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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 ==
== Magic ==
{{Main| 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. [[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.
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
Line 18: Line 19:
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}


: Mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.4994
[[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]]:  
[[Minimax tuning]]:  
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~5/4 = [360.000, 400.000] (3\10 to 1\3)
* 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] (1/4-comma to untempered)
* 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.
[[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, 22, 41 }}
{{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 41, 60, 221cc, 281cc }}


[[Badness]]: 0.039163
[[Badness]] (Sintel): 0.919


=== Overview to extensions ===
=== Overview to extensions ===
Line 49: Line 53:
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.
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 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.  


225/224 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.  
225/224 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.  


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.
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 61: Line 65:
{{Mapping|legend=1| 1 0 2 -1 | 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}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.6512
: [[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]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[Tuning ranges]]:  
[[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 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] (1/4-comma to untempered)
* 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.
[[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, 22, 41, 104, 145c, 186c }}
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}


[[Badness]]: 0.018918
[[Badness]] (Sintel): 0.479


=== 11-limit ===
=== 11-limit ===
Tempering out 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
Subgroup: 2.3.5.7.11


Line 91: Line 94:
Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7200
Optimal tunings:
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}


Minimax tuning:  
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 (unchanged-interval) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
* 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] (1/4-comma to untempered)
* 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: {{Optimal ET sequence| 19, 22, 41, 104, 145c, 249cce }}
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}


Badness: 0.020352
Badness (Sintel): 0.673


==== 13-limit ====
==== 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
Subgroup: 2.3.5.7.11.13


Line 113: Line 119:
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.4354
Optimal tunings:
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}


Tuning ranges:  
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 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 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: {{Optimal ET sequence| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Badness: 0.021509
Badness (Sintel): 0.889


===== Magical =====
===== Magical =====
Line 131: Line 138:
Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}
Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.5103
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: {{Optimal ET sequence| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Badness: 0.020633
Badness (Sintel): 1.05


====== 19-limit ======
====== Magicus ======
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 144: Line 153:
Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.4507
Optimal tunings:
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 41 }}


Badness: 0.020881
Badness (Sintel): 1.27


====== Magica ======
====== Magica ======
Line 157: Line 168:
Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}
Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.5733
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: {{Optimal ET sequence| 19h, 22fh, 41 }}
{{Optimal ET sequence|legend=0| 22fh, 41 }}


Badness: 0.019945
Badness (Sintel): 1.21


===== Magia =====
===== Magia =====
Line 170: Line 183:
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.2706
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: {{Optimal ET sequence| 19g, 41, 101cde, 142cdefg }}
{{Optimal ET sequence|legend=0| 19g, 41, 60 }}


Badness: 0.026232
Badness (Sintel): 1.34


====== 19-limit ======
====== 19-limit ======
Line 183: Line 198:
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.3604
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: {{Optimal ET sequence| 19g, 41 }}
{{Optimal ET sequence|legend=0| 19gh, 41 }}


Badness: 0.023709
Badness (Sintel): 1.44


===== Evening =====
===== Evening =====
'''Evening''' is a remarkable subgroup temperament of 19 &amp; 41 with prime harmonics of 29 and 31.  
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
Subgroup: 2.3.5.7.11.13.29.31
Line 196: Line 213:
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195


Sval mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}
Subgroup-val mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.416
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: {{Optimal ET sequence| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
 
Badness (Sintel): 0.807


==== Sorcery ====
==== Sorcery ====
Line 209: Line 230:
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.6741
Optimal tunings:
 
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
 
Badness: 0.025829
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 52/51, 65/64, 78/77, 91/90, 100/99


Mapping: {{mapping| 1 0 2 -1 6 4 6 | 0 5 1 12 -8 -1 -6 }}
{{Optimal ET sequence|legend=0| 19, 22, 41f }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7839
Badness (Sintel): 1.07
 
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
 
Badness: 0.023768
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 52/51, 65/64, 78/77, 91/90, 100/99, 133/132
 
Mapping: {{mapping| 1 0 2 -1 6 4 6 9 | 0 5 1 12 -8 -1 -6 -15 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.6248
 
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
 
Badness: 0.023232


==== Necromancy ====
==== Necromancy ====
Line 248: Line 245:
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7876
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: {{Optimal ET sequence| 19f, 22, 41, 63, 104 }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}


Badness: 0.025275
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 261: Line 260:
Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}
Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.8373
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: {{Optimal ET sequence| 19f, 22, 41, 63, 104g }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}


Badness: 0.022032
Badness (Sintel): 1.12
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 100/99, 120/119, 133/132, 154/153, 209/208, 225/224
 
Mapping: {{mapping| 1 0 2 -1 6 11 6 9 | 0 5 1 12 -8 -23 -6 -15 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7411
 
Optimal ET sequence: {{Optimal ET sequence| 19f, 22, 41, 63h, 104gh }}
 
Badness: 0.021101


==== Soothsaying ====
==== Soothsaying ====
Line 287: Line 275:
Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}
Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}


Optimal tuning (CTE): ~55/39 = 1\2, ~5/4 = 380.5385
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: {{Optimal ET sequence| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.055443
Badness (Sintel): 2.29


===== 17-limit =====
===== 17-limit =====
Line 300: Line 290:
Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}
Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.5533
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: {{Optimal ET sequence| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.035654
Badness (Sintel): 1.82


===== 19-limit =====
===== 19-limit =====
Line 313: Line 305:
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.4704
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: {{Optimal ET sequence| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.031291
Badness (Sintel): 1.90


=== Telepathy ===
=== Telepathy ===
Line 326: Line 320:
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.2311
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: {{Optimal ET sequence| 19e, 22, 63e, 85ee }}
{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}


Badness: 0.027109
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 339: Line 335:
Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}
Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.1957
Optimal tunings:
 
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
Optimal ET sequence: {{Optimal ET sequence| 19e, 22, 63eff }}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}
 
Badness: 0.025522
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 55/54, 65/64, 85/84, 91/90, 99/98
 
Mapping: {{mapping| 1 0 2 -1 -1 4 -1 | 0 5 1 12 14 -1 16 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.2884
 
Optimal ET sequence: {{Optimal ET sequence| 19eg, 22, 63effg }}
 
Badness: 0.020201
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 55/54, 57/56, 65/64, 76/75, 85/84, 99/98


Mapping: {{mapping| 1 0 2 -1 -1 4 -1 2 | 0 5 1 12 14 -1 16 7 }}
{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.4641
Badness (Sintel): 1.05
 
Optimal ET sequence: {{Optimal ET sequence| 19egh, 22 }}
 
Badness: 0.019004


==== Intuition ====
==== Intuition ====
Line 378: Line 350:
Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.8157
Optimal tunings:
 
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
Optimal ET sequence: {{Optimal ET sequence| 19e, 22f, 41e, 63ef }}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}
 
Badness: 0.026089
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 55/54, 66/65, 85/84, 99/98, 105/104
 
Mapping: {{mapping| 1 0 2 -1 -1 -2 -1 | 0 5 1 12 14 18 16 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.9518
 
Optimal ET sequence: {{Optimal ET sequence| 19eg, 22f, 41eg, 63efg }}
 
Badness: 0.020274
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 55/54, 66/65, 77/76, 85/84, 99/98, 105/104
 
Mapping: {{mapping| 1 0 2 -1 -1 -2 -1 -4 | 0 5 1 12 14 18 16 26 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.8658


Optimal ET sequence: {{Optimal ET sequence| 19egh, 22fh, 41eg, 63efg }}
{{Optimal ET sequence|legend=0| 19e, 22f }}


Badness: 0.019518
Badness (Sintel): 1.08


=== Horcrux ===
=== Horcrux ===
Line 417: Line 365:
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.7959
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: {{Optimal ET sequence| 3de, …, 16d, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.039282
Badness (Sintel): 1.30


==== Tridecimal horcrux ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 430: Line 380:
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.8847
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: {{Optimal ET sequence| 3def, …, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}


Badness: 0.031938
Badness (Sintel): 1.32


===== 17-limit =====
===== 17-limit =====
Line 443: Line 395:
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.2159
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: {{Optimal ET sequence| 3defg, …, 19g, 41eeg }}
{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}


Badness: 0.028074
Badness (Sintel): 1.43


===== Horcruxic =====
===== Horcruxic =====
Line 456: Line 410:
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.5237
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: {{Optimal ET sequence| 3defg, …, 19 }}
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}


Badness: 0.029556
Badness (Sintel): 1.51


==== Glamour ====
==== Glamour ====
Line 469: Line 425:
Mapping: {{mapping| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}
Mapping: {{mapping| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.7574
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: {{Optimal ET sequence| 3de, …, 16d, 19, 41eef, 60eeff }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.033317
Badness (Sintel): 1.38


=== Witchcraft ===
=== Witchcraft ===
Line 482: Line 440:
Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}
Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.4163
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: {{Optimal ET sequence| 19e, 22e, 41 }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}


Badness: 0.030706
Badness (Sintel): 1.02


==== 13-limit ====
==== 13-limit ====
Line 495: Line 455:
Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}
Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.3523
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: {{Optimal ET sequence| 19e, 41, 142cdf }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}


Badness: 0.023547
Badness (Sintel): 0.973


==== 17-limit ====
==== 17-limit ====
Line 508: Line 470:
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.2809
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: {{Optimal ET sequence| 19eg, 41, 101cd, 142cdfg }}
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness: 0.020756
Badness (Sintel): 1.06
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 105/104, 133/132, 154/153, 170/169, 171/169, 196/195
 
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 -4 | 0 5 1 12 33 18 35 26 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.3389
 
Optimal ET sequence: {{Optimal ET sequence| 19egh, 41, 101cdhh, 142cdfghh }}
 
Badness: 0.018625


=== Divination ===
=== Divination ===
Line 534: Line 485:
Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}
Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~5/4 = 380.7317
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: {{Optimal ET sequence| 22, 60e, 82e, 104e }}
{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}


Badness: 0.035864
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 547: Line 500:
Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}
Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~5/4 = 380.4171
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: {{Optimal ET sequence| 22f, 60e, 82ef }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.034551
Badness (Sintel): 1.43


==== 17-limit ====
==== 17-limit ====
Line 560: Line 515:
Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}
Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.4329
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: {{Optimal ET sequence| 22f, 60e, 82ef }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.023775
Badness (Sintel): 1.21


=== Hocus ===
=== Hocus ===
Line 571: Line 528:
Comma list: 225/224, 243/242, 245/242
Comma list: 225/224, 243/242, 245/242


Mapping: {{mapping| 1 5 3 11 12 | 0 -10 -2 -24 -25 }}
Mapping: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.7603
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: {{Optimal ET sequence| 38d, 41 }}
{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}


Badness: 0.038519
Badness (Sintel): 1.27


==== 13-limit ====
==== 13-limit ====
Line 584: Line 543:
Comma list: 105/104, 196/195, 243/242, 245/242
Comma list: 105/104, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 5 3 11 12 16 | 0 -10 -2 -24 -25 -36 }}
Mapping: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.8421
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: {{Optimal ET sequence| 38df, 41 }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}


Badness: 0.030280
Badness (Sintel): 1.25


==== 17-limit ====
==== 17-limit ====
Line 597: Line 558:
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 5 3 11 12 16 14 | 0 -10 -2 -24 -25 -36 -29 }}
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.8958
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: {{Optimal ET sequence| 38df, 41, 120cdg, 161cdg, 202ccddfgg }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d }}


Badness: 0.025491
Badness (Sintel): 1.30


==== 19-limit ====
==== 19-limit ====
Line 610: Line 573:
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 5 3 11 12 16 14 8 | 0 -10 -2 -24 -25 -36 -29 -11 }}
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.8836
Optimal tunings:
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}


Optimal ET sequence: {{Optimal ET sequence| 38df, 41, 120cdgh, 161cdgh, 202ccddfgghh }}
{{Optimal ET sequence|legend=0| 38df, 41, 79dh }}


Badness: 0.020277
Badness (Sintel): 1.23


== Muggles ==
== Muggles ==
Line 629: Line 594:
{{Mapping|legend=1| 1 0 2 5 | 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}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 378.744
: [[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]]:  
[[Tuning ranges]]:  
Line 637: Line 604:
* [[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- 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 }}
{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}


[[Badness]]: 0.056206
[[Badness]] (Sintel): 1.42


=== 11-limit ===
=== 11-limit ===
Line 651: Line 616:
Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}
Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 378.228
Optimal tunings:
* WE: ~2 = 1203.0804{{c}}, ~5/4 = 378.6936{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8174{{c}}


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
* 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 tradeoff: ~5/4 = [347.408, 386.314]
* 11-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947


Optimal ET sequence: {{Optimal ET sequence| 16, 19, 54bd, 73bcd, 92bcdde }}
{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}


Badness: 0.048038
Badness (Sintel): 1.59


==== 13-limit ====
==== 13-limit ====
Line 669: Line 635:
Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}
Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~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: {{Optimal ET sequence| 16, 19, 54bdf, 73bcdff, 92bcddeff }}
{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}


Badness: 0.030386
Badness (Sintel): 1.26


=== Muggloid ===
=== Muggloid ===
Line 682: Line 650:
Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}
Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 377.724
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: {{Optimal ET sequence| 3, 10bd, 13bd, 16, 19e, 35ee }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}


Badness: 0.046970
Badness (Sintel): 1.55


==== 13-limit ====
==== 13-limit ====
Line 695: Line 665:
Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}
Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 377.652
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: {{Optimal ET sequence| 3, 10bd, 13bd, 16, 19e, 35eef }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}


Badness: 0.028732
Badness (Sintel): 1.19


== Darkstone ==
== Brightstone ==
''Darkstone'' (16 &amp; 19d) is a low-accuacy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles. In [http://www.tonalsoft.com/enc/m/magic.aspx ''Encyclopedia of Microtonal Music Theory'', Tonalsoft], this temperament is given a name ''witch''.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 1875/1792
[[Comma list]]: 64/63, 3125/3024


{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
 
{{Multival|legend=1| 5 1 9 -10 0 18 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 377.385
[[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 }}


{{Optimal ET sequence|legend=1| 3d, …, 13b, 16, 19d, 35d }}
{{Optimal ET sequence|legend=1| 3, 19d, 22 }}


[[Badness]]: 0.084213
[[Badness]] (Sintel): 2.23


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 36/35, 45/44, 363/343
Comma list: 55/54, 64/63, 625/616


Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}
Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 377.388
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: {{Optimal ET sequence| 3de, …, 13be, 16, 19d, 35d }}
{{Optimal ET sequence|legend=0| 22, 69b }}


Badness: 0.046775
Badness (Sintel): 1.85


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 36/35, 45/44, 363/343
Comma list: 40/39, 55/54, 64/63, 625/616
 
Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}


Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
Optimal tunings:  
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 376.914
{{Optimal ET sequence|legend=0| 22f, 47bff }}


Optimal ET sequence: {{Optimal ET sequence| 3def, …, 16, 35d }}
Badness (Sintel): 1.99


Badness: 0.038328
== 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.  


== Brightstone ==
This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].
{{see also| Archytas clan }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 3125/3024
[[Comma list]]: 36/35, 1875/1792


{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
 
{{Multival|legend=1| 5 1 -10 -10 -30 -26 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 381.955
[[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| 3, …, 19d, 22 }}
{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}


[[Badness]]: 0.088072
[[Badness]] (Sintel): 2.13


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 64/63, 100/99, 605/588
Comma list: 36/35, 45/44, 363/343


Mapping: {{mapping| 1 0 2 6 6 | 0 5 1 -10 -8 }}
Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.790
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: {{Optimal ET sequence| 3, …, 19d, 22 }}
{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}


Badness: 0.047379
Badness (Sintel): 1.55


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 64/63, 65/63, 100/99, 169/165
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}}


Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 -8 -1 }}
{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.732
Badness (Sintel): 1.58


Optimal ET sequence: {{Optimal ET sequence| 3, …, 19d, 22 }}
Scales: [[User:BudjarnLambeth/Volcanic glass]]


Badness: 0.039703
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)


== Hocum ==
== Hocum ==
Line 792: Line 782:
[[Comma list]]: 3125/3072, 4000/3969
[[Comma list]]: 3125/3072, 4000/3969


{{Mapping|legend=1| 1 5 3 -3 | 0 -10 -2 17 }}
{{Mapping|legend=1| 1 -5 1 14 | 0 10 2 -17 }}


: Mapping generators: ~2, ~63/50
: mapping generators: ~2, ~63/40


{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~63/50 = 409.836
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}


{{Optimal ET sequence|legend=1| 3, …, 35d, 38, 41, 202cc, 243cc, 284cc }}
[[Badness]] (Sintel): 2.71


[[Badness]]: 0.107115
== Trismegistus ==
{{Main| Mabilic and trismegistus }}
{{See also| No-threes subgroup temperaments #Mabilic }}


== Trismegistus ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 3125/3072
[[Comma list]]: 1029/1024, 3125/3072


{{Mapping|legend=1| 1 10 4 0 | 0 -15 -3 5 }}
{{Mapping|legend=1| 1 -5 1 5 | 0 15 3 -5 }}


: Mapping generators: ~2, ~147/100
: mapping generators: ~2, ~168/125


{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~147/100 = 673.187
{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}


{{Optimal ET sequence|legend=1| 16, 25, 41 }}
[[Badness]] (Sintel): 2.49
 
[[Badness]]: 0.098334


=== 11-limit ===
=== 11-limit ===
Line 826: Line 823:
Comma list: 385/384, 441/440, 625/616
Comma list: 385/384, 441/440, 625/616


Mapping: {{mapping| 1 10 4 0 13 | 0 -15 -3 5 -17 }}
Mapping: {{mapping| 1 -5 1 5 -4 | 0 15 3 -5 17 }}


Optimal tuning (CTE): ~2 = 1\1, ~22/15 = 673.241
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: {{Optimal ET sequence| 16, 25e, 41, 221cc, 262ccde }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness: 0.045623
Badness (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 839: Line 838:
Comma list: 105/104, 144/143, 275/273, 625/616
Comma list: 105/104, 144/143, 275/273, 625/616


Mapping: {{mapping| 1 10 4 0 13 11 | 0 -15 -3 5 -17 -13 }}
Mapping: {{mapping| 1 -5 1 5 -4 -2 | 0 15 3 -5 17 13 }}


Optimal tuning (CTE): ~2 = 1\1, ~22/15 = 673.294
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: {{Optimal ET sequence| 16, 25e, 41, 139cf, 180cf, 221ccf }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness: 0.033081
Badness (Sintel): 1.37
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 105/104, 144/143, 441/440, 210/209, 625/616
 
Subgroup-val mapping: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}
 
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 ==
== Quadrimage ==
Line 852: Line 868:
[[Comma list]]: 2401/2400, 3125/3072
[[Comma list]]: 2401/2400, 3125/3072


{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}
{{Mapping|legend=1| 1 -15 -1 -3 | 0 20 4 7 }}
 
: Mapping generators: ~2, ~28/25


{{Multival|legend=1| 20 4 7 -40 -45 5 }}
: mapping generators: ~2, ~25/14


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 204.860
[[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, …, 29b, 35, 41 }}
{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}


[[Badness]]: 0.127422
[[Badness]] (Sintel): 3.22


=== 11-limit ===
=== 11-limit ===
Line 869: Line 887:
Comma list: 245/242, 385/384, 625/616
Comma list: 245/242, 385/384, 625/616


Mapping: {{mapping| 1 5 3 4 5 | 0 -20 -4 -7 -9 }}
Mapping: {{mapping| 1 -15 -1 -3 -4 | 0 20 4 7 9 }}


Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.881
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: {{Optimal ET sequence| 6, …, 29b, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 35, 41 }}


Badness: 0.061572
Badness (Sintel): 2.04


=== 13-limit ===
=== 13-limit ===
Line 882: Line 902:
Comma list: 105/104, 144/143, 245/242, 625/616
Comma list: 105/104, 144/143, 245/242, 625/616


Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}
Mapping: {{mapping| 1 -15 -1 -3 -22 | 0 20 4 7 9 31 }}


Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.956
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: {{Optimal ET sequence| 35f, 41, 199ccdef, 240ccdef, 281ccdeff }}
{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}


Badness: 0.044047
Badness (Sintel): 1.82


== Quinmage ==
== Quinmage ==
Line 897: Line 919:
{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}
{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}


{{Multival|legend=1| 25 5 19 -50 -40 30 }}
: mapping generators: ~2, ~48/35


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~48/35 = 556.123
[[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 }}
{{Optimal ET sequence|legend=1| 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd }}


[[Badness]]: 0.194548
[[Badness]] (Sintel): 4.92


=== 11-limit ===
=== 11-limit ===
Line 912: Line 938:
Mapping: {{mapping| 1 -10 0 -6 3 | 0 25 5 19 1 }}
Mapping: {{mapping| 1 -10 0 -6 3 | 0 25 5 19 1 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/8 = 556.122
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: {{Optimal ET sequence| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}


Badness: 0.101724
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Line 925: Line 953:
Mapping: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}
Mapping: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/8 = 556.106
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: {{Optimal ET sequence| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}


Badness: 0.067742
Badness (Sintel): 2.80


== Warlock ==
== Warlock ==
Line 938: Line 968:
{{Mapping|legend=1| 5 0 10 14 | 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


[[Optimal tuning]] ([[CTE]]): ~8/7 = 1\5, ~5/4 = 380.499
[[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 }}
{{Optimal ET sequence|legend=1| 25, 35, 60 }}


[[Badness]]: 0.287190
[[Badness]] (Sintel): 7.27


[[Category:Temperament families]]
[[Category:Temperament families]]

Latest revision as of 01:34, 12 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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 proper or nearly-proper mos scales in the 5- to 10-note region. Properties may depend on tuning and extension.

Magic

Main article: Magic

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)5 = 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.

Subgroup: 2.3.5

Comma list: 3125/3072

Mapping: [⟨1 0 2], ⟨0 5 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1201.2449 ¢, ~5/4 = 380.4527 ¢
error map: ⟨+1.245 +0.309 -3.371]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2194 ¢
error map: ⟨0.000 -0.858 -6.094]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Tuning ranges:

Algebraic generator: Terzbirat, the positive root of 9x2 - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 cents.

Optimal ET sequence: 3, 13b, 16, 19, 41, 60, 221cc, 281cc

Badness (Sintel): 0.919

Overview to extensions

Apart from magic, we also consider other extensions. The second comma of the 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, quinmage and warlock. Discussed elsewhere are

Septimal magic

Main article: 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 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. 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.

225/224 is the marvel comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the 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.

Subgroup: 2.3.5.7

Comma list: 225/224, 245/243

Mapping: [⟨1 0 2 -1], ⟨0 5 1 12]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1201.0786 ¢, ~5/4 = 380.6939 ¢
error map: ⟨+1.079 +1.514 -3.463 -1.578]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4576 ¢
error map: ⟨0.000 +0.333 -5.856 -3.335]

Minimax tuning:

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] (1/4-comma to untempered)

Algebraic generator: Tirzbirat or Septimage, the real root of 5x5 + 4x - 20, 380.7604 cents.

Optimal ET sequence: 19, 41, 142cd, 183cd, 224ccd

Badness (Sintel): 0.479

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/243

Mapping: [⟨1 0 2 -1 6], ⟨0 5 1 12 -8]]

Optimal tunings:

  • WE: ~2 = 1200.1372 ¢, ~5/4 = 380.7399 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7008 ¢

Minimax tuning:

  • 11-odd-limit: ~5/4 = [1/3 1/9 0 0 -1/18⟩
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] (1/4-comma to untempered)

Optimal ET sequence: 19, 22, 41, 104

Badness (Sintel): 0.673

13-limit

A notable patent val tuning beyond the optimal patent val of 41edo is 19 + 41 = 60edo.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 1 12 -8 18]]

Optimal tunings:

  • WE: ~2 = 1200.0331 ¢, ~5/4 = 380.4377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4284 ¢

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]

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.889

Magical

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 144/143, 154/153

Mapping: [⟨1 0 2 -1 6 -2 6], ⟨0 5 1 12 -8 18 -6]]

Optimal tunings:

  • WE: ~2 = 1199.3584 ¢, ~5/4 = 380.4006 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5896 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 1.05

Magicus

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153

Mapping: [⟨1 0 2 -1 6 -2 6 9], ⟨0 5 1 12 -8 18 -6 -15]]

Optimal tunings:

  • WE: ~2 = 1199.7173 ¢, ~5/4 = 380.3808 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4680 ¢

Optimal ET sequence: 19, 41

Badness (Sintel): 1.27

Magica

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169

Mapping: [⟨1 0 2 -1 6 -2 6 -4], ⟨0 5 1 12 -8 18 -6 26]]

Optimal tunings:

  • WE: ~2 = 1199.3670 ¢, ~5/4 = 380.4681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6541 ¢

Optimal ET sequence: 22fh, 41

Badness (Sintel): 1.21

Magia

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 170/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7], ⟨0 5 1 12 -8 18 35]]

Optimal tunings:

  • WE: ~2 = 1200.1727 ¢, ~5/4 = 380.2982 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2483 ¢

Optimal ET sequence: 19g, 41, 60

Badness (Sintel): 1.34

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7 -4], ⟨0 5 1 12 -8 18 35 26]]

Optimal tunings:

  • WE: ~2 = 1200.2179 ¢, ~5/4 = 380.3942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.3314 ¢

Optimal ET sequence: 19gh, 41

Badness (Sintel): 1.44

Evening

Evening is a remarkable subgroup temperament of 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: [⟨1 0 2 -1 6 -2 2 4], ⟨0 5 1 12 -8 18 9 3]]

Optimal tunings:

  • WE: ~2 = 1200.2802 ¢, ~5/4 = 380.5053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4258 ¢

Optimal ET sequence: 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: [⟨1 0 2 -1 6 4], ⟨0 5 1 12 -8 -1]]

Optimal tunings:

  • WE: ~2 = 1201.2397 ¢, ~5/4 = 380.8698 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5080 ¢

Optimal ET sequence: 19, 22, 41f

Badness (Sintel): 1.07

Necromancy

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 275/273

Mapping: [⟨1 0 2 -1 6 11], ⟨0 5 1 12 -8 -23]]

Optimal tunings:

  • WE: ~2 = 1199.9675 ¢, ~5/4 = 380.7770 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7874 ¢

Optimal ET sequence: 19f, 22, 41, 63, 104

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 154/153, 225/224, 273/272

Mapping: [⟨1 0 2 -1 6 11 6], ⟨0 5 1 12 -8 -23 -6]]

Optimal tunings:

  • WE: ~2 = 1199.6176 ¢, ~5/4 = 380.7053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.8280 ¢

Optimal ET sequence: 19f, 22, 41, 63

Badness (Sintel): 1.12

Soothsaying

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 1352/1331

Mapping: [⟨2 0 4 -2 12 15], ⟨0 5 1 12 -8 -12]]

Optimal tunings:

  • WE: ~55/39 = 600.2918 ¢, ~5/4 = 380.6928 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~5/4 = 380.5121 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 2.29

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5], ⟨0 5 1 12 -8 -12 5]]

Optimal tunings:

  • WE: ~17/12 = 600.2918 ¢, ~5/4 = 380.6927 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.5135 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.82

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5 18], ⟨0 5 1 12 -8 -12 5 -15]]

Optimal tunings:

  • WE: ~17/12 = 600.3301 ¢, ~5/4 = 380.6797 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.4704 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.90

Telepathy

Subgroup: 2.3.5.7.11

Comma list: 55/54, 99/98, 176/175

Mapping: [⟨1 0 2 -1 -1], ⟨0 5 1 12 14]]

Optimal tunings:

  • WE: ~2 = 1200.7724 ¢, ~5/4 = 381.2641 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 381.0913 ¢

Optimal ET sequence: 19e, 22, 41e, 63e

Badness (Sintel): 0.896

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 91/90, 99/98

Mapping: [⟨1 0 2 -1 -1 4], ⟨0 5 1 12 14 -1]]

Optimal tunings:

  • WE: ~2 = 1202.5634 ¢, ~5/4 = 381.3348 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6886 ¢

Optimal ET sequence: 19e, 22, 41ef

Badness (Sintel): 1.05

Intuition

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 99/98, 105/104

Mapping: [⟨1 0 2 -1 -1 -2], ⟨0 5 1 12 14 18]]

Optimal tunings:

  • WE: ~2 = 1201.3172 ¢, ~5/4 = 380.9004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5942 ¢

Optimal ET sequence: 19e, 22f

Badness (Sintel): 1.08

Horcrux

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0], ⟨0 5 1 12 11]]

Optimal tunings:

  • WE: ~2 = 1200.4670 ¢, ~5/4 = 379.7895 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.6889 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.30

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 245/243

Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 11 18]]

Optimal tunings:

  • WE: ~2 = 1200.2953 ¢, ~5/4 = 379.8842 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.8165 ¢

Optimal ET sequence: 3def, 16dff, 19

Badness (Sintel): 1.32

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 85/84, 245/243

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 16]]

Optimal tunings:

  • WE: ~2 = 1200.2484 ¢, ~5/4 = 380.2053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1482 ¢

Optimal ET sequence: 3defg, 16dffgg, 19g

Badness (Sintel): 1.43

Horcruxic

Subgroup: 2.3.5.7.11.13.17

Comma list: 35/34, 45/44, 52/51, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 13]]

Optimal tunings:

  • WE: ~2 = 1199.5457 ¢, ~5/4 = 379.4681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.5713 ¢

Optimal ET sequence: 3defg, 16dff, 19

Badness (Sintel): 1.51

Glamour

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 245/243

Mapping: [⟨1 0 2 -1 0 4], ⟨0 5 1 12 11 -1]]

Optimal tunings:

  • WE: ~2 = 1202.2187 ¢, ~5/4 = 379.8171 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.2709 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.38

Witchcraft

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 441/440

Mapping: [⟨1 0 2 -1 -7], ⟨0 5 1 12 33]]

Optimal tunings:

  • WE: ~2 = 1201.2634 ¢, ~5/4 = 380.6321 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2849 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd, 243ccdde

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243, 275/273

Mapping: [⟨1 0 2 -1 -7 -2], ⟨0 5 1 12 33 18]]

Optimal tunings:

  • WE: ~2 = 1201.0424 ¢, ~5/4 = 380.5193 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2349 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd

Badness (Sintel): 0.973

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 170/169, 196/195, 245/243

Mapping: [⟨1 0 2 -1 -7 -2 -7], ⟨0 5 1 12 33 18 35]]

Optimal tunings:

  • WE: ~2 = 1201.1638 ¢, ~5/4 = 380.4827 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1599 ¢

Optimal ET sequence: 19eg, 41, 60e, 101cd

Badness (Sintel): 1.06

Divination

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 245/243

Mapping: [⟨2 0 4 -2 5], ⟨0 5 1 12 3]]

Optimal tunings:

  • WE: ~99/70 = 600.8306 ¢, ~5/4 = 380.7598 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.3800 ¢

Optimal ET sequence: 22, 38d, 60e, 142cdee, 202ccddeee

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4], ⟨0 5 1 12 3 18]]

Optimal tunings:

  • WE: ~99/70 = 600.9624 ¢, ~5/4 = 380.5297 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.0614 ¢

Optimal ET sequence: 22f, 38df, 60e

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: [⟨2 0 4 -2 5 -4 5], ⟨0 5 1 12 3 18 5]]

Optimal tunings:

  • WE: ~17/12 = 600.8921 ¢, ~5/4 = 380.5094 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.0672 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Sintel): 1.21

Hocus

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13], ⟨0 10 2 24 25]]

Optimal tunings:

  • WE: ~2 = 1201.0749 ¢, ~11/7 = 790.7980 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.1429 ¢

Optimal ET sequence: 38d, 41, 120cd

Badness (Sintel): 1.27

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20], ⟨0 10 2 24 25 36]]

Optimal tunings:

  • WE: ~2 = 1201.2830 ¢, ~11/7 = 790.8409 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0516 ¢

Optimal ET sequence: 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: [⟨1 -5 1 -13 -13 -20 -15], ⟨0 10 2 24 25 36 29]]

Optimal tunings:

  • WE: ~2 = 1201.1557 ¢, ~11/7 = 790.7157 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0057 ¢

Optimal ET sequence: 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

Mapping: [⟨1 -5 1 -13 -13 -20 -3], ⟨0 10 2 24 25 36 29 11]]

Optimal tunings:

  • WE: ~2 = 1201.3558 ¢, ~11/7 = 790.8266 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 789.9880 ¢

Optimal ET sequence: 38df, 41, 79dh

Badness (Sintel): 1.23

Muggles

Main article: 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 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

Comma list: 126/125, 525/512

Mapping: [⟨1 0 2 5], ⟨0 5 1 -7]]

Optimal tunings:

  • WE: ~2 = 1203.9554 ¢, ~5/4 = 379.7269 ¢
error map: ⟨+3.955 -3.321 +1.324 -7.137]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 378.5328 ¢
error map: ⟨0.000 -9.291 -7.781 -18.555]

Tuning ranges:

Optimal ET sequence: 16, 19, 73bcd, 92bcdd, 111bcddd

Badness (Sintel): 1.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 126/125, 385/384

Mapping: [⟨1 0 2 5 0], ⟨0 5 1 -7 11]]

Optimal tunings:

  • WE: ~2 = 1203.0804 ¢, ~5/4 = 378.6936 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8174 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
  • 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]

Optimal ET sequence: 16, 19, 35, 54bd

Badness (Sintel): 1.59

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 126/125

Mapping: [⟨1 0 2 5 0 4], ⟨0 5 1 -7 11 -1]]

Optimal tunings:

  • WE: ~2 = 1203.4291 ¢, ~5/4 = 378.7321 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.7336 ¢

Optimal ET sequence: 16, 19, 35f, 54bdf

Badness (Sintel): 1.26

Muggloid

Subgroup: 2.3.5.7.11

Comma list: 33/32, 126/125, 176/175

Mapping: [⟨1 0 2 5 5], ⟨0 5 1 -7 -5]]

Optimal tunings:

  • WE: ~2 = 1205.6044 ¢, ~5/4 = 379.5966 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8142 ¢

Optimal ET sequence: 3, 16, 19e, 35ee, 54bdeee

Badness (Sintel): 1.55

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 65/64, 105/104, 126/125

Mapping: [⟨1 0 2 5 5 4], ⟨0 5 1 -7 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1205.4897 ¢, ~5/4 = 379.5667 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8118 ¢

Optimal ET sequence: 3, 16, 19e, 35eef

Badness (Sintel): 1.19

Brightstone

Brightstone tempers out 64/63 and may be described as 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: [⟨1 0 2 6], ⟨0 5 1 -10]]

Optimal tunings:

  • WE: ~2 = 1198.1701 ¢, ~5/4 = 381.3741 ¢
error map: ⟨-1.830 +4.915 -8.599 +6.454]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 381.9562 ¢
error map: ⟨0.000 +7.826 -4.358 +11.613]

Optimal ET sequence: 3, 19d, 22

Badness (Sintel): 2.23

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 625/616

Mapping: [⟨1 0 2 6 -1], ⟨0 5 1 -10 14]]

Optimal tunings:

  • WE: ~2 = 1198.5372 ¢, ~5/4 = 381.7556 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 382.1943 ¢

Optimal ET sequence: 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: [⟨1 0 2 6 6 4], ⟨0 5 1 -10 14 -4]]

Optimal tunings:

  • WE: ~2 = 1197.2300 ¢, ~5/4 = 381.6164 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 382.4690 ¢

Optimal ET sequence: 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 Tonalsoft Encyclopedia.

Subgroup: 2.3.5.7

Comma list: 36/35, 1875/1792

Mapping: [⟨1 0 2 0], ⟨0 5 1 9]]

Optimal tunings:

  • WE: ~2 = 1201.7458 ¢, ~5/4 = 377.2996 ¢
error map: ⟨+1.746 -15.457 -5.523 +26.870]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 376.9630 ¢
error map: ⟨0.000 -17.140 -9.351 +23.841]

Optimal ET sequence: 3d, …, 13b, 16

Badness (Sintel): 2.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 363/343

Mapping: [⟨1 0 2 0 0], ⟨0 5 1 9 11]]

Optimal tunings:

  • WE: ~2 = 1201.7428 ¢, ~5/4 = 377.3134 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 376.9735 ¢

Optimal ET sequence: 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: [⟨1 0 2 0 0 -1], ⟨0 5 1 9 11 15]]

Optimal tunings:

  • WE: ~2 = 1201.7428 ¢, ~5/4 = 377.3134 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 376.4221 ¢

Optimal ET sequence: 3def, 13beff, 16

Badness (Sintel): 1.58

Scales: User:BudjarnLambeth/Volcanic glass

Music: Rain in the crystal mirror caves - Budjarn Lambeth (2026)

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 -5 1 14], ⟨0 10 2 -17]]

mapping generators: ~2, ~63/40

Optimal tunings:

  • WE: ~2 = 1200.8375 ¢, ~63/40 = 790.7032 ¢
error map: ⟨+0.838 +0.890 -4.070 +0.944]
  • CWE: ~2 = 1200.0000 ¢, ~63/40 = 790.1542 ¢
error map: ⟨0.000 -0.413 -6.005 -1.447]

Optimal ET sequence: 3, 38, 41, 161c

Badness (Sintel): 2.71

Trismegistus

Main article: Mabilic and trismegistus
See also: No-threes subgroup temperaments § Mabilic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 -5 1 5], ⟨0 15 3 -5]]

mapping generators: ~2, ~168/125

Optimal tunings:

  • WE: ~2 = 1201.0799 ¢, ~168/125 = 527.1841 ¢
error map: ⟨+1.080 +0.408 -3.681 +0.653]
  • CWE: ~2 = 1200.0000 ¢, ~168/125 = 526.7349 ¢
error map: ⟨0.000 -0.932 -6.109 -2.500]

Optimal ET sequence: 16, 25, 41, 139c, 180c, 221cc, 262ccd

Badness (Sintel): 2.49

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 625/616

Mapping: [⟨1 -5 1 5 -4], ⟨0 15 3 -5 17]]

Optimal tunings:

  • WE: ~2 = 1200.8404 ¢, ~15/11 = 527.0289 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6826 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273, 625/616

Mapping: [⟨1 -5 1 5 -4 -2], ⟨0 15 3 -5 17 13]]

Optimal tunings:

  • WE: ~2 = 1200.4759 ¢, ~15/11 = 526.8502 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6548 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.37

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 105/104, 144/143, 441/440, 210/209, 625/616

Subgroup-val mapping: [⟨1 -5 1 5 -4 -2 6], ⟨0 15 3 -5 17 13 -4]]

Optimal tunings:

  • WE: ~2 = 1200.5832 ¢, ~15/11 = 526.8804 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6368 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.26

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 -15 -1 -3], ⟨0 20 4 7]]

mapping generators: ~2, ~25/14

Optimal tunings:

  • WE: ~2 = 1201.2708 ¢, ~25/14 = 996.0669 ¢
error map: ⟨+1.271 +0.322 -3.317 -0.170]
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 995.0515 ¢
error map: ⟨0.000 -0.926 -6.108 -3.466]

Optimal ET sequence: 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd

Badness (Sintel): 3.22

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 625/616

Mapping: [⟨1 -15 -1 -3 -4], ⟨0 20 4 7 9]]

Optimal tunings:

  • WE: ~2 = 1200.6716 ¢, ~25/14 = 995.6009 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 995.0633 ¢

Optimal ET sequence: 6, 35, 41

Badness (Sintel): 2.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/242, 625/616

Mapping: [⟨1 -15 -1 -3 -22], ⟨0 20 4 7 9 31]]

Optimal tunings:

  • WE: ~2 = 1200.6276 ¢, ~25/14 = 995.4920 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 994.9901 ¢

Optimal ET sequence: 6f, 35f, 41, 117c

Badness (Sintel): 1.82

Quinmage

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16875/16807

Mapping: [⟨1 -10 0 -6], ⟨0 25 5 19]]

mapping generators: ~2, ~48/35

Optimal tunings:

  • WE: ~2 = 1201.3334 ¢, ~48/35 = 556.6311 ¢
error map: ⟨+1.333 +0.489 -3.158 -0.835]
  • CWE: ~2 = 1200.0000 ¢, ~48/35 = 556.0504 ¢
error map: ⟨0.000 -0.695 -6.062 -3.868]

Optimal ET sequence: 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: [⟨1 -10 0 -6 3], ⟨0 25 5 19 1]]

Optimal tunings:

  • WE: ~2 = 1200.4252 ¢, ~11/8 = 556.2831 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 556.0951 ¢

Optimal ET sequence: 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: [⟨1 -10 0 -6 3 0], ⟨0 25 5 19 1 8]]

Optimal tunings:

  • WE: ~2 = 1199.8239 ¢, ~11/8 = 556.0389 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 556.1171 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Sintel): 2.80

Warlock

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16807/16384

Mapping: [⟨5 0 10 14], ⟨0 5 1 0]]

mapping generators: ~8/7, ~5/4

Optimal tunings:

  • WE: ~8/7 = 240.3877 ¢, ~5/4 = 380.4267 ¢ (~256/245 = 100.3488 ¢)
error map: ⟨+1.939 +0.178 -2.010 -3.398]
  • CWE: ~8/7 = 240.0000 ¢, ~5/4 = 379.9965 ¢ (~256/245 = 100.0035 ¢)
error map: ⟨0.000 -1.972 -6.317 -8.826]

Optimal ET sequence: 25, 35, 60

Badness (Sintel): 7.27

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