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Division of a ninth (e. g. 9/4) into n equal parts


Division of e. g. the 9:4 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 9:4 or another ninth as a base though, is apparent by being the standard replacement for the root in jazz piano voicings. Also, as a ninth is the double of a fifth, the fifth of normal root position triads will become the common suspension (5-4 or 5-6) of a ninth-based system. However, thirds and sixths are no longer inverses, and thus an octatonic scale (i. e. any of those of the proper Napoli temperament family which are generated by a third or a fourth optionally with a period equivalent to six macrotones, in particular ones at least as wide as 4 degrees of 45edo) takes 1-3-6 as the root position of its regular triad. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 9/4 as an equivalence is the use of the 4:5:6:7:[8]:(9) 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 nine 5/4 to get to the same place as thirteen 7/6 (tempering out the comma |5 -13 9 -13>) or three 7/6 to get between 6/5 and the period of 3/2 (tempering out the comma 1728/1715). So, doing this yields 8 or 10, 12 or 14, and 22 note 2MOS. While the notes are rather farther apart, the scheme is uncannily similar to certain versions of the regularly tempered approximate ("full"-status) shrutis. "Macroshrutis" might be a practically perfect term for it if it hasn't been named yet.

The branches of the Napoli family are named thus:
Bipentachordal:
  • 4&4: Macrodiminshed
  • 6&2: Macroshrutis
5&3: Grandfather
(Difficult to call these names colorful, no? Yet still they are something.)

The temperament family in the Neapolitan temperament area which has an interlaced enneatonic scale is named Fujiyama (i. e. the volcano viewable from practically anywhere in Japan due to the Japanese archipelago consisting of such flat islands).

8edIX
9edIX
13edIX
14edIX
17edIX

Surprisingly, though sort of obviously, the golden tunings of edIXs must be forced to turn out to divide a (nearly) pure 9:4.