Ed9/4: Difference between revisions

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-#redirect [[Ed9/n#Ed9/4]]
+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 ==
+=== 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.
+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]].
+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.
+=== Relation to common practice ===
+/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.
+=== 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.
+== Important ed9/4s ==
+=== 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.
+Every other step of [[5edf]].
+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}}
+=== 9ed9/4 ===
+{{main|9ed9/4}}
+Every other step of [[9edf]] (almost exactly every other step of [[Carlos Alpha]]).
+=== 11ed9/4 ===
+{{main|11ed9/4}}
+Every other step of [[11edf]] (almost exactly every other step of [[Carlos Beta]]).
+=== 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.
+Every other step of [[29edf]].
+{{Harmonics in equal|29|9|4|intervals=prime}}
+=== 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]]).
+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 ==
+{| 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]]
+|}
+== See also ==
+* [[User:Moremajorthanmajor/Ruhf's Ed9/4 theory]]
+[[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.}}