No-threes subgroup temperaments: Difference between revisions

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 * Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
 * Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
-=== Rainy ===
-Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)
-A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
-[[Subgroup]]: 2.5.7
-[[Comma list]]: [[2100875/2097152]]
-[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
-[[Gencom]]: [2 256/245; 2100875/2097152]
-[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
-Optimal tuning ([[POTE]]): ~256/245 = 77.205
-{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
 === Frostburn ===
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 {{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
+=== Mabilic ===
+{{See also| Chromatic pairs #Mabilic }}
+Given below is the no-three version of [[Mabila family#Semabila|semabila]], or equivalently the no-threes version of [[Magic family#Trismegistus|trismegistus]]. It is the 7 &amp; 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
+[[Subgroup]]: 2.5.7
+[[Comma list]]: 1071875/1048576
+{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
+{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
+: [[gencom]]: [2 175/128; 1071875/1048576]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236
+{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents
+=== Rainy ===
+Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)
+A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
+[[Subgroup]]: 2.5.7
+[[Comma list]]: [[2100875/2097152]]
+[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
+[[Gencom]]: [2 256/245; 2100875/2097152]
+[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
+Optimal tuning ([[POTE]]): ~256/245 = 77.205
+{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
 === French decimal ===
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 {{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
+=== Augment ===
+{{See also| Chromatic pairs #Augment }}
+Augment is related to [[augmented]].
+[[Subgroup]]: 2.5.7.11
+[[Comma list]]: 56/55, 128/125
+{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
+{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}
+: [[gencom]]: [5/4 8/7; 56/55 128/125]
+[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275
+{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}
+[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents
 === Ostara ===
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 {{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
+=== Wizz ===
+{{See also| Chromatic pairs #Wizz }}
+Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].
+[[Subgroup]]: 2.5.11
+[[Comma list]]: [[15625/15488]]
+{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
+{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
+: [[gencom]]: [125/88 5/4; 15625/15488]
+[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768
+{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.3997
 === Insect ===
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 {{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
-=== Superquintal ===
+=== Sephiroth ===
-[[Subgroup]]: 2.5.13
+{{See also| Chromatic pairs #Sephiroth }}
+Sephiroth is the no-7 restriction of [[muggles]].
+[[Subgroup]]: 2.5.11.13.17
+[[Comma list]]: 65/64, 170/169, 221/220
+{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
-[[Comma list]]: 64000000/62748517
+{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}
-{{Mapping|legend=2|1 5 6|0 -7 -6}}
+: [[gencom]]: [2 5/4; 65/64 170/169 221/220]
-: Mapping generators, ~2, ~13/10
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
+{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}
-{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
+[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents
 === Trader ===
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 {{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
+=== Superquintal ===
+[[Subgroup]]: 2.5.13
+[[Comma list]]: 64000000/62748517
+{{Mapping|legend=2|1 5 6|0 -7 -6}}
+: Mapping generators, ~2, ~13/10
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
+{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
 === Vengeance ===