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No-fives subgroup temperaments: Difference between revisions

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=== Alphaxenean ===
=== 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.  
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 [[octave]] into two.  


[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11
Line 899: Line 899:


[[Badness]] (Sintel): 0.395
[[Badness]] (Sintel): 0.395
=== Octatonic ===
[[12/11]] is very close to 1 step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.
[[Subgroup]]: 2.3.11
[[Comma list]]: 214990848/214358881
{{Mapping|legend=2| 8 0 15 | 0 1 1 }}
: mapping generators: ~12/11, ~3
[[Optimal tuning]]s:
* [[WE]]: ~12/11 = 149.9917{{c}}, ~3/2 = 701.9324{{c}}
: [[error map]]: {{val| -0.066 -0.089 +0.424 }}
* [[CWE]]: ~12/11 = 150.0000{{c}}, ~3/2 = 701.9147{{c}}
: error map: {{val| 0.000 -0.040 +0.597 }}
{{Optimal ET sequence|legend=1| 8, 16, 24, 104, 128, 152, 176, 200, 1024e, 1224e, 1424e, 1624e }}
[[Badness]] (Sintel): 0.515


=== Infraug ===
=== Infraug ===
Line 918: Line 938:
[[Badness]] (Sintel): 0.734
[[Badness]] (Sintel): 0.734


==== 2.3.11.13 ====
==== 2.3.11.13 subgroup ====
Subgroup: 2.3.11.13
Subgroup: 2.3.11.13


Line 945: Line 965:
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


{{Optimal ET sequence|legend=1| 14, 19, 33 }}
{{Optimal ET sequence|legend=0| 14, 19, 33 }}


Badness (Sintel): 1.59
Badness (Sintel): 1.59
Line 1,043: Line 1,063:


== Temperaments with a 2.3.13 gene ==
== Temperaments with a 2.3.13 gene ==
=== Tridecapyth ===
Sets a stack of 10 (9/8)s equal to 13/4.
[[Subgroup]]: 2.3.13
[[Comma list]]: 3489660928/3486784401
See: [[Tridecapyth comma#Temperaments]].
=== Threedic ===
=== Threedic ===
[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13
Line 1,089: Line 1,100:


[[Badness]] (Sintel): 0.200
[[Badness]] (Sintel): 0.200
=== Tridecapyth ===
Tridecapyth sets a stack of ten [[9/8]]'s equal to [[13/4]]. It shows a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping – for 118edo we get the 118f [[val]] that takes the second-closest, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, [[41edo]] uses this mapping, and so does [[94edo]], the val sum of 41 and 53. Thus it is of interest to flatter tunings of [[garibaldi]]/[[cassandra]] with fifths tending close to pure. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].
[[Subgroup]]: 2.3.13
{{mapping|legend=2| 1 0 -28 | 0 1 20 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9778{{c}}, ~3/2 = 702.0170{{c}}
: [[error map]]: {{val| -0.022 +0.040 -0.011 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0283{{c}}
: error map: {{val| 0.000 +0.070 -0.019 }}
{{Optimal ET sequence|legend=1| 12, 29f, 41, 53, 94, 147, 494, 641, 788, 2217, 3005 }}
[[Badness]] (Sintel): 0.211


=== Superflat ===
=== Superflat ===
Line 1,106: Line 1,135:
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
: error map: {{val| 0.000 -8.347 +14.960 }}
: error map: {{val| 0.000 -8.347 -14.960 }}


{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}
Line 1,132: Line 1,161:
Badness (Sintel): 0.135
Badness (Sintel): 0.135


==== 2.3.13.23 ====
==== 2.3.13.23 subgroup ====
Subgroup: 2.3.13.23
Subgroup: 2.3.13.23


Line 1,212: Line 1,241:
[[Badness]] (Sintel): 0.0294
[[Badness]] (Sintel): 0.0294


=== Dog ===
=== Deviaug ===
Dog is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19-subgroup analogue of [[mavila]].
Deviaug tempers out [[6912/6859]] in the 2.3.19 subgroup, setting [[24/19]] to 1/3 of an octave.  


[[Subgroup]]: 2.3.19
[[Subgroup]]: 2.3.19


[[Comma list]]: 81/76
[[Comma list]]: 6912/6859


{{Mapping|legend=2| 1 0 -2 | 0 1 4 }}
{{Mapping|legend=2| 3 0 8 | 0 1 1 }}
: mapping generators: ~2, ~3
: mapping generators: ~24/19, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.3813{{c}}, ~3/2 = 680.5089{{c}}
* [[WE]]: ~24/19 = 399.8571{{c}}, ~3/2 = 701.9795{{c}}
: [[error map]]: {{val| +3.381 -18.065 +31.285 }}
: [[error map]]: {{val| -0.429 -0.404 +2.894 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 680.5856{{c}}
* [[CWE]]: ~24/19 = 400.0000{{c}}, ~3/2 = 701.8810{{c}}
: error map: {{val| 0.000 -21.369 +24.8295 }}
: error map: {{val| 0.000 -0.074 +4.368 }}


{{Optimal ET sequence|legend=1| 2, 5h, 7, 16, 23 }}
{{Optimal ET sequence|legend=1| 9, 12, 75, 87, 99, 111, 123h, 135h }}


[[Badness]] (Sintel): 0.491
[[Badness]] (Sintel): 0.237


=== Lipsett ===
=== Lipsett ===
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