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.
Inverted Major and Minor Intervals: The Anti-Diatonic Scale
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.)
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.
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.
Mavila Modal Harmony
An exploration of Mavila Modal Harmony can be found in Mavila_Temperament_Modal_Harmony.
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.
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.
Regular Temperament Theory
See Pelogic family.
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].
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.
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.
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.
7-EDO can be thought of as a primitive tuning, yielding a totally equal heptatonic scale that is equally diatonic and anti-diatonic.
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.
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. The fifth is 675 cents.
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.
25-EDO also supports mavila, although the tuning is 672 cents and hence very flat, even flatter than 16-EDO.
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.