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{{interwiki
| de = Magische Temperaturen
| en = Magic family
| es =
| ja =
}}
{{Technical data page}}
{{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.
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.
Line 92: Line 86:
Comma list: 100/99, 225/224, 245/243
Comma list: 100/99, 225/224, 245/243


Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
{{Mapping|legend=0| 1 0 2 -1 6 | 0 5 1 12 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 117: Line 111:
Comma list: 100/99, 105/104, 144/143, 196/195
Comma list: 100/99, 105/104, 144/143, 196/195


Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}


Optimal tunings:  
Optimal tunings:  
Line 136: Line 130:
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153


Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 151: Line 145:
Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153
Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153


Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 166: Line 160:
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169


Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}


Optimal tunings:  
Optimal tunings:  
Line 181: Line 175:
Comma list: 100/99, 105/104, 144/143, 170/169, 196/195
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 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}


Optimal tunings:  
Optimal tunings:  
Line 196: Line 190:
Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195
Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195


Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}


Optimal tunings:  
Optimal tunings:  
Line 228: Line 222:
Comma list: 65/64, 78/77, 91/90, 100/99
Comma list: 65/64, 78/77, 91/90, 100/99


Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}
{{Mapping|legend=0| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 243: Line 237:
Comma list: 100/99, 225/224, 245/243, 275/273
Comma list: 100/99, 225/224, 245/243, 275/273


Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}
{{Mapping|legend=0| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}


Optimal tunings:  
Optimal tunings:  
Line 258: Line 252:
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272


Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}
{{Mapping|legend=0| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 273: Line 267:
Comma list: 100/99, 225/224, 245/243, 1352/1331
Comma list: 100/99, 225/224, 245/243, 1352/1331


Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}
{{Mapping|legend=0| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}


Optimal tunings:  
Optimal tunings:  
Line 288: Line 282:
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272


Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}
{{Mapping|legend=0| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}


Optimal tunings:  
Optimal tunings:  
Line 303: Line 297:
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272
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 }}
{{Mapping|legend=0| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 318: Line 312:
Comma list: 55/54, 99/98, 176/175
Comma list: 55/54, 99/98, 176/175


Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
{{Mapping|legend=0| 1 0 2 -1 -1 | 0 5 1 12 14 }}


Optimal tunings:  
Optimal tunings:  
Line 333: Line 327:
Comma list: 55/54, 65/64, 91/90, 99/98
Comma list: 55/54, 65/64, 91/90, 99/98


Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}
{{Mapping|legend=0| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 348: Line 342:
Comma list: 55/54, 66/65, 99/98, 105/104
Comma list: 55/54, 66/65, 99/98, 105/104


Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
{{Mapping|legend=0| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}


Optimal tunings:  
Optimal tunings:  
Line 363: Line 357:
Comma list: 45/44, 56/55, 245/243
Comma list: 45/44, 56/55, 245/243


Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}
{{Mapping|legend=0| 1 0 2 -1 0 | 0 5 1 12 11 }}


Optimal tunings:  
Optimal tunings:  
Line 378: Line 372:
Comma list: 45/44, 56/55, 78/77, 245/243
Comma list: 45/44, 56/55, 78/77, 245/243


Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}
{{Mapping|legend=0| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


Optimal tunings:  
Optimal tunings:  
Line 393: Line 387:
Comma list: 45/44, 56/55, 78/77, 85/84, 245/243
Comma list: 45/44, 56/55, 78/77, 85/84, 245/243


Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}
{{Mapping|legend=0| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}


Optimal tunings:  
Optimal tunings:  
Line 408: Line 402:
Comma list: 35/34, 45/44, 52/51, 56/55, 245/243
Comma list: 35/34, 45/44, 52/51, 56/55, 245/243


Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}
{{Mapping|legend=0| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}


Optimal tunings:  
Optimal tunings:  
Line 423: Line 417:
Comma list: 45/44, 56/55, 65/64, 245/243
Comma list: 45/44, 56/55, 65/64, 245/243


Mapping: {{mapping| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}
{{Mapping|legend=0| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 438: Line 432:
Comma list: 225/224, 245/243, 441/440
Comma list: 225/224, 245/243, 441/440


Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}
{{Mapping|legend=0| 1 0 2 -1 -7 | 0 5 1 12 33 }}


Optimal tunings:  
Optimal tunings:  
Line 453: Line 447:
Comma list: 105/104, 196/195, 245/243, 275/273
Comma list: 105/104, 196/195, 245/243, 275/273


Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}
{{Mapping|legend=0| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}


Optimal tunings:  
Optimal tunings:  
Line 468: Line 462:
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243


Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}
{{Mapping|legend=0| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}


Optimal tunings:  
Optimal tunings:  
Line 483: Line 477:
Comma list: 121/120, 225/224, 245/243
Comma list: 121/120, 225/224, 245/243


Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}
{{Mapping|legend=0| 2 0 4 -2 5 | 0 5 1 12 3 }}


Optimal tunings:  
Optimal tunings:  
Line 498: Line 492:
Comma list: 105/104, 121/120, 196/195, 245/243
Comma list: 105/104, 121/120, 196/195, 245/243


Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}
{{Mapping|legend=0| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}


Optimal tunings:  
Optimal tunings:  
Line 513: Line 507:
Comma list: 105/104, 121/120, 154/153, 196/195, 245/243
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 }}
{{Mapping|legend=0| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}


Optimal tunings:  
Optimal tunings:  
Line 528: Line 522:
Comma list: 225/224, 243/242, 245/242
Comma list: 225/224, 243/242, 245/242


Mapping: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 | 0 10 2 24 25 }}


Optimal tunings:  
Optimal tunings:  
Line 543: Line 537:
Comma list: 105/104, 196/195, 243/242, 245/242
Comma list: 105/104, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}


Optimal tunings:  
Optimal tunings:  
Line 558: Line 552:
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 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}


Optimal tunings:  
Optimal tunings:  
Line 573: Line 567:
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 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}


Optimal tunings:  
Optimal tunings:  
Line 614: Line 608:
Comma list: 45/44, 126/125, 385/384
Comma list: 45/44, 126/125, 385/384


Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}
{{Mapping|legend=0| 1 0 2 5 0 | 0 5 1 -7 11 }}


Optimal tunings:  
Optimal tunings:  
Line 633: Line 627:
Comma list: 45/44, 65/64, 78/77, 126/125
Comma list: 45/44, 65/64, 78/77, 126/125


Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}
{{Mapping|legend=0| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 648: Line 642:
Comma list: 33/32, 126/125, 176/175
Comma list: 33/32, 126/125, 176/175


Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}
{{Mapping|legend=0| 1 0 2 5 5 | 0 5 1 -7 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 663: Line 657:
Comma list: 33/32, 65/64, 105/104, 126/125
Comma list: 33/32, 65/64, 105/104, 126/125


Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}
{{Mapping|legend=0| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 697: Line 691:
Comma list: 55/54, 64/63, 625/616
Comma list: 55/54, 64/63, 625/616


Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}
{{Mapping|legend=0| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Optimal tunings:  
Optimal tunings:  
Line 712: Line 706:
Comma list: 40/39, 55/54, 64/63, 625/616
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|legend=0| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 748: Line 742:
Comma list: 36/35, 45/44, 363/343
Comma list: 36/35, 45/44, 363/343


Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}
{{Mapping|legend=0| 1 0 2 0 0 | 0 5 1 9 11 }}


Optimal tunings:  
Optimal tunings:  
Line 763: Line 757:
Comma list: 27/26, 36/35, 45/44, 363/343
Comma list: 27/26, 36/35, 45/44, 363/343


Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
{{Mapping|legend=0| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}


Optimal tunings:  
Optimal tunings:  
Line 772: Line 766:


Badness (Sintel): 1.58
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 ==
== Hocum ==
Line 819: Line 817:
Comma list: 385/384, 441/440, 625/616
Comma list: 385/384, 441/440, 625/616


Mapping: {{mapping| 1 -5 1 5 -4 | 0 15 3 -5 17 }}
{{Mapping|legend=0| 1 -5 1 5 -4 | 0 15 3 -5 17 }}


Optimal tunings:  
Optimal tunings:  
Line 834: Line 832:
Comma list: 105/104, 144/143, 275/273, 625/616
Comma list: 105/104, 144/143, 275/273, 625/616


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


Optimal tunings:  
Optimal tunings:  
Line 883: Line 881:
Comma list: 245/242, 385/384, 625/616
Comma list: 245/242, 385/384, 625/616


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


Optimal tunings:  
Optimal tunings:  
Line 898: Line 896:
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|legend=0| 1 -15 -1 -3 -22 | 0 20 4 7 9 31 }}


Optimal tunings:  
Optimal tunings:  
Line 932: Line 930:
Comma list: 385/384, 625/616, 2401/2376
Comma list: 385/384, 625/616, 2401/2376


Mapping: {{mapping| 1 -10 0 -6 3 | 0 25 5 19 1 }}
{{Mapping|legend=0| 1 -10 0 -6 3 | 0 25 5 19 1 }}


Optimal tunings:  
Optimal tunings:  
Line 947: Line 945:
Comma list: 196/195, 364/363, 385/384, 625/616
Comma list: 196/195, 364/363, 385/384, 625/616


Mapping: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}
{{Mapping|legend=0| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}


Optimal tunings:  
Optimal tunings:  
Line 976: Line 974:
[[Badness]] (Sintel): 7.27
[[Badness]] (Sintel): 7.27


[[Category:Temperament families]]
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Magic_family"