Mavila temperament modal harmony

Mavila has a fifth generator significantly flat of a just perfect fifth (between 5\9 = 666.7¢ and 4\7 = 685.7¢) so that 4 of them add up to a minor third. Thus we end up with an "antidiatonic" 7-note MOS, such that the LLsLLLs melodic pattern becomes inverted into the ssLsssL pattern. (see 2L 5s) Additionally, while mavila supports 5-note and 7-note MOS's like meantone, it has an additional 9-note "superdiatonic" MOS scale as well, in which the LLsLLLs melodic pattern becomes extended into the LLLsLLLLs pattern. (see 7L 2s) These correspondences can be harnessed to create an intuitive naming system for the modes of mavila[7] and mavila[9] which reference the names of the existing familiar modes of meantone[7].

How to read these charts

• The 6|0, 5|1, 4|2 markings are an instance of Modal UDP Notation.
• Additionally, "diatonic" names make use of the following abbreviations:
  • SM - Supermajor
  • M - Major
  • N - Neutral
  • m - Minor
  • sm - Subminor

2L3s Pentatonic MOS: Pelogic

4|0 - LsLss - Patet deng

3|1 - LssLs - Patet dang

2|2 - sLsLs - Patet dong

3|1 - sLssL - Patet dung

0|4 - ssLsL - Patet ding

In cyclic permutational order: Patet ding | Patet dong | Patet deng | Patet dung | Patet dang

"Diatonic" interval names

UDP	Pattern	(Patet) Name	1	2	3	4	5
4|0	LsLss	deng	P1	M3	P4	M6	N7
3|1	LssLs	dang	P1	M3	P4	P5	N7
2|2	sLsLs	dong	P1	N2	P4	P5	N7
1|3	sLssL	dung	P1	N2	P4	P5	m6
0|4	ssLsL	ding	P1	N2	m3	P5	m6

2L5s Heptatonic MOS: Antidiatonic

6|0 - LssLsss - (Anti)Locrian
5|1 - LsssLss - (Anti)Phrygian
4|2 - sLssLss - (Anti)Aeolian ("major")
3|3 - sLsssLs - (Anti)Dorian
2|4 - ssLssLs - (Anti)Mixolydian ("harmonic minor")
1|5 - ssLsssL - (Anti)Ionian ("minor")
0|6 - sssLssL - (Anti)Lydian

In cyclic permutational order: Ionian | Dorian | Phrygian | Lydian | Mixolydian | Aeolian | Locrian

"Diatonic" Names

UDP	Pattern	(Anti)Name	1	2	3	4	5	6	7
6|0	LssLsss	Locrian	P1	SM2	M3	P4	sm6	M6	N7
5|1	LsssLss	Phrygian	P1	SM2	M3	P4	P5	M6	N7
4|2	sLssLss	Aeolian	P1	N2	M3	P4	P5	M6	N7
3|3	sLsssLs	Dorian	P1	N2	M3	P4	P5	m6	N7
2|4	ssLssLs	Mixolydian	P1	N2	m3	P4	P5	m6	N7
1|5	ssLsssL	Ionian	P1	N2	m3	P4	P5	m6	sm7
0|6	sssLssL	Lydian	P1	N2	m3	SM3	P5	m6	sm7

7L2s Nonatonic MOS: Superdiatonic

8|0 - LLLLsLLLs - (Super)Lydian
7|1 - LLLsLLLLs - (Super)Ionian
6|2 - LLLsLLLsL - (Super)Mixolydian
5|3 - LLsLLLLsL - (Super)Corinthian
4|4 - LLsLLLsLL - (Super)Olympian
3|5 - LsLLLLsLL - (Super)Dorian
2|6 - LsLLLsLLL - (Super)Aeolian
1|7 - sLLLLsLLL - (Super)Phrygian
0|8 - sLLLsLLLL - (Super)Locrian

In cyclic permutational order: Ionian | Corinthian | Dorian | Phrygian | Lydian | Mixolydian | Olympian | Aeolian | Locrian

"Diatonic" interval names

UDP	Pattern	(Super)Name	1	2	3	4	5	6	7	8	9
8|0	LLLLsLLLs	Lydian	P1	N2	m3	SM3	A4	P5	m6	sm7	SM7
7|1	LLLsLLLLs	Ionian	P1	N2	m3	SM3	P4	P5	m6	sm7	SM7
6|2	LLLsLLLsL	Mixolydian	P1	N2	m3	SM3	P4	P5	m6	sm7	N7
5|3	LLsLLLLsL	Corinthian	P1	N2	m3	M3	P4	P5	m6	sm7	N7
4|4	LLsLLLsLL	Olympian	P1	N2	m3	M3	P4	P5	m6	M6	N7
3|5	LsLLLLsLL	Dorian	P1	N2	SM2	M3	P4	P5	m6	M6	N7
2|6	LsLLLsLLL	Aeolian	P1	N2	SM2	M3	P4	P5	sm6	M6	N7
1|7	sLLLLsLLL	Phrygian	P1	sm2	SM2	M3	P4	P5	sm6	M6	N7
0|8	sLLLsLLLL	Locrian	P1	sm2	SM2	M3	P4	d5	sm6	M6	N7

Notes on Naming

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.