# EdXIV

### Division of a fourteenth (e.g. 7/2) into n equal parts

Division of e. g. the 7:2 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. That e. g. the 7:2 is just about an upper limit of what may be useful as a base is apparent by being the absolute widest imperfect interval comfortably writable on a standard staff (which is why I have named the region of intervals between 17 and 20 degrees of 10edo after the "mangan" system of Riichi Mahjong, the proper Mangan temperament family being based on minor fourteenths) and by the fundamental complete sonority of the tonality of such a scale needing more notes than a person has fingers on one hand. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 7/2 as an equivalence is the use of the 3:4:5:6:7:8 chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes two 4/3 to get to the octave, (tempering out the comma 64/63). So, doing this yields 9, 13, 22 and 31 note MOS. While the notes are rather farther apart, the scheme is uncannily similar to Orwell, making the temperament the "Yakuman" that is Macro-Orwell:

(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