Muggles: Difference between revisions
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Tabulate target tunings |
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{{Infobox regtemp | {{Infobox regtemp | ||
| Title = Muggles | | Title = Muggles | ||
| Subgroups = 2.3.5.7, 2.3.5.7.13 | | Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13 | ||
| Comma basis = [[126/125]], [[525/512]] (7-limit);<br>[[45/44]], [[126/125]], [[385/384]] (11-limit);<br>[[45/44]], [[65/64]], [[78/77]], [[126/125]]<br>(13-limit) | | Comma basis = [[126/125]], [[525/512]] (7-limit);<br>[[45/44]], [[126/125]], [[385/384]] (11-limit);<br>[[45/44]], [[65/64]], [[78/77]], [[126/125]]<br>(13-limit) | ||
| Edo join 1 = 16 | Edo join 2 = 19 | | Edo join 1 = 16 | Edo join 2 = 19 | ||
| Line 91: | Line 91: | ||
== Tunings == | == Tunings == | ||
=== Norm-based tunings === | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~5/4 = 378.7441{{c}} | |||
| CWE: ~5/4 = 378.5328{{c}} | |||
| POTE: ~5/4 = 378.4794{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~5/4 = 377.1761{{c}} | |||
| CWE: ~5/4 = 377.7336{{c}} | |||
| POTE: ~5/4 = 377.6530{{c}} | |||
|} | |||
=== Target tunings === | === Target tunings === | ||
{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed" | {| class="wikitable center-1 center-3 mw-collapsible mw-collapsed" | ||
| Line 171: | Line 204: | ||
| | | | ||
| 375.000 | | 375.000 | ||
| | | Lower bound of 7-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
| Line 251: | Line 284: | ||
| | | | ||
| 378.947 | | 378.947 | ||
| | | Upper bound of 7-odd-limit diamond monotone; <br>9-, 11-, and 13-odd-limit diamond monotone (singleton) | ||
|- | |- | ||
| | | | ||
| Line 271: | Line 304: | ||
| | | | ||
| 381.818 | | 381.818 | ||
| | | 22d… val | ||
|- | |- | ||
| | | | ||
Latest revision as of 09:57, 8 April 2026
| Muggles |
45/44, 126/125, 385/384 (11-limit);
45/44, 65/64, 78/77, 126/125
(13-limit)
13-odd-limit: 29.0 ¢
13-odd-limit: 19 notes
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 bd 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 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
Tunings
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~5/4 = 378.7441 ¢ | CWE: ~5/4 = 378.5328 ¢ | POTE: ~5/4 = 378.4794 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~5/4 = 377.1761 ¢ | CWE: ~5/4 = 377.7336 ¢ | POTE: ~5/4 = 377.6530 ¢ |
Target tunings
| Target | Minimax | Least squares | ||
|---|---|---|---|---|
| Generator | Eigenmonzo* | Generator | Eigenmonzo* | |
| 7-odd-limit | ~5/4 = 377.761 ¢ | 7/6 | ~5/4 = 377.640 ¢ | [0 -21 -5 27⟩ |
| 9-odd-limit | ~5/4 = 378.534 ¢ | 9/7 | ~5/4 = 378.554 ¢ | [0 93 -4 -44⟩ |
| 11-odd-limit | ~5/4 = 377.393 ¢ | 11/8 | ~5/4 = 377.758 ¢ | [0 85 -14 -62 46⟩ |
| 13-odd-limit | ~5/4 = 377.393 ¢ | 11/8 | ~5/4 = 377.630 ¢ | [0 113 -12 -68 58 -26⟩ |
| 15-odd-limit | ~5/4 = 377.393 ¢ | 11/8 | ~5/4 = 377.718 ¢ | [0 134 9 -81 63 -33⟩ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
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 | Lower bound of 7-odd-limit diamond monotone | |
| 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 | |
| 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 | Upper bound of 7-odd-limit diamond monotone; 9-, 11-, and 13-odd-limit diamond monotone (singleton) | |
| 9/5 | 379.733 | ||
| 3/2 | 380.391 | 5-odd-limit minimax | |
| 15/8 | 381.378 | ||
| 7\22 | 381.818 | 22d… val | |
| 5/4 | 386.314 |
* Besides the octave