 35, 41, 47

Badness (Sintel): 0.886

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 245/242, 625/616

Subgroup-val mapping: [⟨1 3 4 5], ⟨0 -4 -7 -9]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • WE: ~2 = 1200.6817 ¢, ~28/25 = 205.0734 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 204.8199 ¢

Optimal ET sequence: 6, 23de, 29, 35, 41

Badness (Sintel): 0.463

Mabilic

Main article: Mabilic and trismegistus
See also: Chromatic pairs § Mabilic

Mabilic is the no-3 restriction of armodue, semabila, and trismegistus. It is the 7 & 9 temperament in the 2.5.7 subgroup, and tempers out 1071875/1048576, the mabilisma.

Subgroup: 2.5.7

Comma list: 1071875/1048576

Subgroup-val mapping: [⟨1 1 5], ⟨0 3 -5]]

Gencom mapping: [⟨1 0 1 5], ⟨0 0 3 -5]]

mapping generators: ~2, ~175/128

Optimal tunings:

  • WE: ~2 = 1201.2543 ¢, ~175/128 = 527.7872 ¢
error map: ⟨+1.254 -1.698 -1.491]
  • CWE: ~2 = 1200.0000 ¢, ~175/128 = 527.2041 ¢
error map: ⟨0.000 -4.701 -4.846]

Optimal ET sequence: 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd

Badness (Sintel): 1.70

Huntington

See also: Chromatic pairs § Huntington

Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.

Subgroup: 2.5.7

Comma list: 40960000/40353607

Subgroup-val mapping: [⟨1 -4 0], ⟨0 9 4]]

Gencom mapping: [⟨1 0 -4 0], ⟨0 0 9 4]]

mapping generators: ~2, ~80/49

Optimal tunings:

  • WE: ~2 = 1199.5781 ¢, ~80/49 = 842.6730 ¢
error map: ⟨-0.422 -0.569 +1.866]
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9136 ¢
error map: ⟨0.000 -0.091 +2.828]

Optimal ET sequence: 7c, 10, 27, 37, 84, 121

Badness (Sintel): 1.87

2.5.7.13 subgroup

Subgroup: 2.5.7.13

Comma list: 640/637, 10985/10976

Subgroup-val mapping: [⟨1 -4 0 3], ⟨0 9 4 1]]

Gencom mapping: [⟨1 0 -4 0 0 3], ⟨0 0 9 4 0 1]]

mapping generators: ~2, ~13/8

Optimal tunings:

  • WE: ~2 = 1199.4788 ¢, ~13/8 = 842.6318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.9447 ¢

Optimal ET sequence: 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf

Badness (Sintel): 0.319

Silver

See also: Chromatic pairs § Silver

Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.

Subgroup: 2.5.7.13.17

Comma list: 170/169, 640/637, 5525/5488

Subgroup-val mapping: [⟨1 -4 0 3 9], ⟨0 9 4 1 -7]]

Gencom mapping: [⟨1 0 -4 0 0 3 9], ⟨0 0 9 4 0 1 -7]]

Optimal tunings:

  • WE: ~2 = 1200.0932 ¢, ~13/8 = 842.7764 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.7143 ¢

Optimal ET sequence: 10, 27, 37, 47, 84, 131, 178g

Badness (Sintel): 0.504

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.

Subgroup: 2.5.7.11

Comma list: 8589934592/8544921875, 30691800524/30517578125

Subgroup-val mapping: [⟨1 1 20 -49], ⟨0 3 -39 119]]

mapping generators: ~2, ~5120/3773

Optimal tunings:

  • WE: ~2 = 1199.9115 ¢, ~5120/3773 = 528.9650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5120/3773 = 529.0037 ¢

Optimal ET sequence: 93, 245e, 338, 955c, 1386c

Badness (Sintel): 11.7

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125

Subgroup-val mapping: [⟨1 1 20 -49 35], ⟨0 3 -39 119 -71]]

Optimal tunings:

  • WE: ~2 = 1199.9194 ¢, ~1664/1225 = 528.9681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1664/1225 = 529.0036 ¢

Optimal ET sequence: 93, 245e, 338, 431, 1386c

Badness (Sintel): 3.42

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251

Subgroup-val mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]

Optimal tunings:

  • WE: ~2 = 1199.9054 ¢, ~1664/1225 = 528.9628 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1664/1225 = 529.0046 ¢

Optimal ET sequence: 93, 338, 431, 955c, 1386cg

Badness (Sintel): 1.99

2.5.7.11.13.17.19 subgroup

Subgroup: 2.5.7.11.13.17.19

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

Subgroup-val mapping: [⟨1 1 20 -49 35 42 21], ⟨0 3 -39 119 -71 -86 -38]]

Optimal tunings:

  • WE: ~2 = 1199.9081 ¢, ~19/14 = 528.9639 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/14 = 529.0045 ¢

Optimal ET sequence: 93, 338, 431, 955c, 1386cg

Badness (Sintel): 1.29

French decimal

French decimal is 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

Comma list: [372 -159 -1⟩

Subgroup-val mapping: [⟨1 0 372], ⟨0 1 -159]]

mapping generators: ~2, ~5

Optimal tunings:

  • WE: ~2 = 1199.9901 ¢, ~5/4 = 386.3562 ¢
error map: ⟨-0.010 +0.023 +0.000]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3595 ¢
error map: ⟨0.000 +0.046 +0.019]

Optimal ET sequence: 205, 264, 733, 997, 2258, 3255, 7507, 10762

Badness (Sintel): 148

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: [-49 8 17 -5⟩, [45 -27 10 -3⟩

Subgroup-val mapping: [⟨1 0 372 1255], ⟨0 1 -159 -539]]

Optimal tunings:

  • WE: ~2 = 1200.0130 ¢, ~5/4 = 386.3653 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3611 ¢

Optimal ET sequence: 264, 997e, 1261e, 1525, 1789

Badness (Sintel): 52.2

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625

Subgroup-val mapping: [⟨1 0 372 1255 -398], ⟨0 1 -159 -539 173]]

Optimal tunings:

  • WE: ~2 = 1200.0137 ¢, ~5/4 = 386.3655 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3611 ¢

Optimal ET sequence: 261, 1261e, 1525, 1789

Badness (Sintel): 10.5

Bastille

Main article: Bastille

Bastille is 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 double bastille.

Subgroup: 2.5.7

Comma list: [1426 -596 -15⟩

Subgroup-val mapping: [⟨1 -4 254], ⟨0 15 -596]]

mapping generators: ~2, ~[-380 159 4⟩

Optimal tunings:

  • WE: ~2 = 1199.9911 ¢, ~[-380 159 4⟩ = 505.7532 ¢
error map: ⟨-0.009 +0.020 +0.001]
  • CWE: ~2 = 1200.0000 ¢, ~[-380 159 4⟩ = 505.7570 ¢
error map: ⟨0.000 +0.041 +0.018]

Optimal ET sequence: 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd

Badness (Sintel): 7.18 × 103

No-threes naiad (rank-3)

Todo: review

Devise the permanent name for this temperament; determine its status with respect to its extensions linked below.

See also: Wizardharry clan § Naiad, and Werckismic temperaments § Seminaiad

This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands tridec and naiadec.

Subgroup: 2.5.7.11

Comma list: 5021863/5000000

Subgroup-val mapping: [⟨1 0 -2 3], ⟨0 1 1 1], ⟨0 0 4 -3]]

mapping generators: ~2, ~5, ~77/50

Optimal tunings:

  • WE: ~2 = 1200.0805 ¢, ~5/4 = 386.6593 ¢, ~77/50 = 745.4622 ¢
error map: ⟨+0.080 +0.507 -1.318 -0.643]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.7404 ¢, ~77/50 = 745.4102 ¢
error map: ⟨0.000 +0.427 -0.445 -0.808]

Optimal ET sequence: 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

Subgroup: 2.5.7.11.13

Comma list: 847/845, 1001/1000

Subgroup-val mapping: [⟨1 0 -2 3 2], ⟨0 1 1 1 1], ⟨0 0 4 -3 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0343 ¢, ~5/4 = 386.6098 ¢, ~20/13 = 745.4658 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.6458 ¢, ~20/13 = 745.4431 ¢

Optimal ET sequence: 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e

Badness (Sintel): 0.179

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Comma list: 170/169, 221/220, 847/845

Subgroup-val 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 ¢, ~5/4 = 386.6701 ¢, ~17/11 = 745.3706 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.1074 ¢, ~17/11 = 745.0940 ¢

Optimal ET sequence: 16, 21, 29g, 37, 66g, 87g, 124g

Badness (Sintel): 0.438

Temperaments with a higher 2.5.p gene

Temperaments discussed elsewhere include:

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

Subgroup-val mapping: [⟨2 0 -7], ⟨0 1 3]]

Gencom mapping: [⟨2 0 4 0 5], ⟨0 0 1 0 3]]

mapping generators: ~125/88, ~5/4

Optimal tunings:

  • WE: ~125/88 = 600.1831 ¢, ~5/4 = 383.8848 ¢
error map: ⟨+0.366 -1.697 +1.252]
  • CWE: ~125/88 = 600.0000 ¢, ~5/4 = 383.9977 ¢
error map: ⟨0.000 -2.316 +0.675]

Optimal ET sequence: 6, 16, 22, 28, 50, 122, 172, 222, 394c

Badness (Sintel): 0.266

Insect

Subgroup: 2.5.11

Comma list: 33275/32768

Subgroup-val mapping: [⟨1 0 5], ⟨0 3 -2]]

mapping generators, ~2, ~55/32

Optimal tunings:

  • WE: ~2 = 1201.1238 ¢, ~55/32 = 928.5003 ¢
error map: ⟨+1.124 -0.813 -2.700]
  • CWE: ~2 = 1200.0000 ¢, ~55/32 = 927.7384 ¢
error map: ⟨0.000 -3.099 -6.975]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 3], ⟨0 3 1]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1203.0339 ¢, ~11/8 = 528.4296 ¢
error map: ⟨+3.034 +2.009 -13.787]
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 528.1575 ¢
error map: ⟨0.000 -1.841 -23.160]

Optimal ET sequence: 7, 9, 16, 25, 41e, 66ee

Badness (Sintel): 0.718

Sephiroth

See also: Chromatic pairs § Sephiroth

Sephiroth is the no-7 restriction of muggles.

Subgroup: 2.5.11

Comma list: 34375/32768

Subgroup-val mapping: [⟨1 0 15], ⟨0 1 -5]]

Gencom mapping: [⟨1 0 0 0 15], ⟨0 0 1 0 -5]]

mapping generators: ~2, ~5

Optimal tunings:

  • WE: ~2 = 1203.3290 ¢, ~5/4 = 373.6097 ¢
error map: ⟨+3.329 -6.046 -2.722]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 372.1586 ¢
error map: ⟨0.000 -14.155 -12.111]

Optimal ET sequence: 3, 10, 13, 16, 29, 132cceee

Badness (Sintel): 1.85

2.5.11.13 subgroup

Subgroup: 2.5.11.13

Comma list: 65/64, 6875/6656

Subgroup-val mapping: [⟨1 0 15 6], ⟨0 1 -5 -1]]

Gencom mapping: [⟨1 0 0 0 15 6], ⟨0 0 1 0 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1203.3825 ¢, ~5/4 = 373.6318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 372.1519 ¢

Optimal ET sequence: 3, 10, 13, 16, 29, 132cceeeff

Badness (Sintel): 0.410

2.5.11.13.17 subgroup

Subgroup: 2.5.11.13.17

Comma list: 65/64, 170/169, 221/220

Subgroup-val mapping: [⟨1 0 15 6 11], ⟨0 1 -5 -1 -3]]

Gencom mapping: [⟨1 0 0 0 15 6 11], ⟨0 0 1 0 -5 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1203.6741 ¢, ~5/4 = 373.3775 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 371.6773 ¢

Optimal ET sequence: 3, 10, 13, 16, 29g, 129ccceeffgggg

Badness (Sintel): 0.299

Trader

Subgroup: 2.5.13

Comma list: 26/25

Subgroup-val mapping: [⟨1 2 3], ⟨0 1 2]]

mapping generators, ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1198.0216 ¢, ~5/4 = 410.2152 ¢
error map: ⟨-1.978 +19.945 -26.033]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 408.9029 ¢
error map: ⟨0.000 +22.589 -22.722]

Optimal ET sequence: 3, 20c, 23c, 26c

Badness (Sintel): 0.138

Superquintal

Subgroup: 2.5.13

Comma list: 64000000/62748517

Subgroup-val mapping: [⟨1 -2 0], ⟨0 7 6]]

mapping generators, ~2, ~20/13

Optimal tunings:

  • WE: ~2 = 1199.5925 ¢, ~20/13 = 740.6286 ¢
error map: ⟨-0.408 -1.098 +3.244]
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 740.8058 ¢
error map: ⟨0.000 -0.673 +4.307]

Optimal ET sequence: 8, 13, 21, 34, 81, 115

Badness (Sintel): 1.93

No-threes no-fives subgroup temperaments

Temperaments discussed elsewhere include

Amaranthine

See also: No-fives subgroup temperaments § Chrysanthemum

Amaranthine is the high-accuracy 2.7.11-subgroup strong restriction of undecimal mothra.

Subgroup: 2.7.11

Comma list: 5767168/5764801

Subgroup-val mapping: [⟨1 0 -19], ⟨0 1 8]]

mapping generators: ~2, ~7

Optimal tunings:

  • WE: ~2 = 1199.9846 ¢, ~7/4 = 968.9078 ¢
error map: ⟨-0.015 +0.051 -0.010]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 968.9174 ¢
error map: ⟨0.000 +0.091 +0.021]

Optimal ET sequence: 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd

Badness (Sintel): 0.0309

Argument

Argument tempers out 1372/1331 in the 2.7.11 subgroup. It is the no-3 restriction of augment.

Subgroup: 2.7.11

Comma list: 1372/1331

Subgroup-val mapping: [⟨3 0 2], ⟨0 1 1]]

mapping generators: ~14/11, ~7

Optimal tunings:

  • WE: ~14/11 = 399.8041 ¢, ~7/4 = 963.1666 ¢
error map: ⟨-0.588 -6.835 +10.281]
  • CWE: ~14/11 = 400.0000 ¢, ~4/4 = 962.7466 ¢
error map: ⟨0.000 -6.079 +11.429]

Optimal ET sequence: 6, 9, 15, 36, 51e, 66e

Badness (Sintel): 0.475

Score

See also: Chromatic pairs § Score

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

Subgroup: 2.7.11.13

Comma list: 343/338, 847/832

Subgroup-val mapping: [⟨1 1 3 1], ⟨0 4 1 6]]

Gencom mapping: [⟨1 0 0 1 3 1], ⟨0 0 0 4 1 6]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1201.5484 ¢, ~11/8 = 540.7963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 540.5091 ¢

Optimal ET sequence: 9, 11, 20

Badness (Sintel): 0.368

Bossier

See also: Chromatic pairs § Bossier

Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out 1573/1568 and 15488/15379.

Subgroup: 2.7.11

Comma list: 214358881/210827008

Subgroup-val mapping: [⟨1 0 1], ⟨0 8 7]]

Gencom mapping: [⟨1 0 0 0 1], ⟨0 0 0 8 7]]

mapping generators: ~2, ~14/11

Optimal tunings:

  • WE: ~2 = 1200.1886 ¢, ~14/11 = 421.2661 ¢
error map: ⟨+0.189 +1.303 -2.266]
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 421.2365 ¢
error map: ⟨0.000 +1.066 -2.662]

Optimal ET sequence: 17, 20, 37, 57, 94, 151

Badness (Sintel): 1.73

2.7.11.13 subgroup

Subgroup: 2.7.11.13

Comma list: 1573/1568, 15488/15379

Subgroup-val mapping: [⟨1 0 1 3], ⟨0 8 7 2]]

Gencom mapping: [⟨], ⟨1 0 0 0 1 3], ⟨0 0 0 8 7 2]]

Optimal tunings:

  • WE: ~2 = 1199.8668 ¢, ~14/11 = 421.2623 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 421.2874 ¢

Optimal ET sequence: 17, 20, 37, 57, 94, 225

Badness (Sintel): 0.307

Voltage

Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 71/4 of 842.2065 ¢, which is also the CTC (constrained Tenney–Chebyshevian) tuning.

Subgroup: 2.7.13

Comma list: 28672/28561

Subgroup-val mapping: [⟨1 0 3], ⟨0 4 1]]

Gencom mapping: [⟨1 0 0 0 0 3], ⟨0 0 0 4 0 1]]

mapping generators: ~2, ~13

Optimal tunings:

  • WE: ~2 = 1199.7827 ¢, ~13/8 = 842.1707 ¢
error map: ⟨-0.217 -0.143 +0.991]
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.2568 ¢
error map: ⟨0.000 +0.201 +1.729]

Optimal ET sequence: 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff

Badness (Sintel): 0.115

Ultrakleismic

Subgroup: 2.7.17

Comma list: 4913/4802

Subgroup-val mapping: [⟨1 2 3], ⟨0 3 4]]

mapping generators, ~2, ~17/14

Optimal tunings:

  • WE: ~2 = 1200.1379 ¢, ~17/14 = 324.3440 ¢
error map: ⟨+0.138 +4.482 -7.166]
  • CWE: ~2 = 1200.000 ¢, ~17/14 = 324.3738 ¢
error map: ⟨0.000 +4.295 -7.460]

Optimal ET sequence: 4, 7, 11, 26, 37

Badness (Sintel): 0.460

Counterultrakleismic

Subgroup: 2.7.17

Comma list: 2024782584832/2015993900449

Subgroup-val mapping: [⟨1 0 1], ⟨0 10 11]]

mapping generators, ~2, ~17/14

Optimal tunings:

  • WE: ~2 = 1199.9723 ¢, ~17/14 = 336.8586 ¢
error map: ⟨-0.028 -0.240 +0.462]
  • CWE: ~2 = 1200.000 ¢, ~17/14 = 336.8621 ¢
error map: ⟨0.000 -0.205 +0.528]

Optimal ET sequence: 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g

Badness (Sintel): 0.860

Shipwreck

Subgroup: 2.7.53

Comma list: 1048576/1042139

Subgroup-val mapping: [⟨1 0 6], ⟨0 3 -1]]]

mapping generators, ~2, ~64/53

Optimal tunings:

  • WE: ~2 = 1199.6967 ¢, ~64/53 = 323.1839 ¢
error map: ⟨-0.303 +0.119 +1.491]
  • CWE: ~2 = 1200.0000 ¢, ~64/53 = 323.1959 ¢
error map: ⟨0.000 +0.762 +3.300]

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 3 3], ⟨0 2 3]]

Gencom mapping: [⟨1 0 0 0 3 3], ⟨0 0 0 0 2 3]]

mapping generators, ~2, ~13/11

Optimal tunings:

  • WE: ~2 = 1199.5223 ¢, ~13/11 = 279.2064 ¢
error map: ⟨-0.478 +5.662 -4.341]
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 278.9918 ¢
error map: ⟨0.000 +6.666 -3.552]

Optimal ET sequence: 4, 9, 13, 30, 43, 73, 116e

Badness (Sintel): 0.175

Bluebirds

Not to be confused with Bluebird.
See also: Chromatic pairs § Bluebirds

Subgroup: 2.11.13

Comma list: 265837/262144

Subgroup-val mapping: [⟨1 0 6], ⟨0 3 -2]]

Gencom mapping: [⟨1 0 0 0 3 4], ⟨0 0 0 0 3 -2]]

mapping generators, ~2, ~143/128

Optimal tunings:

  • WE: ~2 = 1200.8795 ¢, ~143/128 = 182.5017 ¢
error map: ⟨+0.880 -1.174 -2.013]
  • CWE: ~2 = 1200.0000 ¢, ~143/128 = 182.4386 ¢
error map: ⟨0.000 -4.002 -5.405]

Optimal ET sequence: 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff

Badness (Sintel): 0.451

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

Subgroup-val mapping: [⟨4 0 1], ⟨0 1 1]]

Gencom mapping: [⟨4 0 0 0 12 13], ⟨0 0 0 0 1 1]]

mapping generators, ~13/11, ~11

Optimal tunings:

  • WE: ~13/11 = 299.9728 ¢, ~11/8 = 546.6107 ¢
error map: ⟨-0.109 -5.033 +5.730]
  • CWE: ~13/11 = 300.0000 ¢, ~11/8 = 546.4664 ¢
error map: ⟨0.000 -4.852 +5.939]

Optimal ET sequence: 4, 12e, 16, 20, 24, 44, 68

Badness (Sintel): 0.668

Yamablu

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 13th Yamablu[13] scale is a linear-temperament version of Gjaeck.

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048, 83521/83486

Subgroup-val mapping: [⟨1 1 8 9 11], ⟨0 4 -7 -8 -11]]

mapping generators: ~2, ~26/17

Optimal tunings:

  • WE: ~2 = 1200.4661 ¢, ~26/17 = 737.3256 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/17 = 737.0014 ¢

Optimal ET sequence: 13, 44, 57, 70, 127, 197eh

Badness (Sintel): 0.386

Berylic

Berylic tempers out the berylisma in the 2.11.37 subgroup, representing the fact that 44/37 is a continued fraction convergent to 21/4 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 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

Subgroup-val mapping: [⟨4 0 7], ⟨0 1 1]]

mapping generators: ~44/37, ~11

Optimal tunings:

  • WE: ~44/37 = 300.0003 ¢, ~11/8 = 551.3211 ¢
error map: ⟨+0.001 +0.007 -0.017]
  • CWE: ~44/37 = 300.0000 ¢, ~11/8 = 551.3237 ¢
error map: ⟨0.000 +0.006 -0.020]

Optimal ET sequence: 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448

Badness (Sintel): 0.00188

Mavericks

Subgroup: 2.13.19

Comma list: 47525504/47045881

Subgroup-val mapping: [⟨1 1 2], ⟨0 6 5]]

mapping generators: ~2, ~26/19

Optimal tunings:

  • WE: ~2 = 1199.8817 ¢, ~26/19 = 539.9150 ¢
error map: ⟨-0.118 -1.156 +1.825]
  • CWE: ~2 = 1200.0000 ¢, ~26/19 = 539.9280 ¢
error map: ⟨0.000 -0.960 +2.127]

Optimal ET sequence: 9, 11, 20

Badness (Sintel): 0.559

Yer (rank 3)

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048

Subgroup-val mapping: [⟨1 0 0 11 4], ⟨0 1 0 -2 -1], ⟨0 0 1 0 1]]

mapping generators: ~2, ~11, ~13

Optimal tunings:

  • WE: ~2 = 1200.4447 ¢, ~11/8 = 548.4929 ¢, ~13/8 = 841.3613 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 548.2193 ¢, ~13/8 = 841.4707 ¢

Optimal ET sequence: 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh

Badness (Sintel): 0.106

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