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Division of a thirteenth (e. g. 10/3) into n equal parts

Division of e. g. the 10:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of equivalence has not even been posed yet. The utility of 10:3 or another thirteenth as a base though, is apparent by being the top of the upper structure of jazz voicings and the complete ambitus of three, later five, of the church modes (Dorian G below low D to E above high D, Phrygian A below low E to F above high E and Mixolydian C below low G to A above above high G and later Aeolian D below low A to B(b) above high A and Ionian F below low C to D above high C; it is unknown whether a scale on Bb was within the question before the Baroque period). Most importantly, a minor thirteenth is the quadruple of a fourth while a major thirteenth is the triple of a fifth, so diatonic scales will not generate prime edXIIIs though these have 1-3-5-7-10 pentads rather than the tone clusters of an equal division of a (perfect) fourth or fifth. Although they no longer count as equivalent, 2-3 and 4-3 are still as valid suspensions of normal root position pentads as 9-10 and 11-10. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 10/3 as an equivalence is the use of the 4:5:6:7:10 chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Just as in meantone it takes four 3/2 to get to 5/1, tempering out the comma 81/80. So, doing this yields 9, 12, 21 and 33 tone 3MOS. While the notes are rather farther apart, the scheme is uncannily similar to augmented temperament. "Macro-augmented" might be a practically perfect term for it if it hasn't been named yet.

The branches of the Bijou family are named thus:

2&10: Macro-Injera and Macro-Shrutar and Macro-srutal/pajara (Quadrifold Symmetric and Hexachordal Major)

3&9: Macro-augmented (Trifold Symmetric and Pentachordal Major)

4&8: Macro-diminished (Bifold Symmetric and Tetrachordal Major)

5&7: (Contra-alto) Chromatic Major

6&6: Macro-Hexe

10/3 being a major thirteenth, any way to treat it as an equivalence is a member of the Kiriage Mangan family:

(Tetrad and Pentatonic - Mangan Temperament

Hexa- and Heptatonic - Haneman Temperament

Enneatonic plus or minus one - Baiman Temperament

Hen- and dodecatonic - Sanbaiman Temperament)

Triskaidekatonic - Yakuman Temperament List

(1L 12s and 12L 1s - Kazoe Yakuman)

7L 6s and 6L 7s - Daichīsei and Daisharin

9L 4s and 4L 9s - Shōsūshī and Daisūshī

10L 3s and 3L 10s - Shōsangen and Daisangen

5L 8s and 8L 5s - Ryūīsō

2L 11s and 11L 2s - Kokushimusō