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

Antidiatonic Names

UDP	Imprint	Name	1	2	3	4	5
0	LsLss	Jue	P1	M3	P4	M6	M7
1	LssLs	Yu	P1	M3	P4	P5	M7
2	sLsLs	Shang	P1	m2	P4	P5	M7
3	sLssL	Zhi	P1	m2	P4	P5	m6
4	ssLsL	Gong	P1	m2	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	Name	1	2	3	4	5	6	7
0	LssLsss	Antilocrian	1/1	8/7	5/4	4/3	14/9	5/3	11/6
1	LsssLss	Antiphrygian	1/1	8/7	5/4	4/3	3/2	5/3	11/6
2	sLssLss	Antiaeolian	1/1	12/11	5/4	4/3	3/2	5/3	11/6
3	sLsssLs	Antidorian	1/1	12/11	5/4	4/3	3/2	8/5	11/6
4	ssLssLs	Antimixolydian	1/1	12/11	6/5	4/3	3/2	8/5	11/6
5	ssLsssL	Antiionian	1/1	12/11	6/5	4/3	3/2	8/5	7/4
6	sssLssL	Antilydian	1/1	12/11	6/5	9/7	3/2	8/5	7/4

Antidiatonic Names

UDP	Imprint	Name	1	2	3	4	5	6	7
0	LssLsss	Antilocrian	P1	M2	M3	P4	A5	M6	M7
1	LsssLss	Antiphrygian	P1	M2	M3	P4	P5	M6	M7
2	sLssLss	Antiaeolian	P1	m2	M3	P4	P5	M6	M7
3	sLsssLs	Antidorian	P1	m2	M3	P4	P5	m6	M7
4	ssLssLs	Antimixolydian	P1	m2	m3	P4	P5	m6	M7
5	ssLsssL	Antiionian	P1	m2	m3	P4	P5	m6	m7
6	sssLssL	Antilydian	P1	m2	m3	d4	P5	m6	m7

7L 2s Nonatonic MOS: Superdiatonic

8|0 - LLLLsLLLs - Superlydian
7|1 - LLLsLLLLs - Adlocrian
6|2 - LLLsLLLsL - Adphrygian
5|3 - LLsLLLLsL - Adaeolian
4|4 - LLsLLLsLL - Addorian
3|5 - LsLLLLsLL - Admixolydian
2|6 - LsLLLsLLL - Adionian
1|7 - sLLLLsLLL - Adlydian
0|8 - sLLLsLLLL - Superlocrian

In cyclic permutational order: Adlocrian - Adaeolian - Admixolydian - Adlydian - Superlydian - Adphrygian - Addorian - Adionian - Superlocrian

Transversal

UDP	Imprint	Name	1	2	3	4	5	6	7	8	9
8|0	LLLLsLLLs	Superlydian	1/1	12/11	6/5	9/7	7/5	3/2	8/5	7/4	21/11
7|1	LLLsLLLLs	Adlocrian	1/1	12/11	6/5	9/7	4/3	3/2	8/5	7/4	21/11
6|2	LLLsLLLsL	Adphrygian	1/1	12/11	6/5	9/7	4/3	3/2	8/5	7/4	11/6
5|3	LLsLLLLsL	Adaeolian	1/1	12/11	6/5	5/4	4/3	3/2	8/5	7/4	11/6
4|4	LLsLLLsLL	Addorian	1/1	12/11	6/5	5/4	4/3	3/2	8/5	5/3	11/6
3|5	LsLLLLsLL	Admixolydian	1/1	12/11	8/7	5/4	4/3	3/2	8/5	5/3	11/6
2|6	LsLLLsLLL	Adionian	1/1	12/11	8/7	5/4	4/3	3/2	14/9	5/3	11/6
1|7	sLLLLsLLL	Adlydian	1/1	22/21	8/7	5/4	4/3	3/2	14/9	5/3	11/6
0|8	sLLLsLLLL	Superlocrian	1/1	22/21	8/7	5/4	4/3	10/7	14/9	5/3	11/6

"Diatonic" Names

UDP	Imprint	Name	1	2	3	4	5	6	7	8	9
8|0	LLLLsLLLs	Superlydian	P1	N2	m3	SM3	A4	P5	m6	(s)m7	(S)M7
7|1	LLLsLLLLs	Adlocrian	P1	N2	m3	SM3	P4	P5	m6	(s)m7	(S)M7
6|2	LLLsLLLsL	Adphrygian	P1	N2	m3	SM3	P4	P5	m6	(s)m7	N7
5|3	LLsLLLLsL	Adaeolian	P1	N2	m3	M3	P4	P5	m6	(s)m7	N7
4|4	LLsLLLsLL	Addorian	P1	N2	m3	M3	P4	P5	m6	M6	N7
3|5	LsLLLLsLL	Admixolydian	P1	N2	(S)M2	M3	P4	P5	m6	M6	N7
2|6	LsLLLsLLL	Adionian	P1	N2	(S)M2	M3	P4	P5	sm6	M6	N7
1|7	sLLLLsLLL	Adlydian	P1	(s)m2	(S)M2	M3	P4	P5	sm6	M6	N7
0|8	sLLLsLLLL	Superlocrian	P1	(s)m2	(S)M2	M3	P4	d5	sm6	M6	N7

Superdiatonic Qualities

UDP	Imprint	Name	1	2	3	4	5	6	7	8	9
8|0	LLLLsLLLs	Superlydian	P	M	M	M	A	P	M	M	M
7|1	LLLsLLLLs	Adlocrian	P	M	M	M	P	P	M	M	M
6|2	LLLsLLLsL	Adphrygian	P	M	M	M	P	P	M	M	m
5|3	LLsLLLLsL	Adaeolian	P	M	M	m	P	P	M	M	m
4|4	LLsLLLsLL	Addorian	P	M	M	m	P	P	M	m	m
3|5	LsLLLLsLL	Admixolydian	P	M	m	m	P	P	M	m	m
2|6	LsLLLsLLL	Adionian	P	M	m	m	P	P	m	m	m
1|7	sLLLLsLLL	Adlydian	P	m	m	m	P	P	m	m	m
0|8	sLLLsLLLL	Superlocrian	P	m	m	m	P	d	m	m	m

Notes on Naming

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. 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.