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Review optimal GPV sequences; +missing CTE and CWE tunings, +error maps, and misc. cleanup (1/2)
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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 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.
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.


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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]]:  
Line 56: Line 57:
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
Line 65: Line 66:


: 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:  
Line 76: Line 75:
[[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]]:  
Line 96: Line 95:


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:  
Line 105: Line 102:
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:  
Line 363: Line 360:


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:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 378.744
* [[CTE]]: ~2 = 1200.000, ~5/4 = 378.744
: [[error map]]: {{val| 0.000 -8.235 -7.570 -20.034 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 378.479
* [[POTE]]: ~2 = 1200.000, ~5/4 = 378.479
: error map: {{val| 0.000 -9.558 -7.834 -18.181 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 745: Line 740:
* 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]
* 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]


{{Optimal ET sequence|legend=0| 16, 19, 54bd }}
{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}


Badness (Smith): 0.048038
Badness (Smith): 0.048038
Line 760: Line 755:
* POTE: ~2 = 1200.000, ~5/4 = 377.653
* POTE: ~2 = 1200.000, ~5/4 = 377.653


{{Optimal ET sequence|legend=0| 16, 19, 54bdf }}
{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}


Badness (Smith): 0.030386
Badness (Smith): 0.030386
Line 775: Line 770:
* POTE: ~2 = 1200.000, ~5/4 = 377.832
* POTE: ~2 = 1200.000, ~5/4 = 377.832


{{Optimal ET sequence|legend=0| 3, 10bd, 13bd, 16, 19e, 35ee }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}


Badness (Smith): 0.046970
Badness (Smith): 0.046970
Line 790: Line 785:
* POTE: ~2 = 1200.000, ~5/4 = 377.838
* POTE: ~2 = 1200.000, ~5/4 = 377.838


{{Optimal ET sequence|legend=0| 3, 10bd, 13bd, 16, 19e, 35eef }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}


Badness (Smith): 0.028732
Badness (Smith): 0.028732
Line 802: Line 797:


{{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:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 377.385
* [[CTE]]: ~2 = 1200.000, ~5/4 = 377.385
: [[error map]]: {{val| 0.000 -15.028 -8.928 +27.643 }}
* [[CWE]]: ~2 = 1200.000, ~5/4 = 376.963
: error map: {{val| 0.000 -18.198 -9.562 +21.937 }}


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


[[Badness]] (Smith): 0.084213
[[Badness]] (Smith): 0.084213
Line 821: Line 817:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 377.388
* CTE: ~2 = 1200.000, ~5/4 = 377.388
* CWE: ~2 = 1200.000, ~5/4 = 376.973


{{Optimal ET sequence|legend=0| 3de, …, 13be, 16, 19d, 35d }}
{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}


Badness (Smith): 0.046775
Badness (Smith): 0.046775
Line 835: Line 832:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 376.914
* CTE: ~2 = 1200.000, ~5/4 = 376.914
* CWE: ~2 = 1200.000, ~5/4 = 376.422


{{Optimal ET sequence|legend=0| 3def, …, 16, 35d }}
{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}


Badness (Smith): 0.038328
Badness (Smith): 0.038328
Line 846: Line 844:


{{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:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 381.955
* [[CTE]]: ~2 = 1200.000, ~5/4 = 381.955
: [[error map]]: {{val| 0.000 +7.818 -4.359 +11.627 }}
* [[CWE]]: ~2 = 1200.000, ~5/4 = 381.956
: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}


{{Optimal ET sequence|legend=1| 3, …, 19d, 22 }}
{{Optimal ET sequence|legend=1| 3, 19d, 22 }}


[[Badness]] (Smith): 0.088072
[[Badness]] (Smith): 0.088072
Line 865: Line 864:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 381.790
* CTE: ~2 = 1200.000, ~5/4 = 381.790
* CWE: ~2 = 1200.000, ~5/4 = 381.712


{{Optimal ET sequence|legend=0| 3, …, 19d, 22 }}
{{Optimal ET sequence|legend=0| 3, 19d, 22 }}


Badness (Smith): 0.047379
Badness (Smith): 0.047379
Line 879: Line 879:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 381.732
* CTE: ~2 = 1200.000, ~5/4 = 381.732
* CWE: ~2 = 1200.000, ~5/4 = 381.736


{{Optimal ET sequence|legend=0| 3, …, 19d, 22 }}
{{Optimal ET sequence|legend=0| 3, 22 }}


Badness (Smith): 0.039703
Badness (Smith): 0.039703
Line 892: Line 893:


: 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:  
* [[CTE]]: ~2 = 1200.000, ~63/50 = 409.836
* [[CTE]]: ~2 = 1200.000, ~63/50 = 409.836
: [[error map]]: {{val| 0.000 -0.316 -5.986 -1.612 }}
* [[POTE]]: ~2 = 1200.000, ~63/50 = 410.108
* [[POTE]]: ~2 = 1200.000, ~63/50 = 410.108
: error map: {{val| 0.000 -0.437 -6.010 -1.406 }}


{{Optimal ET sequence|legend=1| 3, …, 35d, 38, 41, 202cc, 243cc, 284cc }}
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}


[[Badness]] (Smith): 0.107115
[[Badness]] (Smith): 0.107115


== 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
Line 913: Line 914:


: 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:  
* [[CTE]]: ~2 = 1200.000, ~147/100 = 673.187
* [[CTE]]: ~2 = 1200.000, ~147/100 = 673.187
: [[error map]]: {{val| 0.000 +0.240 -5.875 -2.891 }}
* [[POTE]]: ~2 = 1200.000, ~147/100 = 673.290
* [[POTE]]: ~2 = 1200.000, ~147/100 = 673.290
: error map: {{val| 0.000 -0.932 -6.109 -2.500 }}


{{Optimal ET sequence|legend=1| 16, 25, 41 }}
{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}


[[Badness]] (Smith): 0.098334
[[Badness]] (Smith): 0.098334
Line 935: Line 936:
* POTE: ~2 = 1200.000, ~22/15 = 673.340
* POTE: ~2 = 1200.000, ~22/15 = 673.340


{{Optimal ET sequence|legend=0| 16, 25e, 41, 221cc, 262ccde }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness (Smith): 0.045623
Badness (Smith): 0.045623
Line 950: Line 951:
* POTE: ~2 = 1200.000, ~22/15 = 673.359
* POTE: ~2 = 1200.000, ~22/15 = 673.359


{{Optimal ET sequence|legend=0| 16, 25e, 41, 139cf, 180cf, 221ccf }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


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 ==
Line 961: Line 972:
{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}
{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}


: 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:  
* [[CTE]]: ~2 = 1200.000, ~28/25 = 204.860
* [[CTE]]: ~2 = 1200.000, ~28/25 = 204.860
: [[error map]]: {{val| 0.000 +0.853 -5.752 -2.843 }}
* [[POTE]]: ~2 = 1200.000, ~28/25 = 204.987
* [[POTE]]: ~2 = 1200.000, ~28/25 = 204.987
: error map: {{val| 0.000 -1.691 -6.261 -3.733 }}


{{Optimal ET sequence|legend=1| 6, …, 29b, 35, 41 }}
{{Optimal ET sequence|legend=1| 6, 29b, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}


[[Badness]] (Smith): 0.127422
[[Badness]] (Smith): 0.127422
Line 984: Line 995:
* POTE: ~2 = 1200.000, ~28/25 = 204.956
* POTE: ~2 = 1200.000, ~28/25 = 204.956


{{Optimal ET sequence|legend=0| 6, …, 29b, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 35, 41 }}


Badness (Smith): 0.061572
Badness (Smith): 0.061572
Line 999: Line 1,010:
* POTE: ~2 = 1200.000, ~28/25 = 205.028
* POTE: ~2 = 1200.000, ~28/25 = 205.028


{{Optimal ET sequence|legend=0| 35f, 41, 199ccdef, 240ccdef, 281ccdeff }}
{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}


Badness (Smith): 0.044047
Badness (Smith): 0.044047
Line 1,010: Line 1,021:
{{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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~48/35 = 556.123
* [[CTE]]: ~2 = 1200.000, ~48/35 = 556.123
: [[error map]]: {{val| 0.000 +1.132 -5.696 -2.480 }}
* [[CWE]]: ~2 = 1200.000, ~48/35 = 556.050
: 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]] (Smith): 0.194548
[[Badness]] (Smith): 0.194548
Line 1,028: Line 1,042:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/8 = 556.122
* CTE: ~2 = 1200.000, ~11/8 = 556.122
* CWE: ~2 = 1200.000, ~11/8 = 556.095


{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
Line 1,042: Line 1,057:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/8 = 556.106
* CTE: ~2 = 1200.000, ~11/8 = 556.106
* CWE: ~2 = 1200.000, ~11/8 = 556.117


{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
Line 1,058: Line 1,074:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 240.000, ~5/4 = 380.499 (~256/245 = 99.501)
* [[CTE]]: ~8/7 = 240.000, ~5/4 = 380.499 (~256/245 = 99.501)
* [[POTE]]: ~8/7 = 240.000, ~5/4 = 379.713 (~256/245 = 100.287)
: [[error map]]: {{val| 0.000 +0.542 -5.814 -8.826 }}
* [[CWE]]: ~8/7 = 240.000, ~5/4 = 379.997 (~256/245 = 100.003)
: 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 }}
Line 1,065: 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"