ED5: Difference between revisions

An '''equal division of the 5th harmonic''' ('''ed5''') is a [[tuning]] obtained by dividing the [[5/1|5th harmonic]] in a certain number of [[equal]] steps.  

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 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-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 ==

One reason for taking the ''n''-th root of 5 include finding temperaments like [[orwell]], [[meantone]], and [[thuja]]. This approach can of course be used indiscriminately. The ed5's serve as generator chains for

* [[Hyperpyth]] tuning (e.g. [[17ed5]])

{| class="wikitable center-all"

| [[10ed5|10]]

| [[18ed5|18]]

| [[19ed5|19]]

| [[20ed5|20]]

| [[22ed5|22]]

| [[23ed5|23]]

| [[24ed5|24]]

| [[25ed5|25]]

| [[26ed5|26]]

| [[30ed5|30]]

| [[32ed5|32]]

| [[33ed5|33]]

| [[34ed5|34]]

| [[35ed5|35]]

| [[38ed5|38]]

| [[40ed5|40]]

| [[42ed5|42]]

| [[43ed5|43]]

| [[44ed5|44]]

| [[45ed5|45]]

| [[46ed5|46]]

| [[50ed5|50]]

| [[52ed5|52]]

| [[53ed5|53]]

| [[54ed5|54]]

| [[55ed5|55]]

| [[56ed5|56]]

| [[60ed5|60]]

| [[62ed5|62]]

| [[63ed5|63]]

| [[64ed5|64]]

| [[65ed5|65]]

| [[66ed5|66]]

| [[70ed5|70]]

| [[72ed5|72]]

| [[73ed5|73]]

| [[74ed5|74]]

| [[75ed5|75]]

| [[76ed5|76]]

| [[80ed5|80]]

| [[82ed5|82]]

| [[83ed5|83]]

| [[84ed5|84]]

| [[85ed5|85]]

| [[86ed5|86]]

| [[90ed5|90]]

 

 

| [[105ed5|105]]

| [[106ed5|106]]

| [[107ed5|107]]

| [[108ed5|108]]

 

 

| 7ed5 is 3edo with ~5.9 cent compressed octaves. Equivalently, 3edo is 7ed5 with pentaves stretched by ~13.7 cents. Patent vals match through the 67-limit.

| Very rough correspondence (~38 cent octave stretch), but patent vals agree through the 7-limit.

| [[14ed5]]

| [[6edo]]

| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.

| [[16ed5]]

| [[7edo]]

| Only rough correspondence (~19 cent octave stretch). Patent vals differ in the 7-limit.

| [[21ed5]]

| [[9edo]]

| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 59-limit.

| [[23ed5]]

| [[10edo]]

| 23ed5 is 10edo with ~11 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 11 and 17.

| [[28ed5]]

| [[12edo]]

| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 19-limit, with the exception of 13 (as expected).

| [[30ed5]]

| [[13edo]]

| 30ed5 is 13edo with ~7.4 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 3 (which neither represents well).

| [[35ed5]]

| [[15edo]]

| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 13-limit.

| [[37ed5]]

| [[16edo]]

| 37ed5 is 16edo with ~4.9 cent stretched octaves. Patent vals match through the 23-limit.

| [[42ed5]]

| [[18edo]]

| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 17-limit, with the exception of 11.

| [[44ed5]]

| [[19edo]]

| 44ed5 is 19edo with ~3.2 cent stretched octaves. Patent vals match through the 13-limit.

| [[49ed5]]

| [[21edo]]

| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.

| [[51ed5]]

| [[22edo]]

| 51ed5 is 22edo with ~1.9 cent stretched octaves. Patent vals match through the 19-limit.

| [[56ed5]]

| [[24edo]]

| This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence begins to break down. Patent vals match through the 13-limit, with the exception of 7.

| [[58ed5]]

| [[25edo]]

| 58ed5 is 25edo with ~1.0 cent stretched octaves. Patent vals match through the 73-limit, with the exception of 13.

| [[63ed5]]

| [[27edo]]

| This is a rough correspondence, but the (7n)ed5 ~ (3n)edo sequence is momentarily fixed. Patent vals match through the 13-limit.

| [[65ed5]]

| [[28edo]]

| 65ed5 is 28edo with ~0.26 cent stretched octaves. Patent vals match through the 43-limit.

| [[72ed5]]

| [[31edo]]

| 72ed5 is 31edo with ~0.34 cent compressed octaves. Patent vals match through the 31-limit.

| [[74ed5]]

| [[32edo]]

| Same 4.9 cent octave stretch as 37ed5~16edo. Patent vals match through the 13-limit with the exception of 7 and 11.

| [[79ed5]]

| [[34edo]]

| 79ed5 is 34edo with ~0.83 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7 (which neither represents well).

| [[81ed5]]

| [[35edo]]

| 81ed5 is 35edo with ~4.0 cent stretched octaves. Patent vals match through the 11-limit.

| [[84ed5]]

| [[36edo]]

| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 11-limit, with the exception of 7.

| [[86ed5]]

| [[37edo]]

| 86ed5 is 37edo with ~1.2 cent compressed octaves. Patent vals match through the 19-limit.

| [[88ed5]]

| [[38edo]]

| Same 3.2 cent octave stretch as 44ed5~19edo. Patent vals differ in the 7-limit.

| [[93ed5]]

| [[40edo]]

| 93ed5 is 40edo with ~1.6 cent compressed octaves. Patent vals match through the 7-limit.

| [[95ed5]]

| [[41edo]]

| 95ed5 is 41edo with ~2.5 cent stretched octaves. Patent vals match through the 11-limit.

| [[100ed5]]

| [[43edo]]

| 100ed5 is 43edo with ~1.9 cent compressed octaves. Patent vals match through the 37-limit.

| [[102ed5]]

| [[44edo]]

| Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 23-limit, with the exception of 7.

| [[107ed5]]

| [[46edo]]

| 107ed5 is 46edo with ~2.1 cent compressed octaves. Patent vals match through the 11-limit.

| [[109ed5]]

| [[47edo]]

| 109ed5 is 47edo with ~1.4 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 11.

| [[114ed5]]

| [[49edo]]

| 114ed5 is 49edo with ~2.4 cent compressed octaves. Patent vals match through the 11-limit.

| [[116ed5]]

| [[50edo]]

| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals agree through the 43-limit.

| [[121ed5]]

| [[52edo]]

| 121ed5 is 52edo with ~2.6 cent compressed octaves, but the correspondence is rough. Patent vals match through the 11-limit, with the exception of 3 (which neither represents well).

| [[123ed5]]

| [[53edo]]

| 123ed5 is 53edo with ~0.61 cent stretched octaves. Patent vals match through the 29-limit.

| [[128ed5]]

| [[55edo]]

| 128ed5 is 55edo with ~2.8 cent compressed octaves, but the correspondence is rough. Patent vals differ in the 7-limit.

| [[130ed5]]

| [[56edo]]

| Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 103-limit.

| [[135ed5]]

| [[58edo]]

| 135ed5 is 58edo with ~2.9 cent compressed octaves. Patent vals match through the 13-limit.

| [[137ed5]]

| [[59edo]]

| 137ed5 is 59edo with ~0.05 cent compressed octaves. Patent vals match through the 179-limit.

| [[139ed5]]

| [[60edo]]

| 139ed5 is 60edo with ~2.7 cent stretched octaves. Pantent vals match through the 31-limit with the exception of 11.

| [[144ed5]]

| [[62edo]]

| Same 0.34 cent octave compression as 72ed5~31edo. Patent vals match through the 19-limit, with the exception of 11.

| [[146ed5]]

| [[63edo]]

| 146ed5 is 63edo with ~2.3 cent stretched octaves. Patent vals match through the 13-limit.

| [[151ed5]]

| [[65edo]]

| 151ed5 is 65edo with ~0.59 cent compressed octaves. Patent vals match through the 53-limit, with the exception of 7 (as expected).

| [[153ed5]]

| [[66edo]]

| Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 13-limit, with the exception of 3.

| [[158ed5]]

| [[68edo]]

| Same 0.83 cent octave compression as 79ed5~34edo. Patent vals agree through the 23-limit.

| [[160ed5]]

| [[69edo]]

| 160ed5 is 69edo with ~1.6 cent stretched octaves. Patent vals differ in the 7-limit.

| [[165ed5]]

| [[71edo]]

| 165ed5 is 71edo with ~1.0 cent compressed octaves. Patent vals match through the 37-limit.

| [[167ed5]]

| [[72edo]]

| 167ed5 is 72edo with ~1.3 cent stretched octaves. Patent vals agree through the 19-limit.

| [[172ed5]]

| [[74edo]]

| Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 13-limit.

| [[174ed5]]

| [[75edo]]

| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals differ in the 7-limit.

| [[179ed5]]

| [[77edo]]

| 179ed5 is 77edo with ~1.4 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 11.

| [[181ed5]]

| [[78edo]]

| 181ed5 is 78edo with ~0.73 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 13.

| [[186ed5]]

| [[80edo]]

| Same 1.6 cent octave compression as 93ed5~40edo. Patent vals agree through the 29-limit.

| [[188ed5]]

| [[81edo]]

| 188ed5 is 81edo with ~0.49 cent stretched octaves. Patent vals match through the 41-limit.

| [[193ed5]]

| [[83edo]]

| 193ed5 is 83edo with ~1.7 cent compressed octaves. Patent vals match through the 7-limit.

| [[195ed5]]

| [[84edo]]

| Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 31-limit.

| [[200ed5]]

| [[86edo]]

| Same 1.9 cent octeave compression as 100ed5~43edo. Patent vals match alternate primes through the 17-limit.

| [[202ed5]]

| [[87edo]]

| 202ed5 is 87edo with ~0.05 cent stretched octaves. Patent vals match through the 71-limit.

| [[204ed5]]

| [[88edo]]

| Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 7-limit.

| [[209ed5]]

| [[90edo]]

| 209ed5 is 90edo with ~0.15 cent compressed octaves. Patent vals match through the 53-limit.

| [[211ed5]]

| [[91edo]]

| 211ed5 is 91edo with ~1.7 cent stretched octaves. Patent vals match through the 7-limit.

| [[216ed5]]

| [[93edo]]

| Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 31-limit.

| [[218ed5]]

| [[94edo]]

| Same 1.4 cent octave stretch as 109ed5~47edo. Patent vals match through the 23-limit, with the exception of 13.

| [[223ed5]]

| [[96edo]]

| 223ed5 is 96edo with ~0.51 cent compressed octaves. Patent vals match through the 13-limit.

| [[225ed5]]

| [[97edo]]

| 225ed5 is 97edo with ~1.2 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 11.

| [[230ed5]]

| [[99edo]]

| 230ed5 is 99edo with ~0.67 cent compressed octaves. Patent vals match through the 7-limit.

| [[232ed5]]

| [[100edo]]

| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 13-limit.

| [[237ed5]]

| [[102edo]]

| Same 0.83 cent octave compression as 79ed5~34edo. Patent vals differ in the 7-limit.

| [[239ed5]]

| [[103edo]]

| 239ed5 is 103edo with ~0.80 cent stretched octaves. Patent vals match through the 17-limit.

| [[244ed5]]

| [[105edo]]

| 244ed5 is 105edo with ~0.97 cent compressed octaves. Patent vals disagree in the 5-limit and higher prime limit.

| [[246ed5]]

| [[106edo]]

| Same 0.61 cent octave stretch as 123ed5~53edo. Patent vals match through the 43-limit, with the exception of 7.

| [[251ed5]]

| [[108edo]]

| 251ed5 is 108edo with ~1.1 cent compressed octaves. Patent vals match through the 13-limit.

| [[253ed5]]

| [[109edo]]

| 253ed5 is 109edo with ~0.43 cent stretched octaves. Patent vals match through the 13-limit.

| [[258ed5]]

| [[111edo]]

| Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 19-limit.

| [[260ed5]]

| [[112edo]]

| Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals match through the 47-limit, with the exception of 3.

| [[265ed5]]

| [[114edo]]

| 265ed5 is 114edo with ~2.2 cent compressed octaves. Patent vals match through the 43-limit, with the exception of 11, 19 and 37.

| [[267ed5]]

| [[115edo]]

| 267ed5 is 115edo with ~0.10 cent stretched octaves. Patent vals match through the 17-limit.

| [[274ed5]]

| [[118edo]]

| Same 0.05 cent octave compression as 137ed5~59edo. Patent vals agree through the 103-limit.

| [[276ed5]]

| [[119edo]]

| 276ed5 is 119edo with ~1.35 cent stretched octaves. Patent vals match through the 23-limit with the exception of 13 and 19.

| [[281ed5]]

| [[121edo]]

| 281ed5 is 121edo with ~0.20 cent compressed octaves. Patent vals match through the 29-limit.

| [[283ed5]]

| [[122edo]]

| 283ed5 is 122edo with ~1.2 cent stretched octaves. Patent vals match through the 13-limit.

| [[288ed5]]

| [[124edo]]

| Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 23-limit.

| [[290ed5]]

| [[125edo]]

| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 37-limit, with the exception of 13.

| [[295ed5]]

| [[127edo]]

| 295ed5 is 127edo with ~0.47 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 11.

| [[297ed5]]

| [[128edo]]

| 297ed5 is 128edo with ~0.84 cent stretched octaves. Patent vals match through the 7-limit.

| [[302ed5]]

| [[130edo]]

| Same 0.59 cent octave compression as 151ed5~65edo. Patent vals match through the 41-limit, with the exception of 11, 19 and 23.

| [[304ed5]]

| [[131edo]]

| 304ed5 is 131edo with ~0.68 cent stretched octaves. Patent vals match through the 41-limit, with the exception of 13 and 23.

| [[309ed5]]

| [[133edo]]

| 309ed5 is 133edo with ~0.73 cent compressed octaves. Patent vals match through the 71-limit, with the exception of 7 and 59.

| [[311ed5]]

| [[134edo]]

| 311ed5 is 134edo with ~0.53 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 11, 19 and 53.

| [[316ed5]]

| [[137edo]]

| [[143edo]]

| 32ed5 is 143edo with ~0.13 cent stretched octaves. Patent vals match through the 29-limit, with the exception of 7, 17 and 19.

| [[337ed5]]

| [[339ed5]]

| [[146edo]]

| 339ed5 is practically identical to 146edo, with a slight octave stretch (~0.0053 cents). Patent vals match through the 827-limit.

<references />

* [http://www.nonoctave.com/tuning/fifth_harmonic.html| Nonoctave.com: tuning: equal division of the fifth harmonic]

 

[[Category:Ed5's| ]]

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