Smate family: Difference between revisions

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Normalize mappings. Review optimal ET sequences.
 
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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.
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 5/4s used here.
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 ==
Line 9: Line 9:
[[Comma list]]: 2048/1875
[[Comma list]]: 2048/1875


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


: mapping generators: ~2, ~5/4
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 420.855
{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37cc, 57bccc }}


{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }}
[[Badness]] (Sintel): 4.19
 
[[Badness]] (Smith): 0.178624


== Septimal smate ==
== Septimal smate ==
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[[Comma list]]: 36/35, 2048/1875
[[Comma list]]: 36/35, 2048/1875


{{Mapping|legend=1| 1 3 2 6 | 0 -4 1 -9 }}
{{Mapping|legend=1| 1 -1 3 -3 | 0 4 -1 9 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 }}


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


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


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


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


Optimal tuning (POTE): ~2 = 1200.000, ~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}}


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


Badness (Smith): 0.042518
Badness (Sintel): 1.41


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


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


Optimal tuning (POTE): ~2 = 1200.000, ~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}}


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


Badness (Smith): 0.036836
Badness (Sintel): 1.52


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


{{Mapping|legend=1| 1 3 2 3 | 0 -8 2 -1 }}
{{Mapping|legend=1| 1 -5 4 2 | 0 8 -2 1 }}
: Mapping generators: ~2, ~7/4


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 76: Line 92:
Comma list: 56/55, 77/75, 256/245
Comma list: 56/55, 77/75, 256/245


Mapping: {{mapping| 1 3 2 3 4 | 0 -8 2 -1 -3 }}
Mapping: {{mapping| 1 -5 4 2 1 | 0 8 -2 1 3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~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}}


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


Badness (Smith): 0.065528
Badness (Sintel): 2.17


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


Mapping: {{mapping| 1 3 2 3 4 3 | 0 -8 2 -1 -3 4 }}
Mapping: {{mapping| 1 -5 4 2 1 7 | 0 8 -2 1 3 -4 }}


Optimal tuning (POTE): ~2 = 1200.000, ~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}}


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


Badness (Smith): 0.050472
Badness (Sintel): 2.09


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