ED5: Difference between revisions
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The 5th harmonic, quintuple, or pentave, is particularly wide as far as [[equivalence]]s go, as there are at absolute most about 4.8 instances of the 5th harmonic within the [[human hearing range]]. If one does indeed deal with equivalence of the 5th harmonic, this range restriction is a crucial consideration. | The 5th harmonic, quintuple, or pentave, is particularly wide as far as [[equivalence]]s go, as there are at absolute most about 4.8 instances of the 5th harmonic within the [[human hearing range]]. If one does indeed deal with equivalence of the 5th harmonic, this range restriction is a crucial consideration. | ||
One | One of the ways to arrive at equal divisions of the 5th harmonic would be to take another step up in exoticness from [[EDT|equal divisions of the tritave]] such as [[Bohlen-Pierce]]–the usual approach to EDTs is to eliminate the 2nd harmonic, thus you could view ED5s as removing both the 2nd and 3rd harmonics, leaving the next lowest prime, 5, as the simplest available equave. This creates a system based exclusively on [https://oeis.org/wiki/5-rough_numbers 5-rough numbers] (numbers not divisible by 2 or 3). In this case, the [[5.7.11 subgroup]] and 5:7:11:(25) chord would be analogous to the [[5-limit]] / 4:5:6:(8) chord for octaves and 3.5.7 subgroup / 3:5:7:(9) chord for tritaves. But it is of even more questionable practicality than tritave-equivalent music, due to the aforementioned range restriction caused by the 5th harmonic, as well as the higher complexity of intervals making consonance harder. A possible alternative to solve at least the former problem would be to use [[Ed5/2|5/2-equivalence]] and the [[5/2.7/2.11/2 subgroup]] instead, which would have the same chord 5:7:11 (but adding 5/2 to the chord makes it the more complex 10:14:22:25). | ||
The quintessential example of a 5th-harmonic based tuning is [[hyperpyth]] (see [[17ed5]]). However, perhaps the more common reason to use these systems is in approximation with lower harmonic factors than 5. This approach is highlighted by Hieronymus ([[20ed5]]) which itself is a zeta peak tuning (not "no- | The quintessential example of a 5th-harmonic based tuning is [[hyperpyth]] (see [[17ed5]]). However, perhaps the more common reason to use these systems is in approximation with lower harmonic factors than 5. This approach is highlighted by Hieronymus ([[20ed5]]) which itself is a zeta peak tuning (not "no-2's", full on zeta). | ||
[[96ed5]] is noticeable for its extremely high accuracy in the 8.9.5.7.11.13.17.23 subgroup, while still being reasonably sized. | |||
== As generator chains for temperaments == | == As generator chains for temperaments == | ||
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; 100 and beyond | ; 100 and beyond | ||
* [[116ed5|116]], [[139ed5|139]], [[256ed5|256]] | * [[116ed5|116]], [[139ed5|139]], [[175ed5|175]], [[256ed5|256]] | ||
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* [http://www.nonoctave.com/tuning/fifth_harmonic.html| Nonoctave.com: tuning: equal division of the fifth harmonic] | * [http://www.nonoctave.com/tuning/fifth_harmonic.html| Nonoctave.com: tuning: equal division of the fifth harmonic] | ||
[[Category:Ed5| ]] <!-- main article --> | [[Category:Ed5's| ]] | ||
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[[Category:Lists of scales]] | [[Category:Lists of scales]] | ||
[[Category:Pentave]] | [[Category:Pentave]] | ||
{{Todo|add sound example}} | {{Todo|add sound example}} | ||