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''This page is about the regular temperament. For the scale structures sometimes associated with it, see [[7L 2s]] and [[2L 5s]].''  
: ''This page is about the regular temperament. For the scale structures sometimes associated with it, see [[7L 2s]] and [[2L 5s]].''  


'''Mavila''' is a [[regular temperament|temperament]] where the [[135/128|major chroma (135/128)]] is tempered out. It was first discovered by [[Erv Wilson]], possibly in 1989<ref>A ''Linear Tuning of 4-"5"-"6" Artihmetic Mean (−3=5)'' paper from 1989 was referenced in Erv Wilson's ''Meta Meantone & Meta Mavila'' paper.</ref>, after studying the tuning of the timbila music of the Chopi tribe in Mozambique.  
{{Infobox regtemp
| Title = Mavila
| Subgroups = 2.3.5, 2.3.5.11
| Comma basis = [[135/128]] (2.3.5)<br>[[33/32]], [[45/44]] (2.3.5.11)
| Mapping = 1; 1 -3 -1
| Edo join 1 = 7 | Edo join 2 = 9
| Generators = 3/2
| Generators tuning = 679.0
| Optimization method = CWE
| Pergen = (P8, P5)
| Color name = Layobiti
| MOS scales = [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[7L&nbsp;2s]]
| Odd limit 1 = 5 | Mistuning 1 = 23.0 | Complexity 1 = 5
| 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 tunings range from [[9edo]] to 7edo, with [[16edo]], [[23edo]], and [[25edo]] being typical. These tunings detune 5/4 and 3/2 by significant amounts; it is thus reasonable to call mavila an [[exotemperament]], though it is certainly more accurate than the archetypal exotemperaments such as [[father]].
 
Mavila's antidiatonic scale is similar to [[Pelog]] scales used in Indonesian gamelan music. While Pelog's exact tuning is subject to significant regional variation and usually has unequal intervals throughout the scale (as opposed to having exactly two interval sizes), it can be well approximated by the antidiatonic scales of 9edo and 16edo.
 
Mavila was first discovered by [[Erv Wilson]], possibly in 1989<ref>A ''Linear Tuning of 4-"5"-"6" Artihmetic Mean (−3=5)'' paper from 1989 was referenced in Erv Wilson's ''Meta Meantone & Meta Mavila'' paper.</ref>, after studying the tuning of the timbila music of the Chopi tribe in Mozambique.  


See [[Mavila family #Mavila]] for more technical data.
See [[Mavila family #Mavila]] for more technical data.


== Inverted major and minor intervals: the antidiatonic scale ==
== Notation ==
As a result of tempering out 135/128 rather than 81/80, the fifths are 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 "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 five, three, etc. steps in the antidiatonic scale.
{{Mavila}}


This has some very strange implications for music. The mavila antidiatonic scale is similar to the normal diatonic scale, except interval classes are flipped. Wherever there was a major third, you'll find a minor third, and vice versa. Half steps become whole steps and whole steps become half steps (closer to neutral second range, however). When you sharpen the leading tone in minor, you end up sharpening it down instead, meaning you flatten it. Also, minor is now major—you end up with three parallel natural/harmonic/melodic major scales, and only one minor scale. Instead of a diminished triad in the major scale, there is now an augmented triad.
== Interval chain ==
In the following table, odd harmonics 1–11 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 679.0
| '''3/2''', '''16/11''', 22/15
|-
| 2
| 158.0
| '''9/8''', 11/10, 12/11, 16/15
|-
| 3
| 836.9
| '''8/5''', 18/11
|-
| 4
| 315.9
| 6/5
|-
| 5
| 994.9
| 9/5
|-
| 6
| 473.9
| 27/20, 32/25
|-
| 7
| 1152.8
| 48/25, 108/55
|}
<nowiki/>* In 2.3.5.11-subgroup CWE tuning, octave reduced


As an example, the anti-Ionian scale has steps of ssLsssL, which looks like the regular Ionian scale except the "L" intervals are now "s" and vice versa.
== Chords and harmony ==
{{See also| Mavila temperament modal harmony }}


Because of the structure of this unique tuning, every existing piece of common practice music has, effectively, a "shadow" version in mavila temperament. That is, when Beethoven wrote Für Elise, he actually wrote two compositions—the one that you know, and the antidiatonic equivalent in mavila temperament. It's only that the antidiatonic versions have never been heard before until now. Examples of this are provided below.
Mavila's tuning has some very strange implications for music. The mavila antidiatonic scale is similar to the normal [[5L 2s|diatonic]] scale, except interval classes are flipped. Wherever there was a major third, you will find a minor third, and vice versa. Half steps become whole steps and whole steps become half steps (closer to neutral second range, however). When you sharpen the leading tone in minor, you end up sharpening it down instead, meaning you flatten it. Also, minor is now major – instead of a diminished triad in the major scale, there is now an augmented triad.


Mavila's antidiatonic scale is similar to the [[pelog]] scale used in Indonesian gamelan music. However, Pelog's exact tuning is subject to significant regional variation and usually has unequal intervals throughout the scale (as opposed to having exactly two interval sizes), but it can be well approximated by the antidiatonic scale of [[9edo]] and [[16edo]].
As an example, the anti-Ionian scale has steps of ssLsssL, which looks like the regular Ionian scale except the "L" intervals are now "s" and vice versa.


== Modal harmony ==
Because of the structure of this unique tuning, every existing piece of common practice music has, effectively, a shadow version in antidiatonic. That is, with {{w|Ludwig van Beethoven|Beethoven}}'s {{w|Für Elise}}, there are actually two compositions – the one that you know, and the antidiatonic equivalent that has never been heard before until now. Examples of this are provided in the [[#Music]] section.
{{Main| Mavila temperament modal harmony }}


== Scales ==
== Scales ==
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== Tunings ==
== Tunings ==
The fifths of mavila are very flat—16edo (675.0{{c}}) and 23edo (678.3{{c}}) are typical tunings. As a result, mavila is best played with [[Stretched and compressed tuning|stretched octaves]] and/or specialized timbres: either timbres with high rolloff (e.g. sine waves, marimba, and ocarina) or high inharmonicity (i.e. detuned partials, such as Gamelans, bells, or Timbila instruments).
The fifths of mavila are very flat – 16edo (675.0{{c}}) and 23edo (678.3{{c}}) are typical tunings. As a result, mavila is best played with [[stretched and compressed tuning|stretched octaves]] and/or specialized timbres: either timbres with high rolloff (e.g. sine waves, marimba, and ocarina) or high inharmonicity (i.e. detuned partials, such as Gamelans, bells, or Timbila instruments).


As with meantone, mavila has its own tuning spectrum. 7edo, with its 685.714{{c}} fifth, is often thought of as an informal dividing line between meantone and mavila, in which case it forms the sharpmost endpoint on the mavila tuning spectrum and the flatmost endpoint of the meantone spectrum: if the fifth is flatter than this, it will generate anti-diatonic scales, and if it is sharper than this, it will generate diatonic scales. The fifth of 9edo is also often thought of as the other (flatmost) endpoint on the mavila spectrum.
As with meantone, mavila has its own tuning spectrum. 7edo, with its 685.714{{c}} fifth, is often thought of as an informal dividing line between meantone and mavila, in which case it forms the sharpmost endpoint on the mavila tuning spectrum and the flatmost endpoint of the meantone spectrum: if the fifth is flatter than this, it will generate anti-diatonic scales, and if it is sharper than this, it will generate diatonic scales. The fifth of 9edo is also often thought of as the other (flatmost) endpoint on the mavila spectrum.
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7edo can be thought of as a primitive tuning, yielding a completely equal heptatonic scale that is equally diatonic and anti-diatonic.
7edo can be thought of as a primitive tuning, yielding a completely equal heptatonic scale that is equally diatonic and anti-diatonic.


The next edo supporting mavila is [[9edo]], which has a fifth of 666.67{{c}} and approximates the [[Pelog]] tuning commonly found in Indonesian gamelan music. 9edo can be thought of as the first mavila edo (and the first edo in general) differentiating between 4:5:6 major and 10:12:15 minor chords. This is fairly interesting, as there is no real equivalent in meantone terms. It is larger than the "diatonic" sized mos, but smaller than the 16-tone "chromatic" mos. It is best thought of as a "superdiatonic" scale.
The next edo supporting mavila is 9edo, which has a fifth of 666.67{{c}} and approximates the Pelog tuning commonly found in Indonesian gamelan music. 9edo can be thought of as the first mavila edo (and the first edo in general) differentiating between 4:5:6 major and 10:12:15 minor chords. This is fairly interesting, as there is no real equivalent in meantone terms. It is larger than the "diatonic" sized mos, but smaller than the 16-tone "chromatic" mos. It is best thought of as a "superdiatonic" scale.


It is also supported by 16edo, which is probably the most common tuning for mavila temperament. This can be thought of as the first edo offering the potential for chromatic mavila harmony, similar to 12edo for meantone. This is also the usual setting for the aforementioned Armodue theory, although the Armodue theory can easily be extended to larger mavila scales such as mavila[23].
It is also supported by 16edo, which is probably the most common tuning for mavila temperament. This can be thought of as the first edo offering the potential for chromatic mavila harmony, similar to 12edo for meantone. This is also the usual setting for the aforementioned Armodue theory, although the Armodue theory can easily be extended to larger mavila scales such as Mavila[23].


The next edo supporting mavila is 23edo, which is the second-most common tuning for mavila temperament, used frequently by [[Igliashon Jones]] in his [[Cryptic Ruse]] albums. The fifth is in the sharper range for a mavila fifth at 678{{c}}, and is consequently closer to 3/2 than in 16edo, although still fairly inharmonic compared to meantone. The anti-diatonic scale is more "quasi-equal" in this tuning than in 16edo.
The next edo supporting mavila is 23edo, which is the second-most common tuning for mavila temperament, used frequently by [[Igliashon Jones]] in his [[Cryptic Ruse]] albums. The fifth is in the sharper range for a mavila fifth at 678{{c}}, and is consequently closer to 3/2 than in 16edo, although still fairly inharmonic compared to meantone. The anti-diatonic scale is more "quasi-equal" in this tuning than in 16edo.


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"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 677.145{{c}}
| CWE: ~3/2 = 679.111{{c}}
| POTE: ~3/2 = 679.806{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 676.039{{c}}
| CWE: ~3/2 = 678.978{{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 = 1\1, ~3/2 = 675.456


== Music ==
== Music ==
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; [[Herman Miller]]
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/kosma-jumis-lul Kôsma jumiś lul] (2017)
* [https://soundcloud.com/morphosyntax-1/kosma-jumis-lul ''Kôsma jumiś lul''] (2017)


; [[John Moriarty]]
; [[John Moriarty]]
* [https://web.archive.org/web/20201127014303/http://clones.soonlabel.com/public/micro/j_l_moriat/Mavila.mp3 ''Mavila'']
* [https://www.youtube.com/watch?v=QzZw-KCn2ig ''Netbeans''] (2019)
* [https://www.youtube.com/watch?v=QzZw-KCn2ig ''Netbeans''] (2019)


; [[Sevish]]
; [[Sevish]]
* from ''Sean but not Heard'' (2012)
* from ''Sean but not Heard'' (2012)
** "Sea Poem" – [https://sevish.bandcamp.com/track/sea-poem Bandcamp] | [https://www.youtube.com/watch?v=2p3z9YEpW1k YouTube] – mavila[9] in an unknown non-edo tuning
** "Sea Poem" – [https://sevish.bandcamp.com/track/sea-poem Bandcamp] | [https://www.youtube.com/watch?v=2p3z9YEpW1k YouTube] – in Mavila[9], an unknown non-edo tuning
** "Marooned at Home" – [https://sevish.bandcamp.com/track/marooned-at-home Bandcamp] | [https://www.youtube.com/watch?v=1tdHPqKPOWc YouTube]
** "Marooned at Home" – [https://sevish.bandcamp.com/track/marooned-at-home Bandcamp] | [https://www.youtube.com/watch?v=1tdHPqKPOWc YouTube]
; [[Gene Ward Smith]]
* ''Mysterious Mush'' – [https://web.archive.org/web/20201127014704/http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/mush.ogg unmapped version] · [https://web.archive.org/web/20201127013337/http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/mushc.ogg spectrally mapped version]{{clarify}}
* [https://web.archive.org/web/20201127015551/http://micro.soonlabel.com/gene_ward_smith/transformers/hopper.mp3 ''Hopper''] by Singer-Medora-White-Smith{{clarify}}; in {{nowrap|''f''<sup>4</sup> − 10''f'' + 10}} =&nbsp;0 equal-beating mavila


; [[Starshine]]
; [[Starshine]]
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=== Experiments ===
=== Experiments ===
Mike Battaglia has "translated" several common practice pieces into mavila by using Graham Breed's Lilypond code to tune the generators flat. Musical examples are provided in 9edo, 16edo, 23edo, and 25edo, for comparison. Note that the melodic and/or intonational properties differ slightly for each tuning.
Mike Battaglia has translated several common practice pieces into mavila by using Graham Breed's Lilypond code to tune the generators flat. Musical examples are provided in 9edo, 16edo, 23edo, and 25edo, for comparison. Note that the melodic and/or intonational properties differ slightly for each tuning.


* 9edo: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-9-edo</soundcloud>
* [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-9-edo 9edo version] · [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-16-edo 16edo version] · [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments 23edo version] · [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-25-edo 25edo version]
* 16edo: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-16-edo</soundcloud>
* 23edo: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments</soundcloud>
* 25edo: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-25-edo</soundcloud>


== See also ==
== See also ==
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[[Category:Mavila| ]] <!-- Main article -->
[[Category:Mavila| ]] <!-- Main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Mavila family]]
[[Category:Mavila family]]
[[Category:Starling temperaments]]
Retrieved from "https://en.xen.wiki/w/Mavila"