Ed9/4: Difference between revisions
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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. | |||
== 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 === | |||
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. | |||
=== 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.}} | |||