Mavila temperament modal harmony

Introduction

Mavila is the regular temperament eliminating 135/128. As a consequence, this means that it will have a generator significantly flat of 3/2 so that 4 of them add up to a minor third. For any decent tuning of mavila, another consequence of this tempering is that 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.
• Unless otherwise specified, all transversals occur in the 2.3.5.7.11 extension of Mavila which tempers out 135/128, 36/35, 45/44, 33/32, called 11-limit Armodue temperament.
• Additionally, "diatonic" names make use of the following abbreviations:
  • SM - Supermajor
  • M - Major
  • N - Neutral
  • m - Minor
  • sm - Subminor

2L 3s Pentatonic MOS: Pentic

4|0 - LsLss - Jue

3|1 - LssLs - Yu

2|2 - sLsLs - Shang

3|1 - sLssL - Zhi

0|4 - ssLsL - Gong

In cyclic permutational order: Gong, Shang, Jue, Zhi, Yu

Transversal

UDP	Imprint	Name	1	2	3	4	5
0	LsLss	Jue	1/1	5/4	4/3	5/3	11/6
1	LssLs	Yu	1/1	5/4	4/3	3/2	11/6
2	sLsLs	Shang	1/1	12/11	4/3	3/2	11/6
3	sLssL	Zhi	1/1	12/11	4/3	3/2	8/5
4	ssLsL	Gong	1/1	12/11	6/5	3/2	8/5

"Diatonic" Names

UDP	Imprint	Name	1	2	3	4	5
0	LsLss	Jue	P1	M3	P4	M6	N7
1	LssLs	Yu	P1	M3	P4	P5	N7
2	sLsLs	Shang	P1	N2	P4	P5	N7
3	sLssL	Zhi	P1	N2	P4	P5	m6
4	ssLsL	Gong	P1	N2	m3	P5	m6

Qualities

UDP	Imprint	Name	1	2	3	4	5
4|0	LsLss	Jue	P	M	P	A	M
3|1	LssLs	Yu	P	M	P	P	M
2|2	sLsLs	Shang	P	m	P	P	M
1|3	sLssL	Zhi	P	m	P	P	m
0|4	ssLsL	Gong	P	m	d	P	m

2L 5s Heptatonic MOS: Antidiatonic

0 - LssLsss - Antilocrian
1 - LsssLss - Antiphrygian ("major")
2 - sLssLss - Antiaeolian
3 - sLsssLs - Antidorian
4 - ssLssLs - Antimixolydian ("minor")
5 - ssLsssL - Antiionian
6 - sssLssL - Antiphrygian

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

Transversal

UDP	Imprint	(Anti)Name	1	2	3	4	5	6	7
0	LssLsss	Locrian	1/1	8/7	5/4	4/3	14/9	5/3	11/6
1	LsssLss	Phrygian	1/1	8/7	5/4	4/3	3/2	5/3	11/6
2	sLssLss	Aeolian	1/1	12/11	5/4	4/3	3/2	5/3	11/6
3	sLsssLs	Dorian	1/1	12/11	5/4	4/3	3/2	8/5	11/6
4	ssLssLs	Mixolydian	1/1	12/11	6/5	4/3	3/2	8/5	11/6
5	ssLsssL	Ionian	1/1	12/11	6/5	4/3	3/2	8/5	7/4
6	sssLssL	Lydian	1/1	12/11	6/5	9/7	3/2	8/5	7/4

"Diatonic" Names

UDP	Imprint	(Anti)Name	1	2	3	4	5	6	7
0	LssLsss	Locrian	P1	(S)M2	M3	P4	sm6, A5	M6	N7, M7
1	LsssLss	Phrygian	P1	(S)M2	M3	P4	P5	M6	N7, M7
2	sLssLss	Aeolian	P1	N2, m2	M3	P4	P5	M6	N7, M7
3	sLsssLs	Dorian	P1	N2, m2	M3	P4	P5	m6	N7, M7
4	ssLssLs	Mixolydian	P1	N2, m2	m3	P4	P5	m6	N7, M7
5	ssLsssL	Ionian	P1	N2, m2	m3	P4	P5	m6	(s)m7
6	sssLssL	Lydian	P1	N2, m2	m3	SM3, d4	P5	m6	(s’m7

7L 2s 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

Transversal

UDP	Imprint	(Super)Name	1	2	3	4	5	6	7	8	9
8|0	LLLLsLLLs	Lydian	1/1	12/11	6/5	9/7	7/5	3/2	8/5	7/4	21/11
7|1	LLLsLLLLs	Ionian	1/1	12/11	6/5	9/7	4/3	3/2	8/5	7/4	21/11
6|2	LLLsLLLsL	Mixolydian	1/1	12/11	6/5	9/7	4/3	3/2	8/5	7/4	11/6
5|3	LLsLLLLsL	Corinthian	1/1	12/11	6/5	5/4	4/3	3/2	8/5	7/4	11/6
4|4	LLsLLLsLL	Olympian	1/1	12/11	6/5	5/4	4/3	3/2	8/5	5/3	11/6
3|5	LsLLLLsLL	Dorian	1/1	12/11	8/7	5/4	4/3	3/2	8/5	5/3	11/6
2|6	LsLLLsLLL	Aeolian	1/1	12/11	8/7	5/4	4/3	3/2	14/9	5/3	11/6
1|7	sLLLLsLLL	Phrygian	1/1	22/21	8/7	5/4	4/3	3/2	14/9	5/3	11/6
0|8	sLLLsLLLL	Locrian	1/1	22/21	8/7	5/4	4/3	10/7	14/9	5/3	11/6

"Diatonic" Names

UDP	Imprint	(Super)Name	1	2	3	4	5	6	7	8	9
8|0	LLLLsLLLs	Lydian	P1	N2	m3	SM3	A4	P5	m6	(s)m7	(S)M7
7|1	LLLsLLLLs	Ionian	P1	N2	m3	SM3	P4	P5	m6	(s)m7	(S)M7
6|2	LLLsLLLsL	Mixolydian	P1	N2	m3	SM3	P4	P5	m6	(s)m7	N7
5|3	LLsLLLLsL	Corinthian	P1	N2	m3	M3	P4	P5	m6	(s)m7	N7
4|4	LLsLLLsLL	Olympian	P1	N2	m3	M3	P4	P5	m6	M6	N7
3|5	LsLLLLsLL	Dorian	P1	N2	(S)M2	M3	P4	P5	m6	M6	N7
2|6	LsLLLsLLL	Aeolian	P1	N2	(S)M2	M3	P4	P5	sm6	M6	N7
1|7	sLLLLsLLL	Phrygian	P1	(s)m2	(S)M2	M3	P4	P5	sm6	M6	N7
0|8	sLLLsLLLL	Locrian	P1	(s)m2	(S)M2	M3	P4	d5	sm6	M6	N7

Superdiatonic Qualities

UDP	Imprint	(Super)Name	1	2	3	4	5	6	7	8	9
8|0	LLLLsLLLs	Lydian	P	M	M	M	A	P	M	M	M
7|1	LLLsLLLLs	Ionian	P	M	M	M	P	P	M	M	M
6|2	LLLsLLLsL	Mixolydian	P	M	M	M	P	P	M	M	m
5|3	LLsLLLLsL	Corinthian	P	M	M	m	P	P	M	M	m
4|4	LLsLLLsLL	Olympian	P	M	M	m	P	P	M	m	m
3|5	LsLLLLsLL	Dorian	P	M	m	m	P	P	M	m	m
2|6	LsLLLsLLL	Aeolian	P	M	m	m	P	P	m	m	m
1|7	sLLLLsLLL	Phrygian	P	m	m	m	P	P	m	m	m
0|8	sLLLsLLLL	Locrian	P	m	m	m	P	d	m	m	m

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-Locrian 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.