No-fives subgroup temperaments: Difference between revisions

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 === Leapfrog ===
-{{see also|Gentle region}}
+{{Main| Leapday }}
+{{See also| Gentle region }}
-Subgroup: 2.3.7
+In regular [[13-limit]] [[leapday]], the mapping for prime 5 is very complex at -25 generator steps, which is also in the opposite direction of how all other primes in the [[13-limit]] are reached. Furthermore, adding prime 5 to rank-3 [[parapythic]] is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable [[patent val]] tuning not appearing in the [[optimal ET sequence]] is [[80edo]], which is approximately the just-13's tuning (as [[10edo]] is used as a [[consistent circle]] of [[~]][[16/13]]'s therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris). In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which this case is also a sign of leapday being very efficient.
+Other related temperaments include [[leapweek]] and [[srutal]].
+[[Subgroup]]: 2.3.7
 [[Comma list]]: 14680064/14348907
-[[Gencom]]: [2 3/2; 14680064/14348907]
+{{Mapping|legend=2| 1 0 -21 | 0 1 15 }}
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6}}, {{val|0 1 0 15}}]
+{{Mapping|legend=3| 1 1 0 -6 | 0 1 0 15 }}
-[[Mapping|Sval mapping]]: [{{val|1 0 -21}}, {{val|0 1 15}}]
+: [[gencom]]: [2 3/2; 14680064/14348907]
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents
+[[Optimal tuning]]s:
+* [[POTE]]: ~2 = 1\1, ~3/2 = 704.721
 {{Optimal ET sequence|legend=1| 17, 46, 63 }}
 [[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
-Related temperaments: [[Hemifamity temperaments #Leapday|leapday]], [[Sensamagic clan #Leapweek|leapweek]], [[srutal]]
 ==== 2.3.7.11 ====
-Subgroup: 2.3.7.11
+[[Subgroup]]: 2.3.7.11
 [[Comma list]]: 896/891, 1331/1323
-[[Gencom]]: [2 3/2; 896/891 1331/1323]
+{{Mapping|legend=2| 1 0 -21 -14 | 0 1 15 11 }}
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3}}, {{val|0 1 0 15 11}}]
+{{Mapping|legend=3| 1 1 0 -6 -3 | 0 1 0 15 11 }}
-[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14}}, {{val|0 1 15 11}}]
+: [[gencom]]: [2 3/2; 896/891 1331/1323]
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
+[[Optimal tuning]]s:
+* [[POTE]]: ~3/2 = 704.753
 {{Optimal ET sequence|legend=1| 17, 46, 63 }}
@@ Line 233: / Line 238: @@
 ==== 2.3.7.11.13 ====
-Subgroup: 2.3.7.11.13
+[[Subgroup]]: 2.3.7.11.13
 [[Comma list]]: 169/168, 352/351, 364/363
-[[Gencom]]: [2 3/2; 169/169 352/351 364/363]
+{{Mapping|legend=2| 1 0 -21 -14 -9 | 0 1 15 11 8 }}
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]
+{{Mapping|legend=3| 1 1 0 -6 -3 -1 | 0 1 0 15 11 8 }}
-[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]
+: [[gencom]]: [2 3/2; 169/169 352/351 364/363]
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1\1, ~3/2 = 704.633
+* [[POTE]]: ~2 = 1\1, ~3/2 = 704.745
 {{Optimal ET sequence|legend=1| 17, 46, 63 }}
@@ Line 250: / Line 257: @@
 ===== Skidoo =====
-Subgroup: 2.3.7.11.13.23
+[[Subgroup]]: 2.3.7.11.13.23
 [[Comma list]]: 169/168, 208/207, 352/351, 364/363
-[[Gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
+{{Mapping|legend=2| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1 0 0 1}}, {{val|0 1 0 15 11 8 0 0 6}}]
+{{Mapping|legend=3| 1 1 0 -6 -3 -1 0 0 1 | 0 1 0 15 11 8 0 0 6 }}
-[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9 -5}}, {{val|0 1 15 11 8 6}}]
+: [[gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents
+[[Optimal tuning]]s:
+* [[POTE]]: ~2 = 1\1, ~3/2 = 704.729
 {{Optimal ET sequence|legend=1| 17, 46, 63 }}
@@ Line 267: / Line 275: @@
 ====== 2.3.7.11.13.23.29 ======
-Subgroup: 2.3.7.11.13.23.29
+[[Subgroup]]: 2.3.7.11.13.23.29
 [[Comma list]]: 169/168, 208/207, 232/231, 352/351, 364/363
-[[Gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
+{{Mapping|legend=2| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1 0 0 1 -11}}, {{val|0 1 0 15 11 8 0 0 6 27}}]
+{{Mapping|legend=3| 1 1 0 -6 -3 -1 0 0 1 -11 | 0 1 0 15 11 8 0 0 6 27 }}
-[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9 -5 -38}}, {{val|0 1 15 11 8 6 27}}]
+: [[gencom]]: [2 3/2; 169/169 208/207 352/351 364/363]
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents
+[[Optimal tuning]]s:
+* [[POTE]]: ~2 = 1\1, ~3/2 = 704.729
 {{Optimal ET sequence|legend=1| 17, 46, 63 }}
 ; Music
-* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (in progress):
+* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo]] by [[Inthar]] (in progress):
 ** I. Prelude
 ** II. Allemande
 ** III. Courante
-** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17EDO version]])
+** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17edo version]])
 ** [[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
 ** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]