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.


= Smate =
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 ==
[[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]]: 0.179
[[Badness]] (Sintel): 4.19


= 7-limit =
== Septimal smate ==
{{see also| Mint temperaments #Smate }}
[[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 }}
[[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 }}


[[POTE generator]]: ~5/4 = 422.275
{{Optimal ET sequence|legend=1| 3d, 14, 17c, 37ccdd, 54cccdd }}


{{Val list|legend=1| 3d, 11d, 14, 17c, 37cd }}
[[Badness]] (Sintel): 1.97


[[Badness]]: 0.0779
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
== 11-limit ==


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: 0.0425
Badness (Sintel): 1.41


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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: 0.0368
Badness (Sintel): 1.52


= Hemismate =
== Hemismate ==
[[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]]: 0.1543
[[Badness]] (Sintel): 3.90


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11


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: 0.0655
Badness (Sintel): 2.17


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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: 0.0505
Badness (Sintel): 2.09


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Smate family| ]] <!-- main article -->
[[Category:Smate]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 10:07, 11 March 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The smate family of temperaments tempers out 2048/1875, the smate comma, resulting in equation of four just major thirds (5/4) with the 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

Subgroup: 2.3.5

Comma list: 2048/1875

Mapping: [⟨1 -1 3], ⟨0 4 -1]]

Mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1191.8960 ¢, ~8/5 = 773.8834 ¢
error map: ⟨-8.104 +1.683 +15.491]
  • CWE: ~2 = 1200.000 ¢, ~8/5 = 778.3632 ¢
error map: ⟨0.000 +11.498 +35.323]

Optimal ET sequence: 3, 11, 14, 17c, 20c, 37cc, 57bccc

Badness (Sintel): 4.19

Septimal smate

Subgroup: 2.3.5.7

Comma list: 36/35, 2048/1875

Mapping: [⟨1 -1 3 -3], ⟨0 4 -1 9]]

Optimal tunings:

  • WE: ~2 = 1191.8999 ¢, ~8/5 = 772.4750 ¢
error map: ⟨-8.100 -3.955 +16.911 +7.749]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 776.9542 ¢
error map: ⟨0.000 +5.862 +36.732 +23.762]

Optimal ET sequence: 3d, 14, 17c, 37ccdd, 54cccdd

Badness (Sintel): 1.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 243/242

Mapping: [⟨1 -1 3 -3 -3], ⟨0 4 -1 9 10]]

Optimal tunings:

  • WE: ~2 = 1191.9136 ¢, ~8/5 = 772.5419 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 776.9899 ¢

Optimal ET sequence: 3de, 14, 17c, 37ccddee, 54cccddee

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 243/242

Mapping: [⟨1 -1 3 -3 -3 -5], ⟨0 4 -1 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1194.0453 ¢, ~8/5 = 773.1246 ¢
  • CWE: ~2 = 1200.000 ¢, ~8/5 = 776.5983 ¢

Optimal ET sequence: 3de, 14, 17c

Badness (Sintel): 1.52

Hemismate

Subgroup: 2.3.5.7

Comma list: 256/245, 392/375

Mapping: [⟨1 -5 4 2], ⟨0 8 -2 1]]

Mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1192.0901 ¢, ~7/4 = 983.0251 ¢
error map: ⟨-7.910 +1.795 +15.997 -1.620]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 989.2222 ¢
error map: ⟨0.000 +11.823 +35.242 +20.396]

Optimal ET sequence: 6, 11, 17c, 40bccd, 57bcccd

Badness (Sintel): 3.90

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 256/245

Mapping: [⟨1 -5 4 2 1], ⟨0 8 -2 1 3]]

Optimal tunings:

  • WE: ~2 = 1192.6793 ¢, ~8/7 = 983.4825 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 989.2203 ¢

Optimal ET sequence: 6, 11, 17c, 40bccde

Badness (Sintel): 2.17

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 77/75, 256/245

Mapping: [⟨1 -5 4 2 1 7], ⟨0 8 -2 1 3 -4]]

Optimal tunings:

  • WE: ~2 = 1194.1349 ¢, ~7/4 = 984.1921 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 988.8836 ¢

Optimal ET sequence: 6, 11, 17c

Badness (Sintel): 2.09

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