Maqamic

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Commas: 81/80, 36/35, 121/120

Maqamic temperament is a linear mimicry of maqam music within the regular mapping paradigm, much as pelogic temperament is a linear mimicry of the gamelan pelog scale. It is the temperament that you get if you take, at face value, the simplest harmonic mapping consistent with the scalar structure of the maqamat. It deliberately ignores the issue of whether or not those dyads are resolved by the auditory system as such when played melodically (as maqam music is generally a melodic art form), intending instead to serve as a vehicle for the adventurous microtonalist who wants to explore a harmonic context for maqam music.

Melodically, it is a "linearized" version of the maqam modal system; it elevates the notion of "the diatonic scale with quartertones" out of the realm of 24-equal into a higher-dimensional rank 2 temperament. If the maqamat are taken to be rank 2, they all end up being MODMOS's of the proper 3L4s MOS. This MOS takes a neutral third as a generator, and the usual 5L2s diatonic scale is itself a MODMOS. The "quarter-tone" chroma in this setup is exactly half of the usual chromatic semitone from 5L2s (regardless of how it's intonated).

Harmonically, intervals in this temperament are mapped to reflect some of the intonational choices that actual arabic maqam musicians make when playing maqam music on a fretless instrument such as an oud. Sometimes notes that share the same notational representation are intoned very differently under different circumstances; for example, while the fourths are generally intoned very close to 4/3, it's common to adaptively intone the minor 7th closer to 7/4. When this happens, the two notes are said to share a tempered equivalence class, and the comma between the two versions of the note is said to vanish, no matter how large it is. Real-life intonational differences existing within these equivalence classes can then be viewed as a form of adaptive intonation within this temperament, paralleling how western string quartets will often intone their major chords to be as close to 4:5:6 as possible, despite that 81/80 vanishes over the larger structure of the music.

This temperament will hence appear less accurate than it really is, due to the fact that it approximates the performance of highly adaptive fretless instruments, and equivalence classes should be understood as reflecting a cognitive grouping of intervals rather than demanding any particular "middle of the road" intonational ideal. For fixed pitch instruments, 17-equal and 24-equal support this temperament about as well as can be done, but it should be noted that this temperament was designed particularly with adaptive intonation in mind.

Like mohajira, maqamic tempers out 121/120 and 81/80; unlike it, it eliminates 36/35 (and hence 64/63) instead of 176/175. Harmonically, maqamic temperament maps two whole tones to 5/4, which is an intonational choice sometimes made by actual Arabic maqam performers, and which indicates that 81/80 vanishes. It also maps two fourths to 7/4, which is likewise an intonational choice often made by maqam performers, although they tend to do this adaptively rather than optimizing their 4/3's around this ideal. Finally, we find the simplest harmonic intervals approximating the neutral second and neutral third, which are 11/10 and 11/9. These, when doubled, yield 6/5 and 3/2, respectively, thus indicating that 121/120 and 243/242 vanish in this tuning.

Other attempts to mimic the structure of maqam music within the regular mapping paradigm may exist; this temperament is meant to be the simplest harmonic template possible that is consistent with the scalar structure of maqam music. For example, if one wanted to consider the two neutral thirds and neutral seconds as being different sizes, that would correspond to some sort of rank-3 temperament, and if one wanted to consider the 7/4 as being a "quarter tone" flat of 9/5 (where the quarter tone is the chroma for the 7-note MOS), that would correspond to Mohajira temperament.

POTE generator: 350.934

Map: [<1 1 0 4 2|, <0 2 8 -4 5|]

Generators: 2, 11/9

EDOs: 7, 10c, 17c, 24d, 31d

13-limit

The 13-limit version of this temperament eliminates 144/143 and hence 169/168 as well; this signifies that the generator could also be taken as (or intoned as) 16/13, and also that the 6/5's, which are also 7/6's, are evenly divided into two equal 13/12's.

Commas: 81/80, 36/35, 121/120, 144/143

POTE generator: 350.816

Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|]

Generators: 2, 11/9

EDOs: 7, 10c, 17c, 24d, 31d