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Smate family: Difference between revisions

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The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4|just major thirds (5/4)]] with the [[8/3|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.  
{{Technical data page}}
The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4|just major thirds (5/4)]] with the [[8/3|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.
 
These temperaments resemble [[skwares]], which also splits 8/3 into four major thirds, but those major thirds are flattish [[9/7]]'s rather than the sharp 5/4's used here.  


== Smate ==
== Smate ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 2048/1875
[[Comma list]]: 2048/1875


[[Mapping]]: [{{val| 1 3 2 }}, {{val| 0 -4 1 }}]
{{Mapping|legend=1| 1 -1 3 | 0 4 -1 }}
: Mapping generators: ~2, ~8/5


[[POTE generator]]: ~5/4 = 420.855
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1191.8960{{c}}, ~8/5 = 773.8834{{c}}
: [[error map]]: {{val| -8.104 +1.683 +15.491 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~8/5 = 778.3632{{c}}
: error map: {{val| 0.000 +11.498 +35.323 }}


{{Val list|legend=1| 3, 11, 14, 17c, 20c, 37c }}
{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37cc, 57bccc }}


[[Badness]]: 0.178624
[[Badness]] (Sintel): 4.19


== Septimal smate ==
== Septimal smate ==
{{see also| Mint temperaments #Smate }}
[[Subgroup]]: 2.3.5.7
 
Subgroup: 2.3.5.7


[[Comma list]]: 36/35, 2048/1875
[[Comma list]]: 36/35, 2048/1875


[[Mapping]]: [{{val| 1 3 2 6 }}, {{val| 0 -4 1 -9 }}]
{{Mapping|legend=1| 1 -1 3 -3 | 0 4 -1 9 }}
 
{{Multival|legend=1| 4 -1 9 -11 3 24 }}


[[POTE generator]]: ~5/4 = 422.275
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1191.8999{{c}}, ~8/5 = 772.4750{{c}}
: [[error map]]: {{val| -8.100 -3.955 +16.911 +7.749 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 776.9542{{c}}
: error map: {{val| 0.000 +5.862 +36.732 +23.762 }}


{{Val list|legend=1| 3d, 11d, 14, 17c, 37cd }}
{{Optimal ET sequence|legend=1| 3d, 14, 17c, 37ccdd, 54cccdd }}


[[Badness]]: 0.077871
[[Badness]] (Sintel): 1.97


=== 11-limit ===
=== 11-limit ===
Line 36: Line 44:
Comma list: 36/35, 56/55, 243/242
Comma list: 36/35, 56/55, 243/242


Mapping: [{{val| 1 3 2 6 7 }}, {{val| 0 -4 1 -9 -10 }}]
Mapping: {{mapping| 1 -1 3 -3 -3 | 0 4 -1 9 10 }}


POTE generator: ~5/4 = 422.217
Optimal tunings:  
* WE: ~2 = 1191.9136{{c}}, ~8/5 = 772.5419{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 776.9899{{c}}


{{Val list|legend=1| 14, 17c, 37cde }}
{{Optimal ET sequence|legend=0| 3de, 14, 17c, 37ccddee, 54cccddee }}


Badness: 0.042518
Badness (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
Line 49: Line 59:
Comma list: 26/25, 36/35, 56/55, 243/242  
Comma list: 26/25, 36/35, 56/55, 243/242  


Mapping: [{{val| 1 3 2 6 7 3 }}, {{val| 0 -4 1 -9 -10 2 }}]
Mapping: {{mapping| 1 -1 3 -3 -3 -5 | 0 4 -1 9 10 -2 }}


POTE generator: ~5/4 = 423.020
Optimal tunings:  
* WE: ~2 = 1194.0453{{c}}, ~8/5 = 773.1246{{c}}
* CWE: ~2 = 1200.000{{c}}, ~8/5 = 776.5983{{c}}


{{Val list|legend=1| 14, 17c }}
{{Optimal ET sequence|legend=0| 3de, 14, 17c }}


Badness: 0.036836
Badness (Sintel): 1.52


== Hemismate ==
== Hemismate ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 256/245, 392/375  
[[Comma list]]: 256/245, 392/375  


[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -8 2 -1 }}]
{{Mapping|legend=1| 1 -5 4 2 | 0 8 -2 1 }}
 
: Mapping generators: ~2, ~7/4
{{Multival|legend=1| 8 -2 1 -22 -21 8 }}


[[POTE generator]]: ~8/7 = 210.452
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1192.0901{{c}}, ~7/4 = 983.0251{{c}}
: [[error map]]: {{val| -7.910 +1.795 +15.997 -1.620 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 989.2222{{c}}
: error map: {{val| 0.000 +11.823 +35.242 +20.396 }}


{{Val list|legend=1| 6, 11, 17c, 40bcd }}
{{Optimal ET sequence|legend=1| 6, 11, 17c, 40bccd, 57bcccd }}


[[Badness]]: 0.154301
[[Badness]] (Sintel): 3.90


=== 11-limit ===
=== 11-limit ===
Line 77: Line 92:
Comma list: 56/55, 77/75, 256/245
Comma list: 56/55, 77/75, 256/245


Mapping: [{{val| 1 3 2 3 4 }}, {{val| 0 -8 2 -1 -3 }}]
Mapping: {{mapping| 1 -5 4 2 1 | 0 8 -2 1 3 }}


POTE generator: ~8/7 = 210.481
Optimal tunings:  
* WE: ~2 = 1192.6793{{c}}, ~8/7 = 983.4825{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 989.2203{{c}}


{{Val list|legend=1| 6, 11, 17c, 40bcde }}
{{Optimal ET sequence|legend=0| 6, 11, 17c, 40bccde }}


Badness: 0.065528
Badness (Sintel): 2.17


=== 13-limit ===
=== 13-limit ===
Line 90: Line 107:
Comma list: 26/25, 56/55, 77/75, 256/245
Comma list: 26/25, 56/55, 77/75, 256/245


Mapping: [{{val| 1 3 2 3 4 3 }}, {{val| 0 -8 2 -1 -3 4 }}]
Mapping: {{mapping| 1 -5 4 2 1 7 | 0 8 -2 1 3 -4 }}


POTE generators: ~8/7 = 210.974
Optimal tunings:  
* WE: ~2 = 1194.1349{{c}}, ~7/4 = 984.1921{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 988.8836{{c}}


{{Val list|legend=1| 6, 11, 17c }}
{{Optimal ET sequence|legend=0| 6, 11, 17c }}


Badness: 0.050472
Badness (Sintel): 2.09


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Smate]]
[[Category:Smate family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Smate_family"