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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. | |||
=== Properties === | |||
Division of 9/4 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed9/4 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy. | |||
The structural utility of 9/4 or another major ninth 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. | |||
==== Joseph Ruhf's ed9/4 theory ==== | |||
{{idiosyncratic terms}} | |||
In ed9/4 systems, 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. | |||
One way to approach some ed9/4 tunings 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 superficially similar to certain versions of the regularly tempered approximate ("full"-status) [[A shruti list|shrutis]]. [[Joseph Ruhf]] proposes the name "macroshrutis" for this reason. | |||
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 === | |||
{| class="wikitable center-all" | |||
|+ style=white-space:nowrap | 1…99 | |||
| [[1ed9/4|1]] | |||
| [[3ed9/4|3]] | |||
| [[5ed9/4|5]] | |||
| [[7ed9/4|7]] | |||
| [[9ed9/4|9]] | |||
| [[11ed9/4|11]] | |||
| [[13ed9/4|13]] | |||
| [[15ed9/4|15]] | |||
| [[17ed9/4|17]] | |||
| [[19ed9/4|19]] | |||
|- | |||
| [[21ed9/4|21]] | |||
| [[23ed9/4|23]] | |||
| [[25ed9/4|25]] | |||
| [[27ed9/4|27]] | |||
| [[29ed9/4|29]] | |||
| [[31ed9/4|31]] | |||
| [[33ed9/4|33]] | |||
| [[35ed9/4|35]] | |||
| [[37ed9/4|37]] | |||
| [[39ed9/4|39]] | |||
|- | |||
| [[41ed9/4|41]] | |||
| [[43ed9/4|43]] | |||
| [[45ed9/4|45]] | |||
| [[47ed9/4|47]] | |||
| [[49ed9/4|49]] | |||
| [[51ed9/4|51]] | |||
| [[53ed9/4|53]] | |||
| [[55ed9/4|55]] | |||
| [[57ed9/4|57]] | |||
| [[59ed9/4|59]] | |||
|- | |||
| [[61ed9/4|61]] | |||
| [[63ed9/4|63]] | |||
| [[65ed9/4|65]] | |||
| [[67ed9/4|67]] | |||
| [[69ed9/4|69]] | |||
| [[71ed9/4|71]] | |||
| [[73ed9/4|73]] | |||
| [[75ed9/4|75]] | |||
| [[77ed9/4|77]] | |||
| [[79ed9/4|79]] | |||
|- | |||
| [[81ed9/4|81]] | |||
| [[83ed9/4|83]] | |||
| [[85ed9/4|85]] | |||
| [[87ed9/4|87]] | |||
| [[89ed9/4|89]] | |||
| [[91ed9/4|91]] | |||
| [[93ed9/4|93]] | |||
| [[95ed9/4|95]] | |||
| [[97ed9/4|97]] | |||
| [[99ed9/4|99]] | |||
|} | |||
[[Category:Ed9/4| ]] <!-- main article --> | |||
[[Category:Equal-step tuning]] | |||
[[Category:Edonoi]] | |||