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| 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]] |
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| == Properties ==
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| 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.
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| 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.
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| The branches of the Napoli family are named thus:
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| 5&3: Grandfather
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| Bipentachordal:
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| * 4&4: Macrodiminshed
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| * 6&2: Macroshrutis
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| 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:
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| 3&6: South Mountain Scale
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| 4&5: Hagerstown (particularly in ~9/4)
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| 2&7: Allegany
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| 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).
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| 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).
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| == Individual pages for ed9/4's ==
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| {| class="wikitable center-all"
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| |+ style=white-space:nowrap | 1…99
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| | [[1ed9/4|1]]
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| | [[3ed9/4|3]]
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| | [[5ed9/4|5]]
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| | [[7ed9/4|7]]
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| | [[9ed9/4|9]]
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| | [[11ed9/4|11]]
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| | [[13ed9/4|13]]
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| | [[15ed9/4|15]]
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| | [[17ed9/4|17]]
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| | [[19ed9/4|19]]
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| |-
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| | [[21ed9/4|21]]
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| | [[23ed9/4|23]]
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| | [[25ed9/4|25]]
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| | [[27ed9/4|27]]
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| | [[29ed9/4|29]]
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| | [[31ed9/4|31]]
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| | [[33ed9/4|33]]
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| | [[35ed9/4|35]]
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| | [[37ed9/4|37]]
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| | [[39ed9/4|39]]
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| |-
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| | [[41ed9/4|41]]
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| | [[43ed9/4|43]]
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| | [[45ed9/4|45]]
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| | [[47ed9/4|47]]
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| | [[49ed9/4|49]]
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| | [[51ed9/4|51]]
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| | [[53ed9/4|53]]
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| | [[55ed9/4|55]]
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| | [[57ed9/4|57]]
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| | [[59ed9/4|59]]
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| |-
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| | [[61ed9/4|61]]
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| | [[63ed9/4|63]]
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| | [[65ed9/4|65]]
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| | [[67ed9/4|67]]
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| | [[69ed9/4|69]]
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| | [[71ed9/4|71]]
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| | [[73ed9/4|73]]
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| | [[75ed9/4|75]]
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| | [[77ed9/4|77]]
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| | [[79ed9/4|79]]
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| |-
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| | [[81ed9/4|81]]
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| | [[83ed9/4|83]]
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| | [[85ed9/4|85]]
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| | [[87ed9/4|87]]
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| | [[89ed9/4|89]]
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| | [[91ed9/4|91]]
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| | [[93ed9/4|93]]
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| | [[95ed9/4|95]]
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| | [[97ed9/4|97]]
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| | [[99ed9/4|99]]
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| |}
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| [[Category:Ed9/4| ]] <!-- main article -->
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| [[Category:Edonoi]]
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| [[Category:Lists of scales]]
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