Muggles
Muggles is the rank-2 temperament tempering out 126/125, the starling comma, and 525/512, Avicenna's enharmonic diesis. It is an alternative 7-limit extension to magic and can be described as the 16 & 19 temperament; 16edo, 35edo, and 54edo with the flat-fifth b val all are muggles tunings. As a tuning noted for having both very flat 3rd and 5th harmonics, and supported by 19edo, it is very analogous to flattone. Similarly to flattone, muggles can extend well to the 13-limit by equating 5/4 to both 11/9 and 16/13, thereby tempering out 45/44 and 65/64.
This temperament was named by Gene Ward Smith in 2003[1].
See Magic family #Muggles for more technical data.
Interval chain
Odd harmonics 1–13 and their inverses are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.00 | 1/1 |
| 1 | 378.5 | 5/4, 16/13, 26/21 |
| 2 | 757.0 | 20/13, 32/21 |
| 3 | 1135.4 | 25/13 |
| 4 | 313.9 | 6/5 |
| 5 | 692.4 | 3/2 |
| 6 | 1070.9 | 13/7, 15/8, 24/13 |
| 7 | 249.4 | 8/7, 15/13 |
| 8 | 627.9 | 10/7 |
| 9 | 1006.3 | 9/5 |
| 10 | 184.8 | 9/8 |
| 11 | 563.3 | 18/13 |
| 12 | 941.8 | 12/7 |
| 13 | 120.3 | 15/14 |
* In 2.3.5.7.13 CWE tuning
Tuning spectra
Muggles
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 11/9 | 347.408 | ||
| 13/8 | 359.472 | ||
| 15/11 | 372.610 | ||
| 13/10 | 372.893 | ||
| 11/6 | 374.894 | ||
| 5\16 | 375.000 | ||
| 7/4 | 375.882 | ||
| 13/11 | 375.899 | ||
| 11/10 | 376.500 | ||
| 11/7 | 376.805 | ||
| 13/12 | 376.905 | ||
| 11\35 | 377.143 | ||
| 7/5 | 377.186 | ||
| 11/8 | 377.393 | 11-, 13- and 15-odd-limit minimax | |
| [0 113 -12 -68 58 -26⟩ | 377.630 | 13-odd-limit least squares | |
| [0 -21 -5 27⟩ | 377.640 | 7-odd-limit least squares | |
| [0 134 9 -81 63 -33⟩ | 377.718 | 15-odd-limit least squares | |
| [0 85 -14 -62 46⟩ | 377.758 | 11-odd-limit least squares | |
| 7/6 | 377.761 | 7-odd-limit minimax | |
| 15/13 | 378.249 | ||
| 15/14 | 378.419 | ||
| 13/9 | 378.489 | ||
| 9/7 | 378.534 | 9-odd-limit minimax | |
| [0 93 -4 -44⟩ | 378.554 | 9-odd-limit least squares | |
| 13/7 | 378.617 | ||
| 5/3 | 378.910 | ||
| 6\19 | 378.947 | ||
| 9/5 | 379.733 | ||
| 27/20 | 379.968 | 5-odd-limit least squares | |
| 3/2 | 380.391 | 5-odd-limit minimax | |
| 15/8 | 381.378 | ||
| 7\22 | 381.818 | ||
| 5/4 | 386.314 |
Muggloid
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 13/8 | 359.472 | ||
| 11/8 | 369.736 | ||
| 13/11 | 372.302 | ||
| 11/10 | 372.499 | ||
| 13/10 | 372.893 | ||
| 5\16 | 375.000 | ||
| 11/6 | 375.064 | ||
| 7/4 | 375.882 | ||
| 15/11 | 376.086 | ||
| 11/9 | 376.839 | 11-, 13- and 15-odd-limit minimax | |
| 13/12 | 376.905 | ||
| 11\35 | 377.143 | ||
| 7/5 | 377.186 | ||
| 7/6 | 377.761 | 7-odd-limit minimax | |
| 15/13 | 378.249 | ||
| 15/14 | 378.419 | ||
| 13/9 | 378.489 | ||
| 9/7 | 378.534 | 9-odd-limit minimax | |
| 13/7 | 378.617 | ||
| 5/3 | 378.910 | ||
| 6\19 | 378.947 | ||
| 9/5 | 379.733 | ||
| 3/2 | 380.391 | 5-odd-limit minimax | |
| 15/8 | 381.378 | ||
| 7\22 | 381.818 | ||
| 5/4 | 386.314 | ||
| 11/7 | 391.246 |