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[[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]]:  
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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]] to the list of commas, magic can be extended to an 11-limit version. 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
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: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4
{{Multival|legend=1| 5 1 12 -10 5 25 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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[[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]]:  
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Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
Wedgie: {{Multival| 5 1 12 -8 -10 5 -30 25 -22 -64 }}


Optimal tunings:  
Optimal tunings:  
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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:  
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Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
Wedgie: {{Multival| 5 1 12 14 -10 5 5 25 29 -2 }}


Optimal tunings:  
Optimal tunings:  
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{{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:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
{{Multival|legend=1| 5 1 9 -10 0 18 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
{{Multival|legend=1| 5 1 -10 -10 -30 -26 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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: mapping generators: ~2, ~63/50
: mapping generators: ~2, ~63/50
{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~2, ~147/100
: mapping generators: ~2, ~147/100
{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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Badness (Smith): 0.033081
Badness (Smith): 0.033081
=== 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
Sval mapping: {{mapping| 1 10 4 0 13 11 2| 0 -15 -3 5 -17 -13 4}}
Optimal tunings:
* CWE: ~2 = 1200.000, ~22/15 = 673.363
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


== Quadrimage ==
== Quadrimage ==
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: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~28/25
{{Multival|legend=1| 20 4 7 -40 -45 5 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 1,093: Line 1,083:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Magic_family"