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Very low accuracy temperaments: Difference between revisions

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FloraC (talk | contribs)

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Restore quint. It's present in the 7-limit catalog page, so good just for bookkeeping reasons
Tag: Undo
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:  
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{{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


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== 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]].  


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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:  
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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 ===
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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


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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


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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 313: Line 283:


== Quint ==
== Quint ==
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. 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.  
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
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{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~6/5, ~7
: Mapping generators: ~6/5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) <!-- Please review, last digit may be wrong -->
* [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}})
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
* [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}})
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}


{{Optimal ET sequence|legend=1| 5, 15ccd }}
{{Optimal ET sequence|legend=1| 5, 15ccd, 20cccd }}


[[Badness]] (Sintel): 1.22
[[Badness]] (Sintel): 1.22
== Geryon ==
{{See also| Dicot family }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 8/7, 25/21
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1164.885{{c}}, ~5/4 = 374.162{{c}}
: [[error map]]: {{val| -35.115 +11.253 -82.382 +125.830 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}
{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
[[Badness]] (Sintel): 1.29


== Malacoda ==
== Malacoda ==
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{{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


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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:  
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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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