Ed9/4
The equal division of 9/4 (ed9/4) is a tuning obtained by dividing the Pythagorean ninth (9/4) in a certain number of equal steps. An ed9/4 can be generated by taking every other tone of an edf, so even-numbered ed9/4's are integer edfs.
Properties
Division of 9/4 into equal parts does not necessarily imply directly using this interval as an equivalence. The question of equivalence has not even been posed yet. The utility of 9/4 or another major 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 fourth optionally with a period equivalent to three or six macrotones, in particular ones at least as wide as 101.083 cents) takes 1-3-6, which is not equivalent to a tone cluster as it would be in an edf tuning, as the root position of its regular triad. Many, though not all, of these scales have a false octave, with various degrees of accuracy.
Incidentally, one way to treat 9/4 as an equivalence is the use of the 5:6:8 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 four 9/8 to get to 8/5 (tempering out the schisma). So, doing this yields 6-, 8-, 14- and 20- or 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 has not been named yet.
The branches of the Napoli family are named thus:
5&3: Grandfather
Bipentachordal:
- 4&4: Macrodiminshed
- 6&2: Macroshrutis
The temperament family in the Neapolitan temperament area which has an interlaced enneatonic scale is named for parts of Maryland further west of the Middletown Valley as its generator rises:
3&6: South Mountain Scale
4&5: Hagerstown (particularly in ~9/4)
2&7: Allegany
The temperament family in the Neapolitan temperament area which has an octatonic scale of seven generators and a remainder is named Fujiyama (i. e. the volcano viewable from practically anywhere in Japan due to the Japanese archipelago consisting of such flat islands).
Surprisingly, though sort of obviously, due to 9/4 being the primary attractor for Neapolitan temperaments, the golden and pyrite tunings of edIXs must be forced to turn out to divide a (nearly) pure 9:4 (in particular, using Aeolian mode gives the 2/7-comma meantone major ninth as almost exactly the pyrite tuning of the period, or (8φ+6)/(7φ+5).
Individual pages for ed9/4's
| 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 |
| 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 |
| 41 | 43 | 45 | 47 | 49 | 51 | 53 | 55 | 57 | 59 |
| 61 | 63 | 65 | 67 | 69 | 71 | 73 | 75 | 77 | 79 |
| 81 | 83 | 85 | 87 | 89 | 91 | 93 | 95 | 97 | 99 |