Smate family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,013 edits
No edit summary
Tags: Mobile edit Mobile web edit
Normalize mappings. Review optimal ET sequences.
 
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.
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:  
[[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 ==
== Septimal smate ==
Line 26: Line 27:
[[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]]s:  
[[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 ===
=== 11-limit ===
Line 41: Line 44:
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 tunings:  
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 ===
=== 13-limit ===
Line 56: Line 59:
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 tunings:  
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 }}
{{Optimal ET sequence|legend=0| 3de, 14, 17c }}


Badness (Sintel): 1.522
Badness (Sintel): 1.52


== Hemismate ==
== Hemismate ==
Line 71: Line 74:
[[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, ~8/7
: Mapping generators: ~2, ~7/4


[[Optimal tuning]]s:  
[[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 ===
=== 11-limit ===
Line 87: 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 tunings:  
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 ===
=== 13-limit ===
Line 102: Line 107:
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 tunings:  
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 }}
{{Optimal ET sequence|legend=0| 6, 11, 17c }}


Badness (Sintel): 2.086
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

Retrieved from "https://en.xen.wiki/index.php?title=Smate_family&oldid=225567"