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No-fives subgroup temperaments: Difference between revisions

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=== Archy ===
=== Archy ===
See [[Archytas clan #Archy]].  
See [[Archytas clan #Archy]].  
==== Supra ====
See [[Archytas clan #Supra]].
===== Supraphon =====
See [[Archytas clan #Supraphon]].


==== Suhajira ====
==== Suhajira ====
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=== Slendroschismic ===
=== Slendroschismic ===
See [[5th-octave temperaments #Slendroschismic]].  
See [[5th-octave temperaments #Slendroschismic]].  
==== Pentadecoid ====
See [[15th-octave temperaments #Pentadecoid]].


=== Navy ===
=== Navy ===
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=== Purpleheart ===
=== Purpleheart ===
{{See also| Whitewood }}
{{Main| Whitewood }}


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
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=== Neutral ===
=== Neutral ===
See [[Rastmic clan #Neutral]].  
See [[Rastmic clan #Neutral]].  
==== Namo ====
See [[Rastmic clan #Namo]].


=== Io ===
=== Io ===
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[[Badness]] (Sintel): 0.185
[[Badness]] (Sintel): 0.185
=== Alphaxenean ===
Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It also divides the [[2/1|octave]] into two.
[[Subgroup]]: 2.3.11
[[Comma list]]: 131769/131072
{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11
[[Optimal tuning]]s:
* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}
: [[error map]]: {{val| +0.318 -0.094 -0.897 }}
* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}
: error map: {{val| 0.000 -0.491 -2.050 }}
{{Optimal ET sequence|legend=1| 22, 24, 94, 118, 142, 450e, 592e, 1326beeee }}
[[Badness]] (Sintel): 0.395


=== Infraug ===
=== Infraug ===
Line 915: Line 923:
Comma list: 144/143, 729/704
Comma list: 144/143, 729/704


Subgrouo-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}
Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 363/361, 729/704
Comma list: 363/361, 729/704


Subgrouo-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
: mapping generators: ~2, ~19/11
: mapping generators: ~2, ~19/11


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Badness (Sintel): 0.415
Badness (Sintel): 0.415
=== Profanity ===
Profanity identifies [[11/9]] with 2\7.
[[Subgroup]]: 2.3.11
[[Comma list]]: 19487171/19131876
{{Mapping|legend=2| 7 0 2 | 0 1 2 }}
: mapping generators: ~1458/1331, ~3
[[Optimal tuning]]s:
* [[WE]]: ~1458/1331 = 171.4369{{c}}, ~3/2 = 702.9304{{c}}
: [[error map]]: {{val| +0.058 +1.033 -2.467 }}
* [[CWE]]: ~1458/1331 = 171.4286{{c}}, ~3/2 = 702.9442{{c}}
: error map: {{val| 0.000 -0.989 -2.572 }}
{{Optimal ET sequence|legend=1| 7, … 49, 56, 63, 70 }}
[[Badness]] (Sintel): 3.03


== Temperaments with a 2.3.13 gene ==
== Temperaments with a 2.3.13 gene ==
Line 1,155: Line 1,183:
[[Badness]] (Sintel): 0.0454
[[Badness]] (Sintel): 0.0454


=== Gigapyth ===
=== Boethian ===
[[Subgroup]]: 2.3.85
Boethian is a [[5L 2s|diatonic-based]] temperament that makes [[513/512]] vanish, so that the major third (C–E) is ~[[24/19]] and the minor third (C–E♭) is ~[[19/16]]. As such, it functions as a 2.3.19-subgroup analogue of [[meantone]], though the small size of the comma puts it at [[schismic]] level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to [[12edo]]) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called [[nestoria]], which also tempers out [[361/360]], the difference between 19/18 and 20/19 or between 19/15 and 24/19.
 
[[Subgroup]]: 2.3.19


[[Comma list]]: 2.3.85 {{val| -40 1 6 }}
[[Comma list]]: 513/512


{{Mapping|legend=2| 1 -2 7 | 0 6 -1 }}
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~128/85
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.8260{{c}}, ~128/85 = 717.2589{{c}}
* [[WE]]: ~2 = 1200.2498{{c}}, ~3/2 = 701.4958{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/85 = 716.8156{{c}}
: [[error map]]: {{val| +0.250 -0.209 -0.501 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.3445{{c}}
: error map: {{val| 0.000 -0.610 -1.547 }}


{{Optimal ET sequence|legend=1| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370***, 519b***** }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


<nowiki/>* Wart for 85
[[Badness]] (Sintel): 0.0294
 
==== 2.3.7.85 subgroup ====
Subgroup: 2.3.7.85
 
Comma list: 1029/1024, 7225/7203
 
Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}
 
{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}
 
<nowiki/>* Wart for 85


=== Dog ===
=== Dog ===
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[[Badness]] (Sintel): 0.491
[[Badness]] (Sintel): 0.491
=== Boethian ===
Boethian is a [[5L 2s|diatonic-based]] temperament that makes [[513/512]] vanish, so that the major third (C–E) is ~[[24/19]] and the minor third (C–E♭) is ~[[19/16]]. As such, it functions as a 2.3.19-subgroup analogue of [[meantone]], though the small size of the comma puts it at [[schismic]] level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to [[12edo]]) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called [[nestoria]], which also tempers out [[361/360]], the difference between 19/18 and 20/19 or between 19/15 and 24/19.
[[Subgroup]]: 2.3.19
[[Comma list]]: 513/512
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2498{{c}}, ~3/2 = 701.4958{{c}}
: [[error map]]: {{val| +0.250 -0.209 -0.501 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.3445{{c}}
: error map: {{val| 0.000 -0.610 -1.547 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}
[[Badness]] (Sintel): 0.0294


=== Lipsett ===
=== Lipsett ===
Line 1,233: Line 1,230:
[[Comma list]]: 2187/2116
[[Comma list]]: 2187/2116


{{Mapping|legend=2| 1 0 -1 | 0 2 7}}
{{Mapping|legend=2| 1 0 -1 | 0 2 7 }}
: mapping generators: ~2, ~46/27


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 1,250: Line 1,248:
[[Comma list]]: 24576/24389
[[Comma list]]: 24576/24389


{{Mapping|legend=2| 1 2 5 | 0 3 -1 }}
{{Mapping|legend=2| 1 2 5 | 0 -3 -1 }}
: mapping generators: ~2, ~32/29


[[Optimal tuning]]s ([[CTE]]): ~32/29 = 166.067{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5519{{c}}, ~32/29 = 165.7453{{c}}
: [[error map]]: {{val| -0.448 -0.087 +2.437 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/29 = 165.9004{{c}}
: error map: {{val| 0.000 +0.344 +4.522 }}


{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}
{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}
[[Badness]] (Sintel): 0.317


=== Sematology ===
=== Sematology ===
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[[Comma list]]: 4107/4096
[[Comma list]]: 4107/4096


{{Mapping|legend=2| 1 1 5 | 0 -2 1 }}
{{Mapping|legend=2| 1 0 6 | 0 2 -1 }}
: mapping generators: ~2, ~64/37


[[Optimal tuning]]s ([[POTE]]): ~37/32 = 249.075{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2184{{c}}, ~64/37 = 950.9546{{c}}
: [[error map]]: {{val| +0.218 -0.046 -0.988 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/37 = 950.8250{{c}}
: error map: {{val| 0.000 -0.305 -2.169 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130 }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll }}
 
[[Badness]] (Sintel): 0.0690


=== Reversed mavila ===
=== Reversed mavila ===
Line 1,274: Line 1,286:
[[Comma list]]: 81/74
[[Comma list]]: 81/74


{{Mapping|legend=2| 1 1 0 | 0 -1 12 }}
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3


[[Optimal tuning]]s ([[POTE]]): ~4/3 = 521.397{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.9908{{c}}, ~3/2 = 676.4865{{c}}
: [[error map]]: {{val| +1.991 -23.478 +60.575 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 676.7603{{c}}
: error map: {{val| 0.000 -25.195 +55.697 }}


{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}
{{Optimal ET sequence|legend=1| 2, 5l, 7l, 9, 16l }}
 
[[Badness]] (Sintel): 0.623


=== Reversed meantone ===
=== Reversed meantone ===
Line 1,287: Line 1,306:
[[Comma list]]: 82/81
[[Comma list]]: 82/81


{{Mapping|legend=2| 1 2 7 | 0 -1 -4 }}
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6907{{c}}, ~3/2 = 705.3096{{c}}
: [[error map]]: {{val| -0.309 +3.045 -8.752 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2699{{c}}
: error map: {{val| 0.000 +3.315 -7.983 }}


[[Optimal tuning]]s ([[POTE]]): ~4/3 = 494.509{{c}}
{{Optimal ET sequence|legend=1| 5, 12, 17, 97m, 114m, 131m }}


{{Optimal ET sequence|legend=1| 5, 12, 17 }}
[[Badness]] (Sintel): 0.0841


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"