|
|
| (28 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| The '''equal division of 9/4''' ('''ed9/4''') is a [[tuning]] obtained by dividing the [[9/4|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.
| | #redirect [[Ed9/4]] |
| | |
| == 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 [[Pseudo-traditional harmonic functions of octatonic scale degrees|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) [[A shruti list|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 ==
| |
| * [[8ed9/4]]
| |
| * [[9ed9/4]]
| |
| * [[13ed9/4]]
| |
| * [[14ed9/4]]
| |
| * [[17ed9/4]]
| |
| | |
| [[Category:Ed9/4| ]] <!-- main article -->
| |
| [[Category:Equal-step tuning]]
| |