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.

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 ==

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[[Comma list]]: 2048/1875

: Mapping generators: ~2, ~8/5

: Mapping generators: ~2, ~5/4

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1191.~~896~~{{c}}, ~5/4 = ~~418~~.~~013~~{{c}}

* [[WE]]: ~2 = 1191.8960{{c}}, ~8/5 = 773.8834{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = ~~421~~.~~637~~{{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 ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37cc, 57bccc }}

[[Badness]] (Sintel): 4.~~190~~

[[Badness]] (Sintel): 4.19

== Septimal smate ==

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1191.~~900~~{{c}}, ~5/4 = ~~419~~.~~425~~{{c}}

* [[WE]]: ~2 = 1191.8999{{c}}, ~8/5 = 772.4750{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~5/4 = ~~423~~.~~046~~{{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 }}

[[Badness]] (Sintel): 1.~~971~~

[[Badness]] (Sintel): 1.97

=== 11-limit ===

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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 tunings:

* WE: ~2 = 1191.~~914~~{{c}}, ~5/4 = ~~419~~.~~372~~{{c}}

* WE: ~2 = 1191.9136{{c}}, ~8/5 = 772.5419{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~5/4 = ~~423~~.~~010~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 776.9899{{c}}

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

Badness (Sintel): 1.~~406~~

Badness (Sintel): 1.41

=== 13-limit ===

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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 tunings:

* WE: ~2 = 1194.~~045~~{{c}}, ~5/4 = ~~420~~.~~921~~{{c}}

* WE: ~2 = 1194.0453{{c}}, ~8/5 = 773.1246{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = ~~423~~.~~402~~{{c}}

* CWE: ~2 = 1200.000{{c}}, ~8/5 = 776.5983{{c}}

Badness (Sintel): 1.~~522~~

Badness (Sintel): 1.52

== Hemismate ==

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

: Mapping generators: ~2, ~8/7

: Mapping generators: ~2, ~7/4

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1192.~~090~~{{c}}, ~8/7 = ~~209~~.~~065~~{{c}}

* [[WE]]: ~2 = 1192.0901{{c}}, ~7/4 = 983.0251{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~8/7 = ~~210~~.~~778~~{{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 }}

[[Badness]] (Sintel): 3.~~905~~

[[Badness]] (Sintel): 3.90

=== 11-limit ===

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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 tunings:

* WE: ~2 = 1192.~~679~~{{c}}, ~8/7 = ~~209~~.~~197~~{{c}}

* WE: ~2 = 1192.6793{{c}}, ~8/7 = 983.4825{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~8/7 = ~~210~~.~~780~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 989.2203{{c}}

Badness (Sintel): 2.~~166~~

Badness (Sintel): 2.17

=== 13-limit ===

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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 tunings:

* WE: ~2 = 1194.~~135~~{{c}}, ~8/7 = ~~209~~.~~943~~{{c}}

* WE: ~2 = 1194.1349{{c}}, ~7/4 = 984.1921{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~8/7 = ~~211~~.~~116~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 988.8836{{c}}

Badness (Sintel): 2.~~086~~

Badness (Sintel): 2.09

[[Category:Temperament families]]

[[Category:Smate family| ]]

[[Category:Rank 2]]

@@ Line 2: / Line 2: @@
 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 ==
@@ Line 9: / Line 9: @@
 [[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 = 1191.896{{c}}, ~5/4 = 418.013{{c}}
+* [[WE]]: ~2 = 1191.8960{{c}}, ~8/5 = 773.8834{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 421.637{{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 ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }}
+{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37cc, 57bccc }}
-[[Badness]] (Sintel): 4.190
+[[Badness]] (Sintel): 4.19
 == Septimal smate ==
@@ Line 26: / Line 27: @@
 [[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]]s:
-* [[WE]]: ~2 = 1191.900{{c}}, ~5/4 = 419.425{{c}}
+* [[WE]]: ~2 = 1191.8999{{c}}, ~8/5 = 772.4750{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 423.046{{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]] (Sintel): 1.971
+[[Badness]] (Sintel): 1.97
 === 11-limit ===
@@ Line 41: / Line 44: @@
 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 tunings:
-* WE: ~2 = 1191.914{{c}}, ~5/4 = 419.372{{c}}
+* WE: ~2 = 1191.9136{{c}}, ~8/5 = 772.5419{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~5/4 = 423.010{{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 (Sintel): 1.406
+Badness (Sintel): 1.41
 === 13-limit ===
@@ Line 56: / Line 59: @@
 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 tunings:
-* WE: ~2 = 1194.045{{c}}, ~5/4 = 420.921{{c}}
+* WE: ~2 = 1194.0453{{c}}, ~8/5 = 773.1246{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~5/4 = 423.402{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~8/5 = 776.5983{{c}}
 {{Optimal ET sequence|legend=0| 3de, 14, 17c }}
-Badness (Sintel): 1.522
+Badness (Sintel): 1.52
 == Hemismate ==
@@ Line 71: / Line 74: @@
 [[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, ~8/7
+: Mapping generators: ~2, ~7/4
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1192.090{{c}}, ~8/7 = 209.065{{c}}
+* [[WE]]: ~2 = 1192.0901{{c}}, ~7/4 = 983.0251{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 210.778{{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]] (Sintel): 3.905
+[[Badness]] (Sintel): 3.90
 === 11-limit ===
@@ Line 87: / Line 92: @@
 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 tunings:
-* WE: ~2 = 1192.679{{c}}, ~8/7 = 209.197{{c}}
+* WE: ~2 = 1192.6793{{c}}, ~8/7 = 983.4825{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~8/7 = 210.780{{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 (Sintel): 2.166
+Badness (Sintel): 2.17
 === 13-limit ===
@@ Line 102: / Line 107: @@
 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 tunings:
-* WE: ~2 = 1194.135{{c}}, ~8/7 = 209.943{{c}}
+* WE: ~2 = 1194.1349{{c}}, ~7/4 = 984.1921{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~8/7 = 211.116{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 988.8836{{c}}
 {{Optimal ET sequence|legend=0| 6, 11, 17c }}
-Badness (Sintel): 2.086
+Badness (Sintel): 2.09
 [[Category:Temperament families]]
 [[Category:Smate family| ]] <!-- main article -->
 [[Category:Rank 2]]