@@ Line 5: / Line 5: @@
 | ja =
 }}
-The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. The septimal version of magic is optimal, for some searches, in the [[9-odd-limit]]. It has slightly higher complexity than [[meantone]] and is also closer to just intonation. It is the simplest rank-2 temperament that tunes every 9-odd-limit interval better than is possible in [[12edo]]. The most prominent deficiency is that it lacks [[Rothenberg propriety|proper]] or nearly-proper [[MOS scale]]s in the 5- to 10-note region. Properties may depend on tuning and extension.
+{{Technical data page}}
+The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. The septimal version of magic is locally optimal, for some searches, in the [[9-odd-limit]]. Magic has a slightly higher complexity than [[meantone]] but it is closer to just intonation. It is the simplest rank-2 temperament that tunes every [[9-odd-limit]] interval better than is possible in [[12edo]]. The most prominent deficiency is that it lacks [[Rothenberg propriety|proper]] or nearly-proper [[mos scale]]s in the 5- to 10-note region. Properties may depend on tuning and extension.
 == Magic ==
 {{Main| Magic }}
-The monzo of the magic comma is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 5 1 -10 }} for the [[wedgie]]. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the [[optimal patent val]] generator, also makes a lot of sense, and using [[19edo]] or [[22edo]] is always possible.
+The [[generator]] of magic is a major third, and to get to the interval class of fifths requires five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. [[41edo|13\41]] is a highly recommendable generator, though [[60edo|19\60]], the [[optimal patent val]] generator, also makes a lot of sense, and using [[19edo]] or [[22edo]] is always possible.
-Subgroup: 2.3.5
+[[Subgroup]]: 2.3.5
 [[Comma list]]: 3125/3072
-[[Mapping]]: [{{val| 1 0 2 }}, {{val| 0 5 1 }}]
+{{Mapping|legend=1| 1 0 2 | 0 5 1 }}
-Mapping generators: ~2, ~5/4
+: mapping generators: ~2, ~5/4
-[[POTE generator]]: ~5/4 = 380.058
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.2449{{c}}, ~5/4 = 380.4527{{c}}
+: [[error map]]: {{val| +1.245 +0.309 -3.371 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.2194{{c}}
+: error map: {{val| 0.000 -0.858 -6.094 }}
 [[Minimax tuning]]:
 * [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 [[Tuning ranges]]:
 * 5-odd-limit [[diamond monotone]]: ~5/4 = [360.000, 400.000] (3\10 to 1\3)
-* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314]
+* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
-* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [378.910, 386.314]
 [[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s.
-{{Optimal ET sequence|legend=1|3, 13b, 16, 19, 41, 60, 221c, 281c }}
+{{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 41, 60, 221cc, 281cc }}
-[[Badness]]: 0.039163
+[[Badness]] (Sintel): 0.919
 === Overview to extensions ===
-Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives septimal magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.
+Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives septimal magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator, as well as low-accuracy extensions including darkstone and brightstone.
-Weak extensions considered below are hocum, trismegistus, quadrimage, and warlock. Discussed elsewhere are
+Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are
 * ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
 * ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]
@@ Line 47: / Line 51: @@
 {{Main| Magic }}
-Septimal magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41edo]] is a good magic tuning, and 19- or 22-note MOSes are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.
+Septimal magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41edo]] is a good magic tuning, and 19- or 22-note mosses are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.
-This temperament, with its accurate fifths, works well with [[9-odd-limit]] harmony. It is more accurate than [[meantone]] and simpler than [[Garibaldi temperament|garibaldi]]. It is a little tricky to work with because in its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.
+This temperament, with its accurate fifths, works well with [[9-odd-limit]] harmony. It is more accurate than [[meantone]] and simpler than [[garibaldi]]. It is a little tricky to work with because its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.
-/224 is the [[Marvel family|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.
+/224 is the [[marvel family|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the [[sensamagic family|sensamagic]] comma, leads to another essentially tempered 9-odd-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.
-/243, the [[Sensamagic family|sensamagic]] comma, leads to another essentially tempered 9-odd-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.
+By adding [[100/99]] and [[105/104]] to the list of commas, magic can be extended to the 11-limit and 13-limit. 11-limit magic allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. For this, [[104edo]] provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning. For the 13-limit, 41edo makes for a recommendable tuning.
-By adding [[100/99]] to the list of commas, magic can be extended to an 11-limit version, {{multival| 5 1 12 -8 … }}. For this, [[104edo]] provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7
 [[Comma list]]: 225/224, 245/243
-[[Mapping]]: [{{val| 1 0 2 -1 }}, {{val| 0 5 1 12 }}]
+{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}
-Mapping generators: ~2, ~5/4
+: mapping generators: ~2, ~5/4
-{{Multival|legend=1| 5 1 12 -10 5 25 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
-[[POTE generator]]: ~5/4 = 380.352
+: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}
+: error map: {{val| 0.000 +0.333 -5.856 -3.335 }}
 [[Minimax tuning]]:
 * 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 [[Tuning ranges]]:
 * 7- and 9-odd-limit [[diamond monotone]]: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
-* 7- and 9-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314]
+* 7- and 9-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]
 [[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents.
-{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224cd }}
+{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}
-[[Badness]]: 0.018918
+[[Badness]] (Sintel): 0.479
 === 11-limit ===
-Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)
 Subgroup: 2.3.5.7.11
 Comma list: 100/99, 225/224, 245/243
-Mapping: [{{val| 1 0 2 -1 6 }}, {{val| 0 5 1 12 -8 }}]
+Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
-POTE generator: ~5/4 = 380.696
+Optimal tunings:
+* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}
 Minimax tuning:
-* [[11-odd-limit]]: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
+* 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Tuning ranges:
 * 11-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
-* 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314]
+* 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
-* 11-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]
-{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145c }}
+{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}
-Badness: 0.020352
+Badness (Sintel): 0.673
 ==== 13-limit ====
+A notable [[patent val]] tuning beyond the [[optimal patent val]] of 41edo is [[19edo|19]] + [[41edo|41]] = [[60edo]].
 Subgroup: 2.3.5.7.11.13
 Comma list: 100/99, 105/104, 144/143, 196/195
-Mapping: [{{val| 1 0 2 -1 6 -2 }}, {{val| 0 5 1 12 -8 18 }}]
+Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}
-POTE generator: ~5/4 = 380.427
+Optimal tunings:
+* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}
 Tuning ranges:
 * 13- and 15-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
 * 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]
-{{Optimal ET sequence|legend=1| 19, 22f, 41, 265cdef }}
+{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
-Badness: 0.021509
+Badness (Sintel): 0.889
-===== 17-limit =====
+===== Magical =====
 Subgroup: 2.3.5.7.11.13.17
 Comma list: 100/99, 105/104, 120/119, 144/143, 154/153
-Mapping: [{{val| 1 0 2 -1 6 -2 6 }}, {{val| 0 5 1 12 -8 18 -6 }}]
+Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}
-POTE generator: ~5/4 = 380.604
+Optimal tunings:
+* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}
-{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
+{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
-Badness: 0.020633
+Badness (Sintel): 1.05
-===== Evening =====
+====== Magicus ======
-'''Evening''' is a remarkable subgroup temperament of 19&amp;41 with prime harmonics of 29 and 31.
+Subgroup: 2.3.5.7.11.13.17.19
-Subgroup: 2.3.5.7.11.13.29.31
+Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153
-Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
+Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
-Mapping: [{{val| 1 0 2 -1 6 -2 2 4 }}, {{val| 0 5 1 12 -8 18 9 3 }}]
+Optimal tunings:
+* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}
-POTE generator: ~5/4 = 380.416
+{{Optimal ET sequence|legend=0| 19, 41 }}
-{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
+Badness (Sintel): 1.27
-==== Sorcery ====
+====== Magica ======
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 65/64, 78/77, 91/90, 100/99
+Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169
-Mapping: [{{val| 1 0 2 -1 6 4 }}, {{val| 0 5 1 12 -8 -1 }}]
+Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}
-POTE generator: ~5/4 = 380.477
+Optimal tunings:
+* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}
-{{Optimal ET sequence|legend=1| 19, 22, 41f }}
+{{Optimal ET sequence|legend=0| 22fh, 41 }}
-Badness: 0.025829
+Badness (Sintel): 1.21
-===== 17-limit =====
+===== Magia =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 52/51, 65/64, 78/77, 91/90, 100/99
+Comma list: 100/99, 105/104, 144/143, 170/169, 196/195
+Mapping: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}
+Optimal tunings:
+* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}
+{{Optimal ET sequence|legend=0| 19g, 41, 60 }}
-Mapping: [{{val| 1 0 2 -1 6 4 6 }}, {{val| 0 5 1 12 -8 -1 -6 }}]
+Badness (Sintel): 1.34
-POTE generator: ~5/4 = 380.729
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-{{Optimal ET sequence|legend=1| 19, 22, 41f }}
+Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195
-Badness: 0.023768
+Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}
+Optimal tunings:
+* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}
+{{Optimal ET sequence|legend=0| 19gh, 41 }}
+Badness (Sintel): 1.44
+===== Evening =====
+Evening is a remarkable subgroup temperament of {{nowrap| 19 & 22f }} with prime harmonics of 29 and 31.
+Subgroup: 2.3.5.7.11.13.29.31
+Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
+Subgroup-val mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}
+Optimal tunings:
+* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}
+{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
+Badness (Sintel): 0.807
+==== Sorcery ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 65/64, 78/77, 91/90, 100/99
+Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}
+Optimal tunings:
+* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
+{{Optimal ET sequence|legend=0| 19, 22, 41f }}
+Badness (Sintel): 1.07
 ==== Necromancy ====
@@ Line 183: / Line 243: @@
 Comma list: 100/99, 225/224, 245/243, 275/273
-Mapping: [{{val| 1 0 2 -1 6 11 }}, {{val| 0 5 1 12 -8 -23 }}]
+Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}
-POTE generator: ~5/4 = 380.787
+Optimal tunings:
+* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}
-{{Optimal ET sequence|legend=1| 19f, 22, 41, 63, 104 }}
+{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}
-Badness: 0.025275
+Badness (Sintel): 1.04
 ===== 17-limit =====
@@ Line 196: / Line 258: @@
 Comma list: 100/99, 120/119, 154/153, 225/224, 273/272
-Mapping: [{{val| 1 0 2 -1 6 11 6 }}, {{val| 0 5 1 12 -8 -23 -6 }}]
+Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}
-POTE generator: ~5/4 = 380.827
+Optimal tunings:
+* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}
-{{Optimal ET sequence|legend=1| 19f, 22, 41, 63, 104g }}
+{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}
-Badness: 0.022032
+Badness (Sintel): 1.12
 ==== Soothsaying ====
@@ Line 209: / Line 273: @@
 Comma list: 100/99, 225/224, 245/243, 1352/1331
-Mapping: [{{val| 2 0 4 -2 12 15 }}, {{val| 0 5 1 12 -8 -12 }}]
+Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}
-POTE generator: ~5/4 = 380.508
+Optimal tunings:
+* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{c}}
-{{Optimal ET sequence|legend=1| 22, 60, 82 }}
+{{Optimal ET sequence|legend=0| 22, 60, 82 }}
-Badness: 0.055443
+Badness (Sintel): 2.29
 ===== 17-limit =====
@@ Line 222: / Line 288: @@
 Comma list: 100/99, 221/220, 225/224, 245/243, 273/272
-Mapping: [{{val| 2 0 4 -2 12 15 5 }}, {{val| 0 5 1 12 -8 -12 5 }}]
+Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}
+Optimal tunings:
+* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}
+{{Optimal ET sequence|legend=0| 22, 60, 82 }}
-POTE generator: ~5/4 = 380.508
+Badness (Sintel): 1.82
-{{Optimal ET sequence|legend=1| 22, 60, 82 }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.035654
+Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272
+Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}
+Optimal tunings:
+* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}
+{{Optimal ET sequence|legend=0| 22, 60, 82 }}
+Badness (Sintel): 1.90
 === Telepathy ===
@@ Line 235: / Line 318: @@
 Comma list: 55/54, 99/98, 176/175
-Mapping: [{{val| 1 0 2 -1 -1 }}, {{val| 0 5 1 12 14 }}]
+Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
-POTE generator: ~5/4 = 381.019
+Optimal tunings:
+* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}
-{{Optimal ET sequence|legend=1| 19e, 22, 41e, 63e }}
+{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}
-Badness: 0.027109
+Badness (Sintel): 0.896
 ==== 13-limit ====
@@ Line 248: / Line 333: @@
 Comma list: 55/54, 65/64, 91/90, 99/98
-Mapping: [{{val| 1 0 2 -1 -1 4 }}, {{val| 0 5 1 12 14 -1 }}]
+Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}
-POTE generator: ~5/4 = 380.520
+Optimal tunings:
+* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}
-{{Optimal ET sequence|legend=1| 19e, 22, 41ef }}
+{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}
-Badness: 0.025522
+Badness (Sintel): 1.05
-===== 17-limit =====
+==== Intuition ====
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11.13
-Comma list: 55/54, 65/64, 85/84, 91/90, 99/98
+Comma list: 55/54, 66/65, 99/98, 105/104
-Mapping: [{{val| 1 0 2 -1 -1 4 -1 }}, {{val| 0 5 1 12 14 -1 16 }}]
+Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
-POTE generator: ~5/4 = 380.619
+Optimal tunings:
+* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}
-{{Optimal ET sequence|legend=1| 19eg, 22, 41efg }}
+{{Optimal ET sequence|legend=0| 19e, 22f }}
-Badness: 0.020201
+Badness (Sintel): 1.08
-==== Intuition ====
+=== Horcrux ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 55/54, 66/65, 99/98, 105/104
+Comma list: 45/44, 56/55, 245/243
-Mapping: [{{val| 1 0 2 -1 -1 -2 }}, {{val| 0 5 1 12 14 18 }}]
+Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}
-POTE generator: ~5/4 = 380.483
+Optimal tunings:
+* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}
-{{Optimal ET sequence|legend=1| 19e, 22f, 41e }}
+{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
-Badness: 0.026089
+Badness (Sintel): 1.30
-===== 17-limit =====
+==== 13-limit ====
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11.13
-Comma list: 55/54, 66/65, 85/84, 99/98, 105/104
+Comma list: 45/44, 56/55, 78/77, 245/243
-Mapping: [{{val| 1 0 2 -1 -1 -2 -1 }}, {{val| 0 5 1 12 14 18 16 }}]
+Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}
-POTE generator: ~5/4 = 380.604
+Optimal tunings:
+* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}
-{{Optimal ET sequence|legend=1| 19eg, 22f, 41eg }}
+{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}
-Badness: 0.020274
+Badness (Sintel): 1.32
-=== Horcrux ===
+===== 17-limit =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 45/44, 56/55, 245/243
+Comma list: 45/44, 56/55, 78/77, 85/84, 245/243
-Mapping: [{{val| 1 0 2 -1 0 }}, {{val| 0 5 1 12 11 }}]
+Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}
-POTE generator: ~5/4 = 379.642
+Optimal tunings:
+* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{c}}
-{{Optimal ET sequence|legend=1| 3de, 19, 41ee, 60ee }}
+{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}
-Badness: 0.039282
+Badness (Sintel): 1.43
-==== Charisma ====
+===== Horcruxic =====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 45/44, 56/55, 78/77, 245/243
+Comma list: 35/34, 45/44, 52/51, 56/55, 245/243
-Mapping: [{{val| 1 0 2 -1 0 -2 }}, {{val| 0 5 1 12 11 18 }}]
+Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}
-POTE generator: ~5/4 = 379.791
+Optimal tunings:
+* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{c}}
-{{Optimal ET sequence|legend=1| 3def, 19, 41ee, 60ee }}
+{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}
-Badness: 0.031938
+Badness (Sintel): 1.51
 ==== Glamour ====
@@ Line 326: / Line 423: @@
 Comma list: 45/44, 56/55, 65/64, 245/243
-Mapping: [{{val| 1 0 2 -1 0 4 }}, {{val| 0 5 1 12 11 -1}}]
+Mapping: {{mapping| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}
-POTE generator: ~5/4 = 379.116
+Optimal tunings:
+* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{c}}
-{{Optimal ET sequence|legend=1| 3de, 19 }}
+{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
-Badness: 0.033317
+Badness (Sintel): 1.38
 === Witchcraft ===
@@ Line 339: / Line 438: @@
 Comma list: 225/224, 245/243, 441/440
-Mapping: [{{val| 1 0 2 -1 -7 }}, {{val| 0 5 1 12 33 }}]
+Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}
-POTE generator: ~5/4 = 380.232
+Optimal tunings:
+* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2849{{c}}
-{{Optimal ET sequence|legend=1| 41, 60e, 101cd, 243cde }}
+{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}
-Badness: 0.030706
+Badness (Sintel): 1.02
 ==== 13-limit ====
@@ Line 352: / Line 453: @@
 Comma list: 105/104, 196/195, 245/243, 275/273
-Mapping: [{{val| 1 0 2 -1 -7 -2 }}, {{val| 0 5 1 12 33 18 }}]
+Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}
-POTE generator: ~5/4 = 380.189
+Optimal tunings:
+* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2349{{c}}
-{{Optimal ET sequence|legend=1| 41, 60e, 101cd }}
+{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}
-Badness: 0.023547
+Badness (Sintel): 0.973
 ==== 17-limit ====
@@ Line 365: / Line 468: @@
 Comma list: 105/104, 154/153, 170/169, 196/195, 245/243
-Mapping: [{{val| 1 0 2 -1 -7 -2 -7 }}, {{val| 0 5 1 12 33 18 35 }}]
+Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}
-POTE generator: ~5/4 = 380.114
+Optimal tunings:
+* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1599{{c}}
-{{Optimal ET sequence|legend=1| 41, 60e, 101cd, 161cde }}
+{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}
-Badness: 0.020756
+Badness (Sintel): 1.06
 === Divination ===
@@ Line 378: / Line 483: @@
 Comma list: 121/120, 225/224, 245/243
-Mapping: [{{val| 2 0 4 -2 5 }}, {{val| 0 5 1 12 3 }}]
+Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}
-POTE generator: ~5/4 = 380.233
+Optimal tunings:
+* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{c}}
-{{Optimal ET sequence|legend=1| 22, 38d, 60e, 142cde }}
+{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}
-Badness: 0.035864
+Badness (Sintel): 1.19
 ==== 13-limit ====
@@ Line 391: / Line 498: @@
 Comma list: 105/104, 121/120, 196/195, 245/243
-Mapping: [{{val| 2 0 4 -2 5 -4 }}, {{val| 0 5 1 12 3 18 }}]
+Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}
-POTE generator: ~5/4 = 379.920
+Optimal tunings:
+* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}
-{{Optimal ET sequence|legend=1| 22f, 60e }}
+{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
-Badness: 0.034551
+Badness (Sintel): 1.43
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 105/104, 121/120, 154/153, 196/195, 245/243
+Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}
+Optimal tunings:
+* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.0672{{c}}
+{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
+Badness (Sintel): 1.21
 === Hocus ===
@@ Line 404: / Line 528: @@
 Comma list: 225/224, 243/242, 245/242
-Mapping: [{{val| 1 5 3 11 12 }}, {{val| 0 -10 -2 -24 -25 }}]
+Mapping: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}
-POTE generator: ~14/11 = 409.910
+Optimal tunings:
+* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{c}}
-{{Optimal ET sequence|legend=1| 38d, 41, 120cd, 161cd, 202cd }}
+{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}
-Badness: 0.038519
+Badness (Sintel): 1.27
 ==== 13-limit ====
@@ Line 417: / Line 543: @@
 Comma list: 105/104, 196/195, 243/242, 245/242
-Mapping: [{{val| 1 5 3 11 12 16 }}, {{val| 0 -10 -2 -24 -25 -36 }}]
+Mapping: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}
+Optimal tunings:
+* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{c}}
+{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}
+Badness (Sintel): 1.25
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
+Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}
+Optimal tunings:
+* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}
+{{Optimal ET sequence|legend=0| 38df, 41, 79d }}
+Badness (Sintel): 1.30
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
-POTE generator: ~14/11 = 410.004
+Mapping: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}
-{{Optimal ET sequence|legend=1| 41, 79d, 120cd }}
+Optimal tunings:
+* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}
-Badness: 0.030280
+{{Optimal ET sequence|legend=0| 38df, 41, 79dh }}
+Badness (Sintel): 1.23
 == Muggles ==
-{{main| Muggles }}
+{{Main| Muggles }}
-Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7- or 10-note MOS are reasonable choices, as while the flatter generator compromises the accuracy of the 5-limit intervals, it grants simpler access to some higher-limit ones, and makes the small steps larger and more melodically effective.
+Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo]], in which tuning it is the same thing as magic. Muggles works better for small scales than magic in the sense that 7- or 10-note mosses are reasonable choices, as while the flatter generator compromises the accuracy of the 5-limit intervals, it grants simpler access to some higher-limit ones, and makes the small steps larger and more melodically effective.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 525/512
-[[Mapping]]: [{{val| 1 0 2 5 }}, {{val| 0 5 1 -7 }}]
+{{Mapping|legend=1| 1 0 2 5 | 0 5 1 -7 }}
-{{Multival|legend=1| 5 1 -7 -10 -25 -19 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1203.9554{{c}}, ~5/4 = 379.7269{{c}}
-[[POTE generator]]: ~5/4 = 378.479
+: [[error map]]: {{val| +3.955 -3.321 +1.324 -7.137 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 378.5328{{c}}
+: error map: {{val| 0.000 -9.291 -7.781 -18.555 }}
 [[Tuning ranges]]:
-* 7-odd-limit [[diamond monotone]]: ~5/4 = [375.000, 378.947] (5\16 to 6\19)
+* [[7-odd-limit]] [[diamond monotone]]: ~5/4 = [375.000, 378.947] (5\16 to 6\19)
-* 9-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
+* [[9-odd-limit]] diamond monotone: ~5/4 = 378.947 (6\19)
 * 7- and 9-odd-limit [[diamond tradeoff]]: ~5/4 = [375.882, 386.314]
-* 7-odd-limit diamond monotone and tradeoff: ~5/4 = [375.882, 378.947]
-* 9-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947
-{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcd }}
+{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}
-[[Badness]]: 0.056206
+[[Badness]] (Sintel): 1.42
 === 11-limit ===
@@ Line 456: / Line 614: @@
 Comma list: 45/44, 126/125, 385/384
-Mapping: [{{val| 1 0 2 5 0 }}, {{val| 0 5 1 -7 11 }}]
+Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}
-POTE generator: ~5/4 = 377.724
+Optimal tunings:
+* WE: ~2 = 1203.0804{{c}}, ~5/4 = 378.6936{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8174{{c}}
 Tuning ranges:
 * 11-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
 * 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]
-* 11-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947
-{{Optimal ET sequence|legend=1| 16, 19, 35, 54bd }}
+{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}
-Badness: 0.048038
+Badness (Sintel): 1.59
 ==== 13-limit ====
@@ Line 474: / Line 633: @@
 Comma list: 45/44, 65/64, 78/77, 126/125
-Mapping: [{{val| 1 0 2 5 0 4 }}, {{val| 0 5 1 -7 11 -1 }}]
+Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}
-POTE generator: ~5/4 = 377.653
+Optimal tunings:
+* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}
-{{Optimal ET sequence|legend=1| 16, 19, 35f, 54bdf }}
+{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}
-Badness: 0.030386
+Badness (Sintel): 1.26
 === Muggloid ===
@@ Line 487: / Line 648: @@
 Comma list: 33/32, 126/125, 176/175
-Mapping: [{{val| 1 0 2 5 5 }}, {{val| 0 5 1 -7 -5 }}]
+Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}
-POTE generator: ~5/4 = 377.832
+Optimal tunings:
+* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8142{{c}}
-{{Optimal ET sequence|legend=1| 3, 10bd, 13bd, 16, 19e }}
+{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}
-Badness: 0.046970
+Badness (Sintel): 1.55
 ==== 13-limit ====
@@ Line 500: / Line 663: @@
 Comma list: 33/32, 65/64, 105/104, 126/125
-Mapping: [{{val| 1 0 2 5 5 4 }}, {{val| 0 5 1 -7 -5 -1 }}]
+Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}
+Optimal tunings:
+* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}
+{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}
+Badness (Sintel): 1.19
+== Brightstone ==
+Brightstone tempers out 64/63 and may be described as {{nowrap| 22 & 25 }}. 22edo itself is a good tuning, in which case it is identical to magic. Brightstone can be extended to the 11- and 13-limit in a similar way to muggles.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 64/63, 3125/3024
+{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.1701{{c}}, ~5/4 = 381.3741{{c}}
+: error map: {{val| -1.830 +4.915 -8.599 +6.454 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 381.9562{{c}}
+: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}
-POTE generator: ~5/4 = 377.838
+{{Optimal ET sequence|legend=1| 3, 19d, 22 }}
-{{Optimal ET sequence|legend=1| 3, 10bd, 13bd, 16, 19e }}
+[[Badness]] (Sintel): 2.23
-Badness: 0.028732
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 55/54, 64/63, 625/616
+Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}
+Optimal tunings:
+* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}
+{{Optimal ET sequence|legend=0| 22, 69b }}
+Badness (Sintel): 1.85
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 40/39, 55/54, 64/63, 625/616
+Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}
+Optimal tunings:
+* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}
+{{Optimal ET sequence|legend=0| 22f, 47bff }}
+Badness (Sintel): 1.99
+== Darkstone ==
+Darkstone (16 & 19d) is a low-accuracy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles.
+This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 36/35, 1875/1792
+{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.7458{{c}}, ~5/4 = 377.2996{{c}}
+: [[error map]]: {{val| +1.746 -15.457 -5.523 +26.870 }}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 376.9630{{c}}
+: error map: {{val| 0.000 -17.140 -9.351 +23.841 }}
+{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}
+[[Badness]] (Sintel): 2.13
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 36/35, 45/44, 363/343
+Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}
+Optimal tunings:
+* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}
+{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}
+Badness (Sintel): 1.55
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 27/26, 36/35, 45/44, 363/343
+Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
+Optimal tunings:
+* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{c}}
+{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}
+Badness (Sintel): 1.58
+Scales: [[User:BudjarnLambeth/Volcanic glass]]
+Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)
 == Hocum ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 3125/3072, 4000/3969
-[[Mapping]]: [{{val| 1 5 3 -3 }}, {{val| 0 -10 -2 17 }}]
+{{Mapping|legend=1| 1 -5 1 14 | 0 10 2 -17 }}
-{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
+: mapping generators: ~2, ~63/40
-[[POTE generator]]: ~63/50 = 410.108
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.8375{{c}}, ~63/40 = 790.7032{{c}}
+: [[error map]]: {{val| +0.838 +0.890 -4.070 +0.944 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 790.1542{{c}}
+: error map: {{val| 0.000 -0.413 -6.005 -1.447 }}
-{{Optimal ET sequence|legend=1| 38, 41, 161c, 202c, 243c, 284c }}
+{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}
-[[Badness]]: 0.107115
+[[Badness]] (Sintel): 2.71
 == Trismegistus ==
-Subgroup: 2.3.5.7
+{{Main| Mabilic and trismegistus }}
+{{See also| No-threes subgroup temperaments #Mabilic }}
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 1029/1024, 3125/3072
-[[Mapping]]: [{{val| 1 10 4 0 }}, {{val| 0 -15 -3 5 }}]
+{{Mapping|legend=1| 1 -5 1 5 | 0 15 3 -5 }}
-{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
+: mapping generators: ~2, ~168/125
-[[POTE generator]]: ~147/100 = 673.290
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.0799{{c}}, ~168/125 = 527.1841{{c}}
+: [[error map]]: {{val| +1.080 +0.408 -3.681 +0.653 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~168/125 = 526.7349{{c}}
+: error map: {{val| 0.000 -0.932 -6.109 -2.500 }}
-{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221c, 262c }}
+{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}
-[[Badness]]: 0.098334
+[[Badness]] (Sintel): 2.49
 === 11-limit ===
@@ Line 543: / Line 823: @@
 Comma list: 385/384, 441/440, 625/616
-POTE generator: ~22/15 = 673.340
+Mapping: {{mapping| 1 -5 1 5 -4 | 0 15 3 -5 17 }}
-Mapping: [{{val| 1 10 4 0 13 }}, {{val| 0 -15 -3 5 -17 }}]
+Optimal tunings:
+* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}
-{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139c, 180c }}
+{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
-Badness: 0.045623
+Badness (Sintel): 1.51
 === 13-limit ===
@@ Line 556: / Line 838: @@
 Comma list: 105/104, 144/143, 275/273, 625/616
-Mapping: [{{val| 1 10 4 0 13 11 }}, {{val| 0 -15 -3 5 -17 -13 }}]
+Mapping: {{mapping| 1 -5 1 5 -4 -2 | 0 15 3 -5 17 13 }}
+Optimal tunings:
+* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6548{{c}}
+{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
+Badness (Sintel): 1.37
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
-POTE generator: ~22/15 = 673.359
+Comma list: 105/104, 144/143, 441/440, 210/209, 625/616
-{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139cf }}
+Subgroup-val mapping: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}
-Badness: 0.033081
+Optimal tunings:
+* WE: ~2 = 1200.5832{{c}}, ~15/11 = 526.8804{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6368{{c}}
+{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
+Badness (Sintel): 1.26
 == Quadrimage ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 2401/2400, 3125/3072
-[[Mapping]]: [{{val| 1 5 3 4 }}, {{val| 0 -20 -4 -7 }}]
+{{Mapping|legend=1| 1 -15 -1 -3 | 0 20 4 7 }}
-{{Multival|legend=1| 20 4 7 -40 -45 5 }}
+: mapping generators: ~2, ~25/14
-[[POTE generator]]: ~28/25 = 204.987
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.2708{{c}}, ~25/14 = 996.0669{{c}}
+: [[error map]]: {{val| +1.271 +0.322 -3.317 -0.170 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 995.0515{{c}}
+: error map: {{val| 0.000 -0.926 -6.108 -3.466 }}
-{{Optimal ET sequence|legend=1| 6, 35, 41, 158cd, 199cd, 240cd, 281cd }}
+{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
-[[Badness]]: 0.127422
+[[Badness]] (Sintel): 3.22
 === 11-limit ===
@@ Line 584: / Line 887: @@
 Comma list: 245/242, 385/384, 625/616
-Mapping: [{{val| 1 5 3 4 5 }}, {{val| 0 -20 -4 -7 -9 }}]
+Mapping: {{mapping| 1 -15 -1 -3 -4 | 0 20 4 7 9 }}
-POTE generator: ~28/25 = 204.956
+Optimal tunings:
+* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}
-{{Optimal ET sequence|legend=1| 6, 35, 41, 199cde, 240cde, 281cde }}
+{{Optimal ET sequence|legend=0| 6, 35, 41 }}
-Badness: 0.061572
+Badness (Sintel): 2.04
 === 13-limit ===
@@ Line 597: / Line 902: @@
 Comma list: 105/104, 144/143, 245/242, 625/616
-Mapping: [{{val| 1 5 3 4 5 9 }}, {{val| 0 -20 -4 -7 -9 -31 }}]
+Mapping: {{mapping| 1 -15 -1 -3 -22 | 0 20 4 7 9 31 }}
+Optimal tunings:
+* WE: ~2 = 1200.6276{{c}}, ~25/14 = 995.4920{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 994.9901{{c}}
+{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}
+Badness (Sintel): 1.82
+== Quinmage ==
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 3125/3072, 16875/16807
+{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}
+: mapping generators: ~2, ~48/35
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3334{{c}}, ~48/35 = 556.6311{{c}}
+: error map: {{val| +1.333 +0.489 -3.158 -0.835 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 556.0504{{c}}
+: error map: {{val| 0.000 -0.695 -6.062 -3.868 }}
+{{Optimal ET sequence|legend=1| 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd }}
+[[Badness]] (Sintel): 4.92
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 385/384, 625/616, 2401/2376
+Mapping: {{mapping| 1 -10 0 -6 3 | 0 25 5 19 1 }}
+Optimal tunings:
+* WE: ~2 = 1200.4252{{c}}, ~11/8 = 556.2831{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.0951{{c}}
+{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
+Badness (Sintel): 3.36
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 196/195, 364/363, 385/384, 625/616
+Mapping: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}
-POTE generator: ~28/25 = 205.028
+Optimal tunings:
+* WE: ~2 = 1199.8239{{c}}, ~11/8 = 556.0389{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.1171{{c}}
-{{Optimal ET sequence|legend=1| 41, 117c, 158cd, 199cdef }}
+{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
-Badness: 0.044047
+Badness (Sintel): 2.80
 == Warlock ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 3125/3072, 16807/16384
-[[Mapping]]: [{{val| 5 0 10 14 }}, {{val| 0 5 1 0 }}]
+{{Mapping|legend=1| 5 0 10 14 | 0 5 1 0 }}
-Mapping generators: ~8/7, ~5/4
+: mapping generators: ~8/7, ~5/4
-[[POTE generator]]: ~5/4 = 379.7131
+[[Optimal tuning]]s:
+* [[WE]]: ~8/7 = 240.3877{{c}}, ~5/4 = 380.4267{{c}} (~256/245 = 100.3488{{c}})
+: error map: {{val| +1.939 +0.178 -2.010 -3.398 }}
+* [[CWE]]: ~8/7 = 240.0000{{c}}, ~5/4 = 379.9965{{c}} (~256/245 = 100.0035{{c}})
+: error map: {{val| 0.000 -1.972 -6.317 -8.826 }}
 {{Optimal ET sequence|legend=1| 25, 35, 60 }}
-[[Badness]]: 0.287190
+[[Badness]] (Sintel): 7.27
 [[Category:Temperament families]]
 [[Category:Magic family| ]] <!-- main article -->
+[[Category:Magic| ]] <!-- key article -->
 [[Category:Rank 2]]
-[[Category:Magic]]
 [[Category:Listen]]

Magic family: Difference between revisions