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Very low accuracy temperaments: Difference between revisions

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Delete antonym, antony, and geryon, from the precedent of deletion of {7/6}. Perhaps {9/8} should be the threshold for tempering out superparticulars
 
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Line 31: Line 31:
Comma list: 9/8, 15/14
Comma list: 9/8, 15/14


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


Optimal tunings:  
Optimal tunings:  
Line 46: Line 46:
Comma list: 9/8, 12/11, 15/14
Comma list: 9/8, 12/11, 15/14


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


Optimal tunings:  
Optimal tunings:  
Line 61: Line 61:
Comma list: 9/8, 11/10, 15/14
Comma list: 9/8, 11/10, 15/14


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


Optimal tunings:  
Optimal tunings:  
Line 76: Line 76:
Comma list: 9/8, 35/32
Comma list: 9/8, 35/32


Mapping: {{mapping| 2 3 0 10 | 0 0 1 -1 }}
{{Mapping|legend=0| 2 3 0 10 | 0 0 1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 91: Line 91:
Comma list: 9/8, 12/11, 35/32
Comma list: 9/8, 12/11, 35/32


Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}
{{Mapping|legend=0| 2 3 0 10 7 | 0 0 1 -1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 109: Line 109:


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


Line 124: Line 123:
== Antonian ==
== Antonian ==
{{Main| Antonian }}
{{Main| Antonian }}
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].  
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].  


Line 149: Line 149:
Comma list: 10/9, 15/14
Comma list: 10/9, 15/14


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


Optimal tunings:  
Optimal tunings:  
Line 158: Line 158:


Badness (Sintel): 0.606
Badness (Sintel): 0.606
=== Antonym ===
Subgroup: 2.3.5.7
Comma list: 7/6, 10/9
Mapping: {{mapping| 1 0 -1 1 | 0 1 2 1 }}
Optimal tunings:
* WE: ~2 = 1209.795{{c}}, ~3/2 = 765.995{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 765.949{{c}}
{{Optimal ET sequence|legend=0| 2cd, 3 }}
Badness (Sintel): 0.568
=== Antony ===
Subgroup: 2.3.5.7
Comma list: 8/7, 10/9
Mapping: {{mapping| 1 0 -1 3 | 0 1 2 0 }}
Optimal tunings:
* WE: ~2 = 1151.235{{c}}, ~3/2 = 789.399{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}
{{Optimal ET sequence|legend=0| 1c, 3d }}
Badness (Sintel): 0.720


=== Brutus ===
=== Brutus ===
Line 194: Line 164:
Comma list: 10/9, 28/25
Comma list: 10/9, 28/25


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


Optimal tunings:  
Optimal tunings:  
Line 211: Line 181:
Comma list: 10/9, 35/32
Comma list: 10/9, 35/32


Mapping: {{mapping| 1 0 -1 6 | 0 1 2 -2 }}
{{Mapping|legend=0| 1 0 -1 6 | 0 1 2 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 228: Line 198:
Comma list: 10/9, 49/45
Comma list: 10/9, 49/45


Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
{{Mapping|legend=0| 2 0 -2 -1 | 0 1 2 2 }}
: mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3


Line 246: Line 216:
Comma list: 10/9, 49/48
Comma list: 10/9, 49/48


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


Line 283: Line 253:
Comma list: 10/9, 16/15, 22/21
Comma list: 10/9, 16/15, 22/21


Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}
{{Mapping|legend=0| 3 5 7 0 2 | 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 312: Line 282:
[[Badness]] (Sintel): 1.16
[[Badness]] (Sintel): 1.16


== Geryon ==
== Quint ==
{{See also| Dicot family }}
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is already quite accurately approximated by 5edo for its size, this temperament provides little improvement, so it is only provided below for bookkeeping purposes. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 8/7, 25/21
[[Comma list]]: 16/15, 27/25


{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~6/5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1164.885{{c}}, ~5/4 = 374.162{{c}}
* [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}})
: [[error map]]: {{val| -35.115 +11.253 -82.382 +125.830 }}
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
* [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}})
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}


{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
{{Optimal ET sequence|legend=1| 5, 15ccd, 20cccd }}


[[Badness]] (Sintel): 1.29
[[Badness]] (Sintel): 1.22


== Malacoda ==
== Malacoda ==
Line 339: Line 310:


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


Line 395: Line 365:
Comma list: 15/14, 22/21, 33/32
Comma list: 15/14, 22/21, 33/32


Mapping: {{mapping| 1 0 7 6 5 | 0 1 -3 -2 -1 }}
{{Mapping|legend=0| 1 0 7 6 5 | 0 1 -3 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 449: Line 419:
Comma list: 15/14, 22/21, 81/80
Comma list: 15/14, 22/21, 81/80


Mapping: {{mapping| 1 0 -4 -5 -6 | 0 1 4 5 6 }}
{{Mapping|legend=0| 1 0 -4 -5 -6 | 0 1 4 5 6 }}


Optimal tunings:  
Optimal tunings:  
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