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EdIII means Division of a third interval into n equal parts.

Division of a third (e. g. 9/7, 5/4, 11/9 or 6/5) into n equal parts

Division of e. g. the 9:7 or the 5:4 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 9:7, 5:4, 6:5 or another third as a base though, is apparent by being used at the base of so much harmony in the European tradition from the Renaissance onwards. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy, but which context(s), if any, it is very perceptually important in is as yet an open question.

Incidentally, one way to treat 9/7, 5/4, 11/9, or 6/5 as an equivalence is the use of the 7:8:(9):10, 8:9:(10):11, 9:10:(11):12, or 10:11:(12):13 chord as the fundamental complete sonority in a very similar way to the 4:5:(6):7 chord in Carlos Beta. Whereas in Carlos Beta it takes three 6/5 to get to 7/4, here it takes three 9/8, 10/9, 11/10, or 12/11 to get to 10/7, 11/8, 4/3, or 13/10 (tempering out the comma 5120/5103, 8019/8000, 4000/3993, or 17303/17280). So, doing this yields 15, 17, and 32 note MOS for ED(9/7)s, 17, 19, and 36 note MOS for ED(5/4)s, 19, 21, and 40 note MOS for ED(11/9)s, or 21, 23, and 44 note MOS for ED(6/5)s. While the notes are rather closer together, the scheme is uncannily similar to meantone.

Individual pages for EDIIIs

Equal Divisions of the …
Septimal Supermajor Third (9/7) Just Major Third (5/4) Tridecimal Neutral Third (16/13) Undecimal Neutral Third (11/9) Just Minor Third (6/5) Septimal Subminor Third (7/6)
2 Square Root of 5/4 Square Root of 6/5
3 Cube Root of 9/7 Cube Root of 5/4 Cube Root of 16/13 Cube Root of 11/9
4 Fourth Root of 5/4 Fourth Root of 6/5
5 Fifth Root of 5/4
6 Sixth Root of 5/4
7 Seventh Root of 5/4
8 Eighth Root of 5/4
9 Ninth Root of 5/4