Mavila temperament modal harmony

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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.

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