Muggles: Difference between revisions

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[[de:Magische_Temperaturen#Muggel|Muggel]]
{{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]].
'''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]].


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== 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|Unchanged interval<br>(eigenmonzo)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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|  
|  
| 375.000
| 375.000
|  
| Lower bound of 7-odd-limit diamond monotone
|-
|-
|  
|  
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| 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)
|-
|-
|  
|  
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| 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


== Notes ==
== References ==


[[Category:Muggles| ]] <!-- main article -->
[[Category:Muggles| ]] <!-- main article -->

Latest revision as of 09:57, 8 April 2026

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);
45/44, 126/125, 385/384 (11-limit);
45/44, 65/64, 78/77, 126/125
(13-limit)
Reduced mapping ⟨1; 5 1 -7 11 -1]
ET join 16 & 19
Generators (CWE) ~5/4 = 377.7 ¢
MOS scales 3L 7s, 3L 10s, 3L 13s, 16L 3s
Ploidacot alpha-pentacot
Minimax error 9-odd-limit: 18.6 ¢;
13-odd-limit: 29.0 ¢
Target scale size 9-odd-limit: 19 notes;
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

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 378.7441 ¢ CWE: ~5/4 = 378.5328 ¢ POTE: ~5/4 = 378.4794 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 377.1761 ¢ CWE: ~5/4 = 377.7336 ¢ POTE: ~5/4 = 377.6530 ¢

Target tunings

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

References

  1. ↑ Yahoo! Tuning Group | Poptimal generators
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