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Mavila is an extremely important temperament. It was first discovered by Erv Wilson after studying the tuning of the "Timbila" music of the Chopi tribe in Mozambique. It is also closely related to the "pelog" scale in Indonesian and Balinese Gamelan music.
: ''This page is about the regular temperament. For the scale structures sometimes associated with it, see [[7L 2s]] and [[2L 5s]].''


{{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.


== Inverted Major and Minor Intervals: The Anti-Diatonic 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]].


In mavila, the comma 135/128 vanishes, rather than 81/80. As a result, the fifths are very flat (~675-680 cents or so). They are so flat that they are even flatter than 7-EDO. As a result, stacking 7 of these fifths gives you an "anti-diatonic" 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 anti-diatonic scale.)
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.


This has some very strange implications for music. The mavila diatonic 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.
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.
 
== Notation ==
{{Mavila}}
 
== 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
 
== Chords and harmony ==
{{See also| Mavila temperament modal harmony }}
 
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.


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.
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.


Because of the structure of this unique tuning, it is true that every existing piece of common practice music has a "shadow" version in mavila temperament. That is, when Beethoven wrote Fur Elise, he actually wrote two compositions - the one that you know, and the anti-diatonic equivalent in mavila temperament. It's only that the anti-diatonic versions have never been heard before. Examples of this are provided below.
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.
 
== Scales ==
* [[Mavila-eb]] – 12-tone chromatic scale, equal-beating tuning
 
=== MOS tree ===
In addition to the 7-note anti-diatonic scale described, Mavila also has a 9 note "superdiatonic" mos, the "super-Ionian" mode of which looks LLLsLLLLs. This is the basis for the [[Armodue theory]].  


== Mavila Modal Harmony ==
Mavila generates a 16-tone "chromatic" mos. In a certain sense, much of mavila makes sense if viewed within the lens of a 16-tone chromatic gamut, similarly to how much of meantone is thought of in the setting of a 12-tone chromatic gamut.


An exploration of Mavila Modal Harmony can be found in [[Mavila_Temperament_Modal_Harmony]].
After the 16-tone "chromatic" scale is the 23-tone or 25-tone "enharmonic" mos (depending on the tuning of the fifth), which can be thought of as an "extended mavila" analogous to the "extended meantone" 19-tone enharmonic scale. The two alternative enharmonic scales are similar to how if the fifth is tuned sharper than 12edo, it will generate a 17-tone mos rather than a 19-tone one.  


== 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).
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.
Much like meantone, mavila is [[support]]ed by several low-numbered edos, which will basically be the same size as the mosses listed above.
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.
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 fifths of mavila are very flat - 16-EDO (675.0 cents) and 23-EDO (678.3 cents) are typical tunings, and the optimal 5-limit tuning is 679.8 cents. As a result, mavila is best played with specialized timbres: either timbres with a lot of rolloff (such as marimba, sine waves, ocarina, etc), or timbres with detuned partials (such as Gamelan or Timbila instruments), etc.
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.


Mavila temperament defines a tuning "spectrum," similarly to the meantone spectrum. The fifth of 7-EDO (~686 cents) is often thought of as an informal dividing line between meantone and mavila temperament: 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 9-EDO is also often thought of as the other tuning endpoint on the mavila spectrum.
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.


== Regular Temperament Theory ==
=== 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}}
|}


See [[Pelogic_family#Mavila|Pelogic family]].
{| 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}}
|}


== MOS tree ==
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 675.456{{c}}


In addition to the 7-note anti-diatonic scale described, Mavila also has a 9 note "superdiatonic" MOS, the "super-Ionian" mode of which looks LLLsLLLLs. This is the basis for the [Armodue_theory Armodue theory].
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Comments
|-
| 1\2
|
| 600.000
| Lower bound of 5-odd-limit diamond monotone
|-
|
| 11/8
| 648.682
|
|-
| 6\11
|
| 654.545
|
|-
|
| 15/8
| 655.866
| 1/2 comma
|-
|
| 15/11
| 663.049
|
|-
| 5\9
|
| 666.667
|
|-
|
| 5/4
| 671.229
| 1/3 comma
|-
| 9\16
|
| 675.000
|
|-
|
| 11/6
| 675.319
|
|-
|
| 25/24
| 675.618
| 2/7 comma
|-
|
| ''f''<sup>4</sup> + ''f''<sup>3</sup> - 8 = 0
| 676.337
| 1–3–5 equal-beating tuning, Erv Wilson's meta-mavila
|-
| 13\23
|
| 678.261
|
|-
|
| 5/3
| 678.910
| 1/4 comma, 5-odd-limit minimax
|-
|
| 11/10
| 682.502
|
|-
|
| 9/5
| 683.519
| 1/5 comma, 5-limit 9-odd-limit minimax
|-
|
| 11/9
| 684.197
|
|-
| 4\7
|
| 685.714
| 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)
|-
|
| 3/2
| 701.955
| Pythagorean tuning
|}
<nowiki/>* Besides the octave


Mavila generates a 16 tone chromatic MOS. In a certain sense, much of mavila makes sense if viewed within the lens of a 16-tone chromatic gamut, similarly to how much of meantone is thought of in the setting of a 12-tone chromatic gamut.
== Music ==
; [[City of the Asleep]]
* from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape (Reissue)''] (2007)
** [https://cityoftheasleep.bandcamp.com/track/illegible-red-ink ''Illegible Red Ink''] – in 16edo tuning
** [https://cityoftheasleep.bandcamp.com/track/run-run-red-robot ''Run Run Red Robot''] – in 9edo tuning


After the 16 tone chromatic scale is the 23 tone enharmonic MOS, which can be thought of as an "extended mavila" analogous to the "extended meantone" 19-tone enharmonic scale. If the mavila fifth is flatter than that of 16-EDO (675 cents), it will instead generate an MOS at 25 notes. This is similar to how if the meantone fifth is tuned sharper than 12-EDO, it will instead generate a 17-tone MOS rather than a 19-tone one.
; [[Kraig Grady]]
* [https://anaphoria.bandcamp.com/track/court-music-of-the-mesa-fifth-transcription ''Court Music of the Mesa (fifth-transcription)''] (1989) – first piece written in meta-mavila
* [https://anaphoria.bandcamp.com/track/our-rainy-season ''Our Rainy Season''] (2011)


== Tuning ==
; [[groundfault]]
* [https://www.youtube.com/watch?v=bgnGZCQr5yE ''Wallowing in Madness''] (2020) – in 16edo tuning


Much like meantone temperament, mavila is supported by several low-numbered EDOs, which will basically be the same size as the MOS's listed above.
; [[K9assassin]]
* [https://en.xen.wiki/w/File:Mavila_Jazz_Rhodes_1.mp3 ''Mavila Jazz Rhodes 1''] (archived 2014)


7-EDO can be thought of as a primitive tuning, yielding a totally equal heptatonic scale that is equally diatonic and anti-diatonic.
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/kosma-jumis-lul ''Kôsma jumiś lul''] (2017)


The next EDO supporting Mavila is 9-EDO, which 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 fifth is 667 cents.
; [[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)


It is also supported by 16-EDO, 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 12-EDO 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 fifth is 675 cents.
; [[Sevish]]
* from ''Sean but not Heard'' (2012)
** "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]


The next EDO supporting mavila is 23-EDO, which is the second-most common tuning for mavila temperament, used frequently by Igliashon Jones in his Cryptic Ruse albums. The fifth is 678 cents, and as a result the harmonic properties are slightly better than 16-EDO, although still fairly inharmonic compared to meantone. The anti-diatonic scale is more "quasi-equal" in this tuning than in 16-EDO.
; [[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


25-EDO also supports mavila, although the tuning is 672 cents and hence very flat, even flatter than 16-EDO.
; [[Starshine]]
* [https://soundcloud.com/starshine99/undercity-ft-hatsune-miku ''Undercity'' ft. Hatsune Miku] (2020) – in 16edo tuning


== Musical Examples ==
=== 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 Temperament by using Graham Breed's Lilypond code to tune the generators flat. Musical examples are provided in 9-EDO, 16-EDO, 23-EDO, and 25-EDO, for comparison. Note that the melodic and/or intonational properties differ slightly for each tuning.
* [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]


9-EDO: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-9-edo</soundcloud>
== See also ==
* [[African music]] – contains a discussion about the original tuning that inspired the discovery of this temperament
* [[Pitch Set Harmonization|Pitch set harmonization]] – includes instructions for applying pitch set harmonization to mavila


16-EDO: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-16-edo</soundcloud>
== External links ==
* [https://anaphoria.com/metameantone-mavila.pdf ''Meta Meantone & Meta Mavila''] by Erv Wilson


23-EDO: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments</soundcloud>
== References ==
<references />


25-EDO: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-25-edo</soundcloud>
[[Category:Mavila| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Mavila family]]
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