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No-threes subgroup temperaments: Difference between revisions

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+ some missing entries. "Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups." this is simply not how we do rtt
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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 ===
Line 80: Line 145:
[[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: 640/637, 10985/10976


[[Comma list]]: {{monzo| 372 -159 -1 }}
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}


{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~5
: mapping generators: ~2, ~13/8
 
[[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 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 ===
Line 281: Line 271:
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.
{{Todo|inline=1| complete section | comment = Add missing data from 2.5.7 to 2.5.7.11.19.29. }}
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: {{monzo| 372 -159 -1 }}


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. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.
{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
: mapping generators: ~2, ~5


==== 2.5.7.11.19.29.31 subgroup ====
[[Optimal tuning]]s:
Subgroup: 2.5.7.11.19.29.31
* [[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 }}


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}


Subgroup-val mapping: {{mapping| 1 -42 -2 -15 -12 -61 -61 | 0 461 50 192 169 685 686 }}
[[Badness]] (Sintel): 148


Optimal tunings:
==== 2.5.7.11 subgroup ====
* WE: ~2 = 1200.0079{{c}}, ~31/29 = 115.3723{{c}}
Subgroup: 2.5.7.11
* CWE: ~2 = 1200.0000{{c}}, ~31/29 = 115.3716{{c}}


{{Optimal ET sequence|legend=0| 52, 1737, 1789 }}
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}


Badness (Sintel): 4.51
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}


=== Huntington ===
Optimal tunings:
{{See also| Chromatic pairs #Huntington }}
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}


[[Subgroup]]: 2.5.7
Badness (Sintel): 52.2


[[Comma list]]: 40960000/40353607
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
: mapping generators: ~2, ~80/49


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
: error map: {{val| 0.000 -0.091 +2.828 }}


{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}
{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}


Badness (Sintel): 1.87
Badness (Sintel): 10.5


==== 2.5.7.13 subgroup ====
=== Bastille ===
Subgroup: 2.5.7.13
{{Main| Bastille }}


Comma list: 640/637, 10985/10976
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-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
[[Subgroup]]: 2.5.7


Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
[[Comma list]]: {{monzo| 1426 -596 -15 }}
: mapping generators: ~2, ~13/8


Optimal tunings:
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}
[[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 }}


Badness (Sintel): 0.319
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


==== Silver ====
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
{{See also| Chromatic pairs #Silver }}


Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.  
=== Tricesimoprimal miracloid ===
{{See also| Tricesimoprimal miracloid/Eliora's approach }}
{{Todo|inline=1| complete section | comment = Add missing data from 2.5.7 to 2.5.7.11.19.29. }}


Subgroup: 2.5.7.13.17
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. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.


Comma list: 170/169, 640/637, 5525/5488
==== 2.5.7.11.19.29.31 subgroup ====
Subgroup: 2.5.7.11.19.29.31


Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688


Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
Subgroup-val mapping: {{mapping| 1 -42 -2 -15 -12 -61 -61 | 0 461 50 192 169 685 686 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* WE: ~2 = 1200.0079{{c}}, ~31/29 = 115.3723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/29 = 115.3716{{c}}


{{Optimal ET sequence|legend=1| 10, 27, 37, 47, 84, 131, 178g }}
{{Optimal ET sequence|legend=0| 52, 1737, 1789 }}


Badness (Sintel): 0.504
Badness (Sintel): 4.51


=== Pakkanen ===
=== Pakkanen ===
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