Smate family: Difference between revisions

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Line 11: Line 11:
{{Mapping|legend=1| 1 3 2 | 0 -4 1 }}
{{Mapping|legend=1| 1 3 2 | 0 -4 1 }}


: mapping generators: ~2, ~5/4
: Mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 420.855
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1191.896{{c}}, ~5/4 = 418.013{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 421.637{{c}}


{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }}
{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }}


[[Badness]] (Smith): 0.178624
[[Badness]] (Sintel): 4.190


== Septimal smate ==
== Septimal smate ==
Line 26: Line 28:
{{Mapping|legend=1| 1 3 2 6 | 0 -4 1 -9 }}
{{Mapping|legend=1| 1 3 2 6 | 0 -4 1 -9 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 422.275
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1191.900{{c}}, ~5/4 = 419.425{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 423.046{{c}}


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


[[Badness]] (Smith): 0.077871
[[Badness]] (Sintel): 1.971


=== 11-limit ===
=== 11-limit ===
Line 39: Line 43:
Mapping: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }}
Mapping: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }}


Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 422.217
Optimal tunings:
* WE: ~2 = 1191.914{{c}}, ~5/4 = 419.372{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 423.010{{c}}


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


Badness (Smith): 0.042518
Badness (Sintel): 1.406


=== 13-limit ===
=== 13-limit ===
Line 52: Line 58:
Mapping: {{mapping| 1 3 2 6 7 3 | 0 -4 1 -9 -10 2 }}
Mapping: {{mapping| 1 3 2 6 7 3 | 0 -4 1 -9 -10 2 }}


Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 423.020
Optimal tunings:
* WE: ~2 = 1194.045{{c}}, ~5/4 = 420.921{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 423.402{{c}}


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


Badness (Smith): 0.036836
Badness (Sintel): 1.522


== Hemismate ==
== Hemismate ==
Line 64: Line 72:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 210.452
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1192.090{{c}}, ~8/7 = 209.065{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 210.778{{c}}


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


[[Badness]] (Smith): 0.154301
[[Badness]] (Sintel): 3.905


=== 11-limit ===
=== 11-limit ===
Line 78: Line 89:
Mapping: {{mapping| 1 3 2 3 4 | 0 -8 2 -1 -3 }}
Mapping: {{mapping| 1 3 2 3 4 | 0 -8 2 -1 -3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 210.481
Optimal tunings:
* WE: ~2 = 1192.679{{c}}, ~8/7 = 209.197{{c}}
* CWE: ~2 = 1200.000{{c}}, ~8/7 = 210.780{{c}}


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


Badness (Smith): 0.065528
Badness (Sintel): 2.166


=== 13-limit ===
=== 13-limit ===
Line 91: Line 104:
Mapping: {{mapping| 1 3 2 3 4 3 | 0 -8 2 -1 -3 4 }}
Mapping: {{mapping| 1 3 2 3 4 3 | 0 -8 2 -1 -3 4 }}


Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 210.974
Optimal tunings:
* WE: ~2 = 1194.135{{c}}, ~8/7 = 209.943{{c}}
* CWE: ~2 = 1200.000{{c}}, ~8/7 = 211.116{{c}}


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


Badness (Smith): 0.050472
Badness (Sintel): 2.086


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

Revision as of 12:05, 29 January 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 9/7s rather than the sharp 5/4s used here.

Smate

Subgroup: 2.3.5

Comma list: 2048/1875

Mapping: [⟨1 3 2], ⟨0 -4 1]]

Mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1191.896 ¢, ~5/4 = 418.013 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 421.637 ¢

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

Badness (Sintel): 4.190

Septimal smate

Subgroup: 2.3.5.7

Comma list: 36/35, 2048/1875

Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]

Optimal tunings:

  • WE: ~2 = 1191.900 ¢, ~5/4 = 419.425 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 423.046 ¢

Optimal ET sequence: 3d, 11d, 14, 17c, 37ccdd

Badness (Sintel): 1.971

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 2 6 7], ⟨0 -4 1 -9 -10]]

Optimal tunings:

  • WE: ~2 = 1191.914 ¢, ~5/4 = 419.372 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 423.010 ¢

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

Badness (Sintel): 1.406

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 2 6 7 3], ⟨0 -4 1 -9 -10 2]]

Optimal tunings:

  • WE: ~2 = 1194.045 ¢, ~5/4 = 420.921 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 423.402 ¢

Optimal ET sequence: 3de, 14, 17c

Badness (Sintel): 1.522

Hemismate

Subgroup: 2.3.5.7

Comma list: 256/245, 392/375

Mapping: [⟨1 3 2 3], ⟨0 -8 2 -1]]

Mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1192.090 ¢, ~8/7 = 209.065 ¢
  • CWE: ~2 = 1200.000 ¢, ~8/7 = 210.778 ¢

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

Badness (Sintel): 3.905

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1192.679 ¢, ~8/7 = 209.197 ¢
  • CWE: ~2 = 1200.000 ¢, ~8/7 = 210.780 ¢

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

Badness (Sintel): 2.166

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1194.135 ¢, ~8/7 = 209.943 ¢
  • CWE: ~2 = 1200.000 ¢, ~8/7 = 211.116 ¢

Optimal ET sequence: 6, 11, 17c

Badness (Sintel): 2.086

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