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Mavila: Difference between revisions

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| Mapping = 1; 1 -3 -1
| Mapping = 1; 1 -3 -1
| Edo join 1 = 7 | Edo join 2 = 9
| Edo join 1 = 7 | Edo join 2 = 9
| Generator = 3/2
| Generators = 3/2
| Generator tuning = 679.0
| Generators tuning = 679.0
| Optimization method = CWE
| Optimization method = CWE
| Pergen = (P8, P5)
| Pergen = (P8, P5)
| Color name = Layobiti
| Color name = Layobiti
| MOS scales = [[2L 3s]], [[2L 5s]], [[7L 2s]]
| MOS scales = [[2L 3s]], [[2L 5s]], [[7L 2s]]
| Odd limit 1 = 5 | Mistuning 1 = 23.0 | Complexity 1 = 9
| Odd limit 1 = 5 | Mistuning 1 = 23.0 | Complexity 1 = 5
| Odd limit 2 = (5-limit) 9 | Mistuning 2 = 36.9 | Complexity 2 = 16
| Odd limit 2 = 2.3.5.11 11 | Mistuning 2 = 36.9 | Complexity 2 = 7
}}
}}
'''Mavila''' is a [[regular temperament|temperament]] where the major chroma, [[135/128]], is [[tempering out|tempered out]]. Like [[meantone]], mavila is based on the [[chain of fifths]], but as a result of tempering out 135/128 rather than [[81/80]], the fifths are supposedly very flat ({{nowrap|~{{dash|670, 680}}}}{{c}} or so), flatter than even that of [[7edo]] (4\7). Consequently, stacking 7 of these fifths gives you an [[2L 5s|antidiatonic]] [[mos scale]], where in a certain sense, major and minor intervals get reversed. For example, stacking four fifths and octave-reducing now gets you a [[6/5]] ''minor'' third, whereas stacking three fourths and octave-reducing now gets you a [[5/4]] ''major'' third. Note that since we have a heptatonic scale, terms like ''fifths'', ''thirds'', etc. make perfect sense and really are the fifth, third, etc. steps in the antidiatonic scale.
'''Mavila''' is a [[regular temperament|temperament]] where the major chroma, [[135/128]], is [[tempering out|tempered out]]. Like [[meantone]], mavila is based on the [[chain of fifths]], but as a result of tempering out 135/128 rather than [[81/80]], the fifths are supposedly very flat ({{nowrap|~{{dash|670, 680}}}}{{c}} or so), flatter than even that of [[7edo]] (4\7). Consequently, stacking 7 of these fifths gives you an [[2L 5s|antidiatonic]] [[mos scale]], where in a certain sense, major and minor intervals get reversed. For example, stacking four fifths and octave-reducing now gets you a [[6/5]] ''minor'' third, whereas stacking three fourths and octave-reducing now gets you a [[5/4]] ''major'' third. Note that since we have a heptatonic scale, terms like ''fifths'', ''thirds'', etc. make perfect sense and really are the fifth, third, etc. steps in the antidiatonic scale.


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25edo also supports mavila. The tuning is 672{{c}} and hence very flat, even flatter than 16edo, but not as flat as 9edo. This is 25edo's second-best 3/2; the alternate fifth generates 5edo.
25edo also supports mavila. The tuning is 672{{c}} and hence very flat, even flatter than 16edo, but not as flat as 9edo. This is 25edo's second-best 3/2; the alternate fifth generates 5edo.


=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
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| POTE: ~3/2 = 679.788{{c}}
| POTE: ~3/2 = 679.788{{c}}
|}
|}
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 675.456{{c}}


=== Tuning spectrum ===
=== 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|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| 600.000
| 600.000
| Lower bound of 5-odd-limit diamond monotone
| Lower bound of 5-odd-limit diamond monotone
|-
|
| 11/8
| 648.682
|
|-
| 6\11
|
| 654.545
|
|-
|-
|  
|  
| 15/8
| 15/8
| 655.866
| 655.866
| 1/2 comma
|-
|
| 15/11
| 663.049
|  
|  
|-
|-
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| 5/4
| 5/4
| 671.229
| 671.229
|  
| 1/3 comma
|-
|-
| 9\16
| 9\16
|  
|  
| 675.000
| 675.000
|
|-
|
| 11/6
| 675.319
|  
|  
|-
|-
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| 25/24
| 25/24
| 675.618
| 675.618
|  
| 2/7 comma
|-
|-
|  
|  
|  
| ''f''<sup>4</sup> + ''f''<sup>3</sup> - 8 = 0
| 676.337
| 676.337
| octave mirror to Wilson's 523.662 meta-mavila
| 1–3–5 equal-beating tuning, Erv Wilson's meta-mavila
|-
|-
| 13\23
| 13\23
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| 5/3
| 5/3
| 678.910
| 678.910
| 5-odd-limit minimax
| 1/4 comma, 5-odd-limit minimax
|-
|
| 11/10
| 682.502
|
|-
|-
|  
|  
| 9/5
| 9/5
| 683.519
| 683.519
| 5-limit 9-odd-limit minimax
| 1/5 comma, 5-limit 9-odd-limit minimax
|-
|
| 11/9
| 684.197
|
|-
|-
| 4\7
| 4\7
|  
|  
| 685.714
| 685.714
| Upper bound of 5-odd-limit diamond monotone<br />5-limit 9-odd-limit diamond monotone (singleton)
| Upper bound of 5-odd-limit diamond monotone<br>5-limit 9-odd-limit and 2.3.5.11-subgroup 11-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
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| Pythagorean tuning
| Pythagorean tuning
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave
 
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 675.456{{c}}


== Music ==
== Music ==
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