Hemimean clan: Difference between revisions

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 {{Technical data page}}
 The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].
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 [[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
-[[Badness]] (Dirichlet): 0.091
+[[Badness]] (Sintel): 0.091
 = Strong extensions =
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 Mapping generators: ~2, ~25/14
-{{Multival|legend=1| 16 2 5 -34 -37 6 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
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 {{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
-Badness (Dirichlet): 0.304
+Badness (Sintel): 0.304
 === 11-limit ===
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 === Quadrawürschmidt ===
 This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.
+The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].
 Subgroup: 2.3.5.7.11
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 {{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
-{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
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 * [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
 : {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
 * [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
 : {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 {{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
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 {{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
-{{Multival|legend=1| 10 2 5 -20 -20 6 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
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 : mapping generators: ~2, ~75/49
-{{Multival|legend=1| 17 6 15 -30 -24 18 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
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 : mapping generators: ~2, ~48/35
-{{Multival|legend=1| 27 8 20 -50 -44 24 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
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 {{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
-{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
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 {{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
-{{Multival|legend=1| 29 16 40 -42 -18 48 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
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 {{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
-{{Multival|legend=1| 15 22 55 0 45 66 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
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 {{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
-{{Multival|legend=1| 31 24 60 -34 8 72 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
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 {{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
-{{Multival|legend=1| 11 30 75 22 88 90 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
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 RMS error: 0.5567 cents
-Badness (Dirichlet): 0.195
+Badness (Sintel): 0.195
 === Tridecimal didacus ===
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 Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
-Badness (Dirichlet): 0.324
+Badness (Sintel): 0.324
 ==== Mediantone ====
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 <nowiki />* 117d only appears without prime 19
-Badness (Dirichlet): 0.618
+Badness (Sintel): 0.618
 ==== Roulette ====
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 Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
-Badness (Dirichlet): 0.676
+Badness (Sintel): 0.676
 == Rectified hebrew ==
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 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Hemimean clan| ]] <!-- main article -->
 [[Category:Hemimean| ]] <!-- key article -->
 [[Category:Rank 2]]