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Breedsmic temperaments: Difference between revisions

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* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]


Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, newt, unthirds, septidiasemi, subneutral, maviloid, lockerbie, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, newt, septidiasemi, subneutral, maviloid, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  


== Tertiaseptal ==
== Tertiaseptal ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979)
* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979{{c}})
: [[error map]]: {{val| +0.100 -0.008 -0.123 -0.119 }}
: [[error map]]: {{val| +0.100 -0.008 -0.123 -0.119 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899)
* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899{{c}})
: error map: {{val| 0.000 -0.133 -0.364 -0.396 }}
: error map: {{val| 0.000 -0.133 -0.364 -0.396 }}


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Badness (Sintel): 0.438
Badness (Sintel): 0.438
== Unthirds ==
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 68359375/68024448
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0859{{c}}, ~3969/3125 = 416.7465{{c}}
: [[error map]]: {{val| +0.086 +0.281 -0.431 -0.218 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3969/3125 = 416.7184{{c}}
: error map: {{val| 0.000 +0.220 -0.547 -0.399 }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
[[Badness]] (Sintel): 1.90
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4000/3993
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}
Optimal tunings:
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}
Badness (Sintel): 0.758
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}
Optimal tunings:
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}
Badness (Sintel): 0.863


== Septidiasemi ==
== Septidiasemi ==
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Badness (Sintel): 1.25
Badness (Sintel): 1.25
== Unthirds ==
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 68359375/68024448
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0859{{c}}, ~3969/3125 = 416.7465{{c}}
: [[error map]]: {{val| +0.086 +0.281 -0.431 -0.218 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3969/3125 = 416.7184{{c}}
: error map: {{val| 0.000 +0.220 -0.547 -0.399 }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
[[Badness]] (Sintel): 1.90
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4000/3993
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}
Optimal tunings:
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}
Badness (Sintel): 0.758
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}
Optimal tunings:
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}
Badness (Sintel): 0.863


== Neominor ==
== Neominor ==
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