7L 2s
| ↖ 6L 1s | ↑ 7L 1s | 8L 1s ↗ |
| ← 6L 2s | 7L 2s | 8L 2s → |
| ↙ 6L 3s | ↓ 7L 3s | 8L 3s ↘ |
┌╥╥╥╥┬╥╥╥┬┐ │║║║║│║║║││ │││││││││││ └┴┴┴┴┴┴┴┴┴┘
sLLLsLLLL
The 7L 2s scale step pattern refers to a MOS scale with 7 large and 2 small steps, one mode of which is LLLsLLLLs.
Any interval between 4\7 (four degrees of 7edo = 685.71¢) to 5\9 (five degrees of 9edo = 666.67¢) will generate this scale; the generator turns out to be the "large sixth" within the MOS. In the case of 9edo, L and s are the same size; in the case of 7edo, s becomes so small it disappears (and all that remains are the seven equal L's); the 7L 2s spectrum is between these two extremes.
This scale is sometimes referred to as the superdiatonic scale, which is also the TAMNAMS name for the scale. It is also notable as being one of the MOS's produced by near-optimal tunings for mavila temperament.
The same generator also generates the 2L 5s "anti-diatonic" MOS, which Erv Wilson first studied as being very similar to the tuning of the traditional Timbila music of the Chopi tribe in Mozambique (and from which the name "Mavila" is derived); the superdiatonic scale can be thought of as adding two additional pitches to the anti-diatonic MOS. This generator also generates the 2L 3s MOS, albeit the "hard" tunings in which L/s > 3; these share some resemblance to the "pelog" scale of Gamelan music.
Introduction
In the superdiatonic scale, the fifths 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 *minor* third, whereas stacking three fourths and octave-reducing now gets you a *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 superdiatonic 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.
Notation
In this article we use JKLMNOPQRJ = the symmetric Olympian mode (LLSLLLSLL), J = 261.6255653. &/@ = raise/lower by one chroma. So the fifth chain becomes ... P@ L@ Q M R N J O K P L Q& M& ...
A possible alternative (incompatible with diamond MOS notation) is to keep the 7-note fifth chain FCGDAEB from diatonic and extend it to a 9-note one, YFCGDAEBX, and to use #/b for the 7L 2s chroma. So the extended chain becomes ... Bb Xb Y F C G D A E B X Y# F# ...
Tunings
Much like 5L 2s diatonic, the superdiatonic scale is supported by several low-numbered EDOs, which will basically be the same size as the MOS's listed above.
7edo can be thought of as a degenerate tuning, yielding a totally equal heptatonic scale that is equally diatonic and anti-diatonic.
The next EDO supporting Mavila is 9edo, which can be thought of as the first mavila EDO (and the first EDO in general) differentiating between major and 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 16edo, which is probably the most common tuning for mavila. 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 fifth is 675 cents.
The next EDO supporting mavila is 23edo, which is the second-most common tuning for mavila, 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 16edo, although still fairly inharmonic compared to meantone. The anti-diatonic scale is more "quasi-equal" in this tuning than in 16edo.
25edo also contains mavila, although the tuning is 672 cents and hence very flat, even flatter than 16edo. The major third is 384¢, close to a just 5/4 and the minor third is 288¢, close to a just 13/11.
Mavila defines a tuning "spectrum", similarly to the meantone spectrum. The fifth of 7edo (~686 cents) is often thought of as a dividing line between diatonic and mavila: if the fifth is flatter than this, it will generate anti-diatonic scales but not 7L 2s superdiatonic scales (it will generate 2L 7s instead), and if it is sharper than this, it will generate diatonic scales. The fifth of 9edo is also often thought of as the other tuning endpoint on the mavila spectrum.
Intervals
Modes
- See also Mavila modal harmony
From brightest to darkest, the superdiatonic modes are:
| LLLLsLLLs - (Super)Lydian | |
| LLLsLLLLs - (Super)Ionian | |
| LLLsLLLsL - (Super)Mixolydian | |
| LLsLLLLsL - (Super)Corinthian | |
| LLsLLLsLL - (Super)Olympian | |
| LsLLLLsLL - (Super)Dorian | |
| LsLLLsLLL - (Super)Aeolian | |
| sLLLLsLLL - (Super)Phrygian | |
| sLLLsLLLL - (Super)Locrian |
The modes of the antidiatonic scale are simply named after the existing diatonic scale modes, but with the "Anti-" prefix (e.g. Anti-ionian, Anti-aeolian, etc). The modes of the superdiatonic scale are also named after the existing modes, but contain the "Super" prefix (e.g. Superionian, Superaeolian, etc.). The "anti" and "super" prefixes can be left in to explicitly distinguish which MOS's modes you're talking about, or can be omitted for convention.
Each superdiatonic mode contains its corresponding mode in the antidiatonic scale. Additionally, the superdiatonic modes also resemble the "shape" of their meantone diatonic counterparts. This leads to a pattern: LLLsLLLLs both resembles the meantone LLsLLLs Ionian mode, and contains the mavila ssLsssL anti-Ionian mode as well. Additionally, sLLLsLLLL resembles the diatonic sLLsLLL Locrian mode, and also contains the mavila LssLsss anti-Ionian mode. Furthermore, every mavila mode contains a tonic triad of the -opposite- quality as the corresponding diatonic mode, so that Superionian and Anti-ionian contain a minor triad, and Superphrygian and Antiphrygian contain a major triad.
Since there are only seven diatonic modes, two of the superdiatonic modes need additional names and cannot reference any mode of the diatonic scale. These two modes present themselves as "mixed" modes, which begin with the LLs tetrachord, and so contain both the ~300 cent minor third and the ~375 cent major third (and hence both minor and major triads). These are the only two modes to exhibit this behavior. They're interspersed on the rotational continuum between Ionian and Dorian, and Mixolydian and Aeolian.
As were the original modes named after regions of ancient Greece, so are these new superdiatonic extensions. The one between Ionian and Dorian is called Corinthian, after the Greek island of Corinth, set up so that the Ionian -> Corinthian -> Dorian cyclic sequence will resemble the columns of ancient Greek architecture. The mode cyclically placed between Mixolydian and Aeolian, which is the symmetrical LLsLLLsLL scale, has a number of noteworthy theoretical properties, in that it contains every modal rotation of mavila[5], and hence the entire mavila-tempered 5-limit tonality diamond; it was given the name of Olympian to match its unique status in this regard.
Table of modes
Chords and extended harmony
Primodal theory
Neji versions of superdiatonic modes
- 40:48:52:54:59:64:70:77:80 Pental Superionian
16nejis
23nejis
25nejis
Scale tree
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.
| Generator | Generator size (cents) | L/s | Comments | ||||
|---|---|---|---|---|---|---|---|
| 4\7 | 685.714 | 1/0 | |||||
| 21\37 | 681.08 | 5/1 | |||||
| 17\30 | 680 | 4/1 | |||||
| 13\23 | 678.261 | 3/1 | |||||
| 22\39 | 676.923 | 5/2 | |||||
| 9\16 | 675 | 2/1 | Boundary of propriety; smaller generators are strictly proper | ||||
| 23\41 | 673.171 | 5/3 | |||||
| 672.85 | φ/1 | Golden mavila | |||||
| 14\25 | 672 | 3/2 | |||||
| 19\34 | 670.588 | 4/3 | |||||
| 24\43 | 669.767 | 5/4 | |||||
| 5\9 | 666.667 | 1/1 | |||||