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 - 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

Transversal

UDP Imprint (Patet) Name 1 2 3 4 5
0 LsLss deng 1/1 5/4 4/3 5/3 11/6
1 LssLs dang 1/1 5/4 4/3 3/2 11/6
2 sLsLs dong 1/1 12/11 4/3 3/2 11/6
3 sLssL dung 1/1 12/11 4/3 3/2 8/5
4 ssLsL ding 1/1 12/11 6/5 3/2 8/5

"Diatonic" Names

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

Patet Qualities

UDP Imprint (Patet) Name 1 2 3 4 5
4|0 LsLss deng P M P A M
3|1 LssLs dang P M P P M
2|2 sLsLs dong P m P P M
1|3 sLssL dung P m P P m
0|4 ssLsL ding P m d P m

2L 5s Heptatonic MOS: Antidiatonic

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

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.

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