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Magic family: Difference between revisions

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Spell 1\1 in decimal; note the badness metric being used; misc. cleanup
Review optimal GPV sequences; +missing CTE and CWE tunings, +error maps, and misc. cleanup (1/2)
Line 22: Line 22:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.499
* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.499
: [[error map]]: {{val| 0.000 +0.542 -5.814 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.058
* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.058
: error map: {{val| 0.000 -1.663 -6.255 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 34: Line 36:
[[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]] (Smith): 0.039163
[[Badness]] (Smith): 0.039163
Line 50: Line 52:
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]] 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.
Line 68: Line 70:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.651
* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.651
: [[error map]]: {{val| 0.000 +1.301 -5.662 -1.011 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.352
* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.352
: error map: {{val| 0.000 -0.195 -5.962 -4.602 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 80: Line 84:
[[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]] (Smith): 0.018918
[[Badness]] (Smith): 0.018918
Line 107: Line 111:
* 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)


{{Optimal ET sequence|legend=0| 19, 22, 41, 104, 145c, 249cce }}
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}


Badness (Smith): 0.020352
Badness (Smith): 0.020352
Line 157: Line 161:
* CTE: ~2 = 1200.000, ~5/4 = 380.451
* CTE: ~2 = 1200.000, ~5/4 = 380.451


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


Badness (Smith): 0.020881
Badness (Smith): 0.020881
Line 171: Line 175:
* CTE: ~2 = 1200.000, ~5/4 = 380.573
* CTE: ~2 = 1200.000, ~5/4 = 380.573


{{Optimal ET sequence|legend=0| 19h, 22fh, 41 }}
{{Optimal ET sequence|legend=0| 22fh, 41 }}


Badness (Smith): 0.019945
Badness (Smith): 0.019945
Line 185: Line 189:
* CTE: ~2 = 1200.000, ~5/4 = 380.271
* CTE: ~2 = 1200.000, ~5/4 = 380.271


{{Optimal ET sequence|legend=0| 19g, 41, 101cde, 142cdefg }}
{{Optimal ET sequence|legend=0| 19g, 41, 60 }}


Badness (Smith): 0.026232
Badness (Smith): 0.026232
Line 199: Line 203:
* CTE: ~2 = 1200.000, ~5/4 = 380.360
* CTE: ~2 = 1200.000, ~5/4 = 380.360


{{Optimal ET sequence|legend=0| 19g, 41 }}
{{Optimal ET sequence|legend=0| 19gh, 41 }}


Badness (Smith): 0.023709
Badness (Smith): 0.023709
Line 213: Line 217:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.478
* POTE: ~2 = 1200.000, ~5/4 = 380.416
* POTE: ~2 = 1200.000, ~5/4 = 380.416


Line 243: Line 248:
* POTE: ~2 = 1200.000, ~5/4 = 380.729
* POTE: ~2 = 1200.000, ~5/4 = 380.729


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


Badness (Smith): 0.023768
Badness (Smith): 0.023768
Line 256: Line 261:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.625
* CTE: ~2 = 1200.000, ~5/4 = 380.625
* CWE: ~2 = 1200.000, ~5/4 = 380.621


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


Badness (Smith): 0.023232
Badness (Smith): 0.023232
Line 287: Line 293:
* POTE: ~2 = 1200.000, ~5/4 = 380.827
* POTE: ~2 = 1200.000, ~5/4 = 380.827


{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104g }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}


Badness (Smith): 0.022032
Badness (Smith): 0.022032
Line 300: Line 306:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.741
* CTE: ~2 = 1200.000, ~5/4 = 380.741
* CWE: ~2 = 1200.000, ~5/4 = 380.735


{{Optimal ET sequence|legend=0| 19f, 22, 41, 63h, 104gh }}
{{Optimal ET sequence|legend=0| 19f, 22, 41 }}


Badness (Smith): 0.021101
Badness (Smith): 0.021101
Line 344: Line 351:
Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~5/4 = 380.470
* CTE: ~17/12 = 600.000, ~5/4 = 380.470
* CWE: ~17/12 = 600.000, ~5/4 = 380.470


{{Optimal ET sequence|legend=0| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Line 356: Line 364:
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}


{{Multival|legend=1| 5 1 12 14 -10 5 5 25 29 -2 }}
Wedgie: {{Multival| 5 1 12 14 -10 5 5 25 29 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 362: Line 370:
* POTE: ~2 = 1200.000, ~5/4 = 381.019
* POTE: ~2 = 1200.000, ~5/4 = 381.019


{{Optimal ET sequence|legend=0| 19e, 22, 63e, 85ee }}
{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}


Badness (Smith): 0.027109
Badness (Smith): 0.027109
Line 377: Line 385:
* POTE: ~2 = 1200.000, ~5/4 = 380.520
* POTE: ~2 = 1200.000, ~5/4 = 380.520


{{Optimal ET sequence|legend=0| 19e, 22, 63eff }}
{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}


Badness (Smith): 0.025522
Badness (Smith): 0.025522
Line 392: Line 400:
* POTE: ~2 = 1200.000, ~5/4 = 380.619
* POTE: ~2 = 1200.000, ~5/4 = 380.619


{{Optimal ET sequence|legend=0| 19eg, 22, 63effg }}
{{Optimal ET sequence|legend=0| 19eg, 22, 41efg }}


Badness (Smith): 0.020201
Badness (Smith): 0.020201
Line 405: Line 413:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 381.464
* CTE: ~2 = 1200.000, ~5/4 = 381.464
* CWE: ~2 = 1200.000, ~5/4 = 380.735


{{Optimal ET sequence|legend=0| 19egh, 22 }}
{{Optimal ET sequence|legend=0| 19egh, 22, 41efghh }}


Badness (Smith): 0.019004
Badness (Smith): 0.019004
Line 421: Line 430:
* POTE: ~2 = 1200.000, ~5/4 = 380.483
* POTE: ~2 = 1200.000, ~5/4 = 380.483


{{Optimal ET sequence|legend=0| 19e, 22f, 41e, 63ef }}
{{Optimal ET sequence|legend=0| 19e, 22f }}


Badness (Smith): 0.026089
Badness (Smith): 0.026089
Line 436: Line 445:
* POTE: ~2 = 1200.000, ~5/4 = 380.604
* POTE: ~2 = 1200.000, ~5/4 = 380.604


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


Badness (Smith): 0.020274
Badness (Smith): 0.020274
Line 449: Line 458:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.866
* CTE: ~2 = 1200.000, ~5/4 = 380.866
* CWE: ~2 = 1200.000, ~5/4 = 380.628


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


Badness (Smith): 0.019518
Badness (Smith): 0.019518
Line 465: Line 475:
* POTE: ~2 = 1200.000, ~5/4 = 379.642
* POTE: ~2 = 1200.000, ~5/4 = 379.642


{{Optimal ET sequence|legend=0| 3de, …, 16d, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness (Smith): 0.039282
Badness (Smith): 0.039282
Line 480: Line 490:
* POTE: ~2 = 1200.000, ~5/4 = 379.791
* POTE: ~2 = 1200.000, ~5/4 = 379.791


{{Optimal ET sequence|legend=0| 3def, …, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}


Badness (Smith): 0.031938
Badness (Smith): 0.031938
Line 493: Line 503:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.216
* CTE: ~2 = 1200.000, ~5/4 = 380.216
* CWE: ~2 = 1200.000, ~5/4 = 380.148


{{Optimal ET sequence|legend=0| 3defg, …, 19g, 41eeg }}
{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}


Badness (Smith): 0.028074
Badness (Smith): 0.028074
Line 507: Line 518:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 379.524
* CTE: ~2 = 1200.000, ~5/4 = 379.524
* CWE: ~2 = 1200.000, ~5/4 = 379.571


{{Optimal ET sequence|legend=0| 3defg, …, 19 }}
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}


Badness (Smith): 0.029556
Badness (Smith): 0.029556
Line 523: Line 535:
* POTE: ~2 = 1200.000, ~5/4 = 379.116
* POTE: ~2 = 1200.000, ~5/4 = 379.116


{{Optimal ET sequence|legend=0| 3de, …, 16d, 19, 41eef, 60eeff }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness (Smith): 0.033317
Badness (Smith): 0.033317
Line 538: Line 550:
* POTE: ~2 = 1200.000, ~5/4 = 380.232
* POTE: ~2 = 1200.000, ~5/4 = 380.232


{{Optimal ET sequence|legend=0| 19e, 22e, 41 }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}


Badness (Smith): 0.030706
Badness (Smith): 0.030706
Line 553: Line 565:
* POTE: ~2 = 1200.000, ~5/4 = 380.189
* POTE: ~2 = 1200.000, ~5/4 = 380.189


{{Optimal ET sequence|legend=0| 19e, 41, 142cdf }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}


Badness (Smith): 0.023547
Badness (Smith): 0.023547
Line 568: Line 580:
* POTE: ~2 = 1200.000, ~5/4 = 380.114
* POTE: ~2 = 1200.000, ~5/4 = 380.114


{{Optimal ET sequence|legend=0| 19eg, 41, 101cd, 142cdfg }}
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness (Smith): 0.020756
Badness (Smith): 0.020756
Line 581: Line 593:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.339
* CTE: ~2 = 1200.000, ~5/4 = 380.339
* CWE: ~2 = 1200.000, ~5/4 = 380.205


{{Optimal ET sequence|legend=0| 19egh, 41, 101cdhh, 142cdfghh }}
{{Optimal ET sequence|legend=0| 19egh, 41, 60eh }}


Badness (Smith): 0.018625
Badness (Smith): 0.018625
Line 597: Line 610:
* POTE: ~99/70 = 600.000, ~5/4 = 380.223
* POTE: ~99/70 = 600.000, ~5/4 = 380.223


{{Optimal ET sequence|legend=0| 22, 60e, 82e, 104e }}
{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}


Badness (Smith): 0.035864
Badness (Smith): 0.035864
Line 612: Line 625:
* POTE: ~99/70 = 600.000, ~5/4 = 379.920
* POTE: ~99/70 = 600.000, ~5/4 = 379.920


{{Optimal ET sequence|legend=0| 22f, 60e, 82ef }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness (Smith): 0.034551
Badness (Smith): 0.034551
Line 625: Line 638:
Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~5/4 = 380.433
* CTE: ~17/12 = 600.000, ~5/4 = 380.433
* CWE: ~17/12 = 600.000, ~5/4 = 380.067


{{Optimal ET sequence|legend=0| 22f, 60e, 82ef }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness (Smith): 0.023775
Badness (Smith): 0.023775
Line 641: Line 655:
* POTE: ~2 = 1200.000, ~14/11 = 409.910
* POTE: ~2 = 1200.000, ~14/11 = 409.910


{{Optimal ET sequence|legend=0| 38d, 41 }}
{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}


Badness (Smith): 0.038519
Badness (Smith): 0.038519
Line 656: Line 670:
* POTE: ~2 = 1200.000, ~14/11 = 410.004
* POTE: ~2 = 1200.000, ~14/11 = 410.004


{{Optimal ET sequence|legend=0| 38df, 41 }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}


Badness (Smith): 0.030280
Badness (Smith): 0.030280
Line 669: Line 683:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~14/11 = 409.896
* CTE: ~2 = 1200.000, ~14/11 = 409.896
* CWE: ~2 = 1200.000, ~14/11 = 409.994


{{Optimal ET sequence|legend=0| 38df, 41, 120cdg, 161cdg, 202ccddfgg }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d }}


Badness (Smith): 0.025491
Badness (Smith): 0.025491
Line 683: Line 698:
Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~14/11 = 409.884
* CTE: ~2 = 1200.000, ~14/11 = 409.884
* CWE: ~2 = 1200.000, ~14/11 = 410.012


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


Badness (Smith): 0.020277
Badness (Smith): 0.020277
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