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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.  
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.  


=== 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 ed9/4s — 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 ed9/4s ==


Bipentachordal:
=== 5ed9/4 ===
Completely misses [[3/2]], [[2/1]], [[3/2]] 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.


* 4&4: Macrodiminshed
Every other step of [[5edf]].
* 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:
Its intervals are:
* 281¢ ([[7/6]], [[6/5]])
* 562¢ ([[11/8]], [[7/5]])
* 842¢ ([[phi]], [[13/8]])
* 1123¢ ([[21/11]])
* 1404¢ ([[9/4]])
{{Harmonics in equal|5|9|4|intervals=integer}}


3&6: South Mountain Scale
=== 9ed9/4 ===
{{main|9ed9/4}}
Every other step of [[9edf]] (almost exactly every other step of [[Carlos Alpha]]).


4&5: Hagerstown (particularly in ~9/4)
=== 11ed9/4 ===
{{main|11ed9/4}}
Every other step of [[11edf]] (almost exactly every other step of [[Carlos Beta]]).


2&7: Allegany
=== 29ed9/4 ===
A compromise between [[39edt]] (triple Bohlen-Pierce) and [[25edo]], combining the benefits and drawbacks of both systems while being audibly different from either.


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).
Every other step of [[29edf]].
{{Harmonics in equal|29|9|4|intervals=prime}}


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).
=== 43ed9/4 ===
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]]).


=== Individual pages for ed9/4's ===
A compromise between [[58edt]] and [[37edo]], combining the benefits and drawbacks of both systems while being audibly different from either.


Every other step of [[43edf]].
{{Harmonics in equal|43|9|4|intervals=prime}}
=== 45ed9/4 ===
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.
Every other step of [[45edf]].
{{Harmonics in equal|45|9|4|intervals=prime}}
=== 47ed9/4 ===
A compromise between [[64edt]] and [[40edo]], combining the benefits and drawbacks of both systems while being audibly different from either.
Every other step of [[47edf]].
{{Harmonics in equal|47|9|4|intervals=prime}}
=== 57ed9/4 ===
A compromise between [[77edt]] and [[49edo]], combining the benefits and drawbacks of both systems while being audibly different from either.
Every other step of [[57edf]].
{{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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