Ed7/2

The equal division of 7/2 (ed7/2) is a tuning obtained by dividing the septimal minor fourteenth (7/2) into a number of equal steps.

Properties

Division of 7/2 into equal parts does not necessarily imply directly using this interval as an equivalence. The question of equivalence has not even been posed yet. 7/2 may be an upper limit of what may be useful as a scale period, being the absolute widest imperfect interval comfortably writable on a standard staff.

Due to the above, Joseph Ruhf has named the region of intervals between 17 and 20 degrees of 10edo after the "mangan" system of Riichi Mahjong, creating the Mangan temperament family whose periods are minor fourteenths (e.g. 7/2). The fundamental complete sonority of the tonality of such a scale needs 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.

If one wishes to treat 7/2 as an equivalence, one way 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 scales. While the notes are rather farther apart, the scheme is uncannily similar to orwell. This is the yakuman temperament, named by Joseph Ruhf, that is a kind of macro-orwell.

Proposed names for 7/2-equivalent temperament collections

Joseph Ruhf’s names

• Tetrad and pentatonic - Mangan temperament
• Hexatonic and heptatonic - Haneman temperament
• Enneatonic plus or minus one - Baiman temperament
• Hen- and dodecatonic - Sanbaiman temperament
• Triskaidekatonic - Yakuman temperament

Todo: clarify What do the numbers of notes mean: are they MOS scale sizes? What limit or subgroup does each temperament approximate? What commas does each temperament temper out?

Proposed names for 7/2-equivalent MOS scales

See also: MOS scale.

Joseph Ruhf’s names

• 7L 6s - Daichīsei
• 6L 7s - Daisharin
• 9L 4s - Shōsūshī
• 4L 9s - Daisūshī
• 1L 12s and 12L 1s - Kazoe Yakuman
• 2L 11s and 11L 2s - Kokushimusō
• 5L 8s and 8L 5s - Ryūīsō

Todo: review There probably shouldn’t be instances of two MOSes having the same name. Can we come up with new names for the other one in each of those last three pairs?