No-fives subgroup temperaments: Difference between revisions

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@@ Line 5: / Line 5: @@
 === Archy ===
 See [[Archytas clan #Archy]].
-==== Supra ====
-See [[Archytas clan #Supra]].
-===== Supraphon =====
-See [[Archytas clan #Supraphon]].
 ==== Suhajira ====
@@ Line 55: / Line 49: @@
 === Slendroschismic ===
 See [[5th-octave temperaments #Slendroschismic]].
-==== Pentadecoid ====
-See [[15th-octave temperaments #Pentadecoid]].
 === Navy ===
@@ Line 598: / Line 589: @@
 === Purpleheart ===
-{{See also| Whitewood }}
+{{Main| Whitewood }}
 [[Subgroup]]: 2.3.7
@@ Line 866: / Line 857: @@
 === Neutral ===
 See [[Rastmic clan #Neutral]].
-==== Namo ====
-See [[Rastmic clan #Namo]].
 === Io ===
@@ Line 891: / Line 879: @@
 [[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 ===
@@ Line 915: / Line 923: @@
 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:
@@ Line 930: / Line 938: @@
 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
@@ Line 1,013: / Line 1,021: @@
 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 ==
@@ Line 1,155: / Line 1,183: @@
 [[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.
-[[Comma list]]: 2.3.85 {{val| -40 1 6 }}
+[[Subgroup]]: 2.3.19
-{{Mapping|legend=2| 1 -2 7 | 0 6 -1 }}
+[[Comma list]]: 513/512
-: mapping generators: ~2, ~128/85
+{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
+: mapping generators: ~2, ~3
 [[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 ===
@@ Line 1,205: / Line 1,222: @@
 [[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 ===