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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.  
An '''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.  


=== Properties ===
== 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.  
=== Relation to edfs ===
An ed9/4 can be generated by taking every other tone of an [[edf]], so even-numbered ed9/4's are integer edfs.  


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.
This is the primary use for ed9/4's – to get the same benefits of a particular edf, without having to juggle such a large number of notes per [[period]]. This is a similar principle to using an [[ed4]] in place of a very large [[edo]].


==== Joseph Ruhf's ed9/4 theory ====
Perhaps a composer wanting to explore ''N''-edf but daunted by the number of notes, could instead simply use ''N''-ed9/4. Otherwise, they could also compose for two instruments, both tuned to ''N''-ed9/4, but each tuned one step of ''N''-edf apart, making the piece overall in ''N''-edf, but each individual instrument ''N''-ed9/4. This is a similar strategy to how some composers have approached [[24edo]] – using two [[12edo]] instruments tuned a 24edo-step apart.
{{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.
=== Relation to common practice ===
9/4 or another major ninth is a standard replacement for the [[root]] in jazz piano voicings. Perhaps, then, a composer could approach the period of an ed9/4 not as an [[equivalence]], but as a skeleton for chords to be built out of – potentially encouraging an approach that focuses more on individual chords than an overall scale.


The branches of the Napoli family are named thus:
=== Equivalence ===
Few would argue that 9/4 itself could be heard as an equivalence. Some might argue that some degree of 3/2-equivalence may be possible in a scale which has no 2/1, 3/1, or 4/1, though that claim is controversial. If that is the case though, then perhaps in a similar situation, 9/4 may have some form of faint equivalence as it might sound like two periods of 3/2. This is usually not really the point of using ed9/4 though as discussed above.


5&3: Grandfather
== Important ed9/4's ==
=== 5ed9/4 ===
Every other step of [[5edf]]. Completely misses [[3/2]], [[2/1]], [[3/1]] and [[4/1]], but aproximates [[5/1]], [[6/1]] and [[7/1]] well for its size. This makes it still posess useful consonances, but with no strong [[equivalence interval]] to pull the listener's ear – making it a prime candidate for perceptual 9/4-equivalence, if such a thing is even possible.


Bipentachordal:
Its intervals are:
* 281{{c}} ([[7/6]], [[6/5]])
* 562{{c}} ([[11/8]], [[7/5]])
* 842{{c}} ([[phi]], [[13/8]])
* 1123{{c}} ([[21/11]])
* 1404{{c}} ([[9/4]])
{{Harmonics in equal|5|9|4|intervals=prime}}


* 4&4: Macrodiminshed
=== 9ed9/4 ===
* 6&2: Macroshrutis
{{Main| 9ed9/4 }}


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:
Every other step of [[9edf]] (almost exactly every other step of [[Carlos Alpha]]).


3&6: South Mountain Scale
=== 11ed9/4 ===
{{Main| 11ed9/4 }}


4&5: Hagerstown (particularly in ~9/4)
Every other step of [[11edf]] (almost exactly every other step of [[Carlos Beta]]).


2&7: Allegany
=== 29ed9/4 ===
Every other step of [[29edf]]. A compromise between [[39edt]] (triple Bohlen-Pierce) and [[25edo]], combining the benefits and drawbacks of both systems while being audibly different from either.
{{Harmonics in equal|29|9|4|intervals=prime}}


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).
=== 43ed9/4 ===
Every other step of [[43edf]]. The smallest ed9/4 with a truly great approximation of full [[11-limit]] JI (let alone all the way up to the full [[23-limit]]).


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).
A compromise between [[58edt]] and [[37edo]], combining the benefits and drawbacks of both systems while being audibly different from either.
{{Harmonics in equal|43|9|4|intervals=prime}}


=== Individual pages for ed9/4's ===
=== 45ed9/4 ===
Every other step of [[45edf]]. Very similar to [[61edt]] but improves on its approximations of [[JI]], with slightly better approximations of primes 2, 5, 7, 11, 13, 17, 19 ''and'' 23 compared to 61edt.


Compared to the nearest edo ([[38edo]]), it has a much worse prime 2 or course, but it has dramatically better primes 3, 7 and 11, and slightly better 13, 17, 19, 23 and even 29 compared to 38edo. It does however have a worse 5/1, but only slightly.
{{Harmonics in equal|45|9|4|intervals=prime}}
=== 47ed9/4 ===
Every other step of [[47edf]]. A compromise between [[64edt]] and [[40edo]], combining the benefits and drawbacks of both systems while being audibly different from either.
{{Harmonics in equal|47|9|4|intervals=prime}}
=== 57ed9/4 ===
Every other step of [[57edf]]. A compromise between [[77edt]] and [[49edo]], combining the benefits and drawbacks of both systems while being audibly different from either.
{{Harmonics in equal|57|9|4|intervals=prime}}
== Individual pages for ed9/4's ==
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[[Category:Ed9/4| ]] <!-- main article -->
== See also ==
[[Category:Equal-step tuning]]
* [[User:Moremajorthanmajor/Ruhf's Ed9/4 theory]]
[[Category:Edonoi]]
 
[[Category:Ed9/4's| ]] <!-- main article -->
[[Category:Lists of scales]]
 
{{Todo|inline=1|explain edonoi|text=Most people do not think 9/4 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is.}}
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