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'''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]]. 11-limit extension of the Muggles include:
{{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 (3e&amp;16 or 16&amp;19) - tempering out 45/44 and 385/384
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>.
* Muggloid (3&amp;16 or 16&amp;19e) - tempering out 33/32 and 176/175


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


== Tuning spectra ==
== Interval chain ==
=== Muggles ===
Odd harmonics 1–13 and their inverses are in '''bold'''.
Gencom: [2 5/4; 45/44 65/64 78/77 126/125]


Gencom mapping: [{{val| 1 0 2 5 0 4 }}, {{val| 0 5 1 -7 11 -1 }}]
{| class="wikitable center-1 right-2"
 
|-
{| class="wikitable center-all left-4"
! #
! Cents*
! Approximate ratios
|-
|-
! ET<br>Generator
| 0
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged Interval)]]
| 0.00
! Generator<br>(¢)
| '''1/1'''
! Comments
|-
|-
|  
| 1
| 11/9
| 378.5
| 347.408
| '''5/4''', '''16/13''', 26/21
|  
|-
|-
|  
| 2
| 16/13
| 757.0
| 359.472
| 20/13, 32/21
|  
|-
|-
|  
| 3
| 15/11
| 1135.4
| 372.610
| 25/13
|  
|-
|-
|  
| 4
| 13/10
| 313.9
| 372.893
| 6/5
|  
|-
|-
|  
| 5
| 12/11
| 692.4
| 374.894
| '''3/2'''
|  
|-
|-
| 5\16
| 6
|  
| 1070.9
| 375.000
| 13/7, 15/8, 24/13
|  
|-
|-
|  
| 7
| 8/7
| 249.4
| 375.882
| '''8/7''', 15/13
|  
|-
|-
|  
| 8
| 13/11
| 627.9
| 375.899
| 10/7
|  
|-
|-
|  
| 9
| 11/10
| 1006.3
| 376.500
| 9/5
|  
|-
|-
|  
| 10
| 14/11
| 184.8
| 376.805
| '''9/8'''
|  
|-
|-
|  
| 11
| 13/12
| 563.3
| 376.905
| 18/13
|  
|-
|-
| 11\35
| 12
|  
| 941.8
| 377.143
| 12/7
|  
|-
|-
|  
| 13
| 7/5
| 120.3
| 377.186
| 15/14
|  
 
|}
<nowiki/>* In 2.3.5.7.13 CWE tuning
 
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
|  
! rowspan="2" |  
| 11/8
! colspan="3" | Euclidean
| 377.393
| 11-, 13- and 15-odd-limit minimax
|-
|-
|
! Constrained
| <span style="font-size:0.75em">{{monzo| 0 113 -12 -68 58 -26 }}</span>
! Constrained & skewed
| 377.630
! Destretched
| 13-odd-limit least squares
|-
|-
|  
! Tenney
| {{monzo| 0 -21 -5 27 }}
| CTE: ~5/4 = 378.7441{{c}}
| 377.640
| CWE: ~5/4 = 378.5328{{c}}
| 7-odd-limit least squares
| 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" |  
| <span style="font-size:0.75em">{{monzo| 0 134 9 -81 63 -33 }}</span>
! colspan="3" | Euclidean
| 377.718
| 15-odd-limit least squares
|-
|-
|
! Constrained
| <span style="font-size:0.9em">{{monzo| 0 85 -14 -62 46 }}</span>
! Constrained & skewed
| 377.758
! Destretched
| 11-odd-limit least squares
|-
|-
|  
! Tenney
| 7/6
| CTE: ~5/4 = 377.1761{{c}}
| 377.761
| CWE: ~5/4 = 377.7336{{c}}
| 7-odd-limit minimax
| 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
| 15/13
! colspan="2" | Minimax
| 378.249
! colspan="2" | Least squares
|  
|-
|-
|
! Generator
| 15/14
! Eigenmonzo*
| 378.419
! Generator
|
! Eigenmonzo*
|-
|-
|  
| 7-odd-limit
| 18/13
| ~5/4 = 377.761{{c}}
| 378.489
| 7/6
|  
| ~5/4 = 377.640{{c}}
| {{Monzo| 0 -21 -5 27 }}
|-
|-
|  
| 9-odd-limit
| ~5/4 = 378.534{{c}}
| 9/7
| 9/7
| 378.534
| ~5/4 = 378.554{{c}}
| 9-odd-limit minimax
| {{Monzo| 0 93 -4 -44 }}
|-
|-
|  
| 11-odd-limit
| {{monzo| 0 93 -4 -44 }}
| ~5/4 = 377.393{{c}}
| 378.554
| 11/8
| 9-odd-limit least squares
| ~5/4 = 377.758{{c}}
| {{Monzo| 0 85 -14 -62 46 }}
|-
|-
|  
| 13-odd-limit
| 14/13
| ~5/4 = 377.393{{c}}
| 378.617
| 11/8
|  
| ~5/4 = 377.630{{c}}
| {{Monzo| 0 113 -12 -68 58 -26 }}
|-
|-
|  
| 15-odd-limit
| 6/5
| ~5/4 = 377.393{{c}}
| 378.910
| 11/8
|
| ~5/4 = 377.718{{c}}
|-
| {{Monzo| 0 134 9 -81 63 -33 }}
| 6\19
|
| 378.947
|
|-
|  
| 10/9
| 379.733
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|
| 4/3
| 380.391
| 5-odd-limit minimax
|-
|
| 16/15
| 381.378
|
|-
| 7\22
|
| 381.818
|
|-
|
| 5/4
| 386.314
|
|}
|}


=== Muggloid ===
=== Tuning spectrum ===
Gencom: [2 5/4; 33/32 65/64 105/104 126/125]
 
Gencom mapping: [{{val| 1 0 2 5 5 4 }}, {{val| 0 5 1 -7 -5 -1 }}]
 
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! ET<br>Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged Interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator<br>(¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/13
| 11/9
| 359.472
| 347.408
|  
|  
|-
|-
|  
|  
| 11/8
| 13/8
| 369.736
| 359.472
|  
|  
|-
|-
|  
|  
| 13/11
| 15/11
| 372.302
| 372.610
|  
|  
|-
|-
|  
|  
| 11/10
| 13/10
| 372.499
| 372.893
|  
|  
|-
|-
|  
|  
| 13/10
| 11/6
| 372.893
| 374.894
|  
|  
|-
|-
Line 225: Line 204:
|  
|  
| 375.000
| 375.000
|  
| Lower bound of 7-odd-limit diamond monotone
|-
|-
|  
|  
| 12/11
| 7/4
| 375.064
| 375.882
|  
|  
|-
|-
|  
|  
| 8/7
| 13/11
| 375.882
| 375.899
|  
|  
|-
|-
|  
|  
| 15/11
| 11/10
| 376.086
| 376.500
|  
|  
|-
|-
|  
|  
| 11/9
| 11/7
| 376.839
| 376.805
| 11-, 13- and 15-odd-limit minimax
|  
|-
|-
|  
|  
Line 261: Line 240:
| 377.186
| 377.186
|  
|  
|-
|
| 11/8
| 377.393
| 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
Line 278: Line 262:
|-
|-
|  
|  
| 18/13
| 13/9
| 378.489
| 378.489
|  
|  
Line 288: Line 272:
|-
|-
|  
|  
| 14/13
| 13/7
| 378.617
| 378.617
|  
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
Line 300: Line 284:
|  
|  
| 378.947
| 378.947
|  
| Upper bound of 7-odd-limit diamond monotone; <br>9-, 11-, and 13-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
| 10/9
| 9/5
| 379.733
| 379.733
|  
|  
|-
|-
|
|
| 4/3
| 3/2
| 380.391
| 380.391
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 16/15
| 15/8
| 381.378
| 381.378
|  
|  
Line 320: Line 304:
|  
|  
| 381.818
| 381.818
|  
| 22d… val
|-
|-
|  
|  
| 5/4
| 5/4
| 386.314
| 386.314
|
|-
|
| 14/11
| 391.246
|  
|  
|}
|}
<nowiki/>* Besides the octave
== References ==


{{IoT}}
[[Category:Muggles| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Magic family]]
[[Category:Magic family]]
[[Category:Starling temperaments]]
[[Category:Starling temperaments]]
[[Category:Avicennmic temperaments]]
[[Category:Avicennmic temperaments]]

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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