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edited intro, mentioning its existence as the magic equivalent of flattone; also muggloid is kinda exo and useless since 45/44, 65/64 are well justified in the same way as for flattone
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'''Muggles''' is the rank-2 [[regular temperament|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 [[3/2|3rd]] and [[5/4|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]].
{{Interwiki
| en = Muggles
| de = Magische Temperaturen #Muggel
}}
{{Infobox regtemp
| Title = Muggles
| 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)
| Edo join 1 = 16 | Edo join 2 = 19
| Mapping = 1; 5 1 -7 11 -1
| Generators = 5/4
| Generators tuning = 377.7
| Optimization method = CWE
| MOS scales = [[3L 7s]], [[3L 10s]], [[3L 13s]], [[16L 3s]]
| Odd limit 1 = 9 | Mistuning 1 = 18.6 | Complexity 1 = 19
| Odd limit 2 = 13 | Mistuning 2 = 29.0 | Complexity 2 = 19
}}
'''Muggles''' is the rank-2 [[regular temperament|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 [[3/2|3rd]] and [[5/4|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<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5279.html#5299 Yahoo! Tuning Group | ''Poptimal generators'']</ref>.
This temperament was named by [[Gene Ward Smith]] in 2003<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5279.html#5299 Yahoo! Tuning Group | ''Poptimal generators'']</ref>.


See [[Magic family #Muggles]] for more technical data.  
See [[Magic family #Muggles]] for more technical data.


== Interval chain ==
== Interval chain ==
Odd harmonics 1–13 and their inverses are in '''bold'''.
Odd harmonics 1–13 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
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<nowiki/>* In 2.3.5.7.13 CWE tuning
<nowiki/>* In 2.3.5.7.13 CWE tuning


== Tuning spectra ==
== Tunings ==
=== Muggles ===
=== 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 ===
{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~5/4 = 377.761{{c}}
| 7/6
| ~5/4 = 377.640{{c}}
| {{Monzo| 0 -21 -5 27 }}
|-
| 9-odd-limit
| ~5/4 = 378.534{{c}}
| 9/7
| ~5/4 = 378.554{{c}}
| {{Monzo| 0 93 -4 -44 }}
|-
| 11-odd-limit
| ~5/4 = 377.393{{c}}
| 11/8
| ~5/4 = 377.758{{c}}
| {{Monzo| 0 85 -14 -62 46 }}
|-
| 13-odd-limit
| ~5/4 = 377.393{{c}}
| 11/8
| ~5/4 = 377.630{{c}}
| {{Monzo| 0 113 -12 -68 58 -26 }}
|-
| 15-odd-limit
| ~5/4 = 377.393{{c}}
| 11/8
| ~5/4 = 377.718{{c}}
| {{Monzo| 0 134 9 -81 63 -33 }}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 109: Line 204:
|  
|  
| 375.000
| 375.000
|  
| Lower bound of 7-odd-limit diamond monotone
|-
|-
|  
|  
Line 150: Line 245:
| 377.393
| 377.393
| 11-, 13- and 15-odd-limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 113 -12 -68 58 -26 }}
| 377.630
| 13-odd-limit least squares
|-
|
| {{monzo| 0 -21 -5 27 }}
| 377.640
| 7-odd-limit least squares
|-
|
| {{monzo| 0 134 9 -81 63 -33 }}
| 377.718
| 15-odd-limit least squares
|-
|
| {{monzo| 0 85 -14 -62 46 }}
| 377.758
| 11-odd-limit least squares
|-
|-
|  
|  
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| 378.534
| 378.534
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|
| {{monzo| 0 93 -4 -44 }}
| 378.554
| 9-odd-limit least squares
|-
|-
|  
|  
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|  
|  
| 378.947
| 378.947
|  
| Upper bound of 7-odd-limit diamond monotone; <br>9-, 11-, and 13-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
Line 220: Line 290:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
|
|
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|  
|  
| 381.818
| 381.818
|  
| 22d… val
|-
|-
|  
|  
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|  
|  
|}
|}
<nowiki/>* Besides the octave


=== Muggloid ===
== References ==
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(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
|
|}
 
== Notes ==


[[Category:Muggles| ]] <!-- main article -->
[[Category:Muggles| ]] <!-- main article -->
Retrieved from "https://en.xen.wiki/w/Muggles"