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 way to treat 5/1 as an equivalence is by eliminating the [[prime harmonics|primes]] [[2/1|2]] and [[3/1|3]]. The most fundamental chord in this paradigm is [[5:7:11]]. This chord can be approximated in a 5.7.11-subgroup [[regular temperament]] by eliminating the comma 859375/823543, equating a stack of seven [[7/5]] generators with [[11/5]]. Other equivalences that could be used for such no-2's no-3's music include [[ed11/5|equal divisions of 11/5]] and [[ed11/7|equal divisions of 11/7]].

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-5's", full on zeta).

== 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

* [[3ed5]] – [[orwell]] generator

* [[4ed5]] – [[meantone]] generator

* [[5ed5]] – [[thuja]] generator

* [[6ed5]] – [[uncle]] generator

* [[8ed5]] – [[mohajira]] generator

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

* [[20ed5]] – Hieronymus Tuning

* [[25ed5]] – Stockhausen, McLaren

== Individual pages for ed5's ==

{| class="wikitable center-all"

|+ style=white-space:nowrap | 0…99

| [[1ed5|1]]

| [[3ed5|3]]

 

{| class="wikitable center-all mw-collapsible mw-collapsed"

| [[110ed5|110]]

| [[118ed5|118]]

| [[119ed5|119]]

| [[120ed5|120]]

| [[122ed5|122]]

| [[123ed5|123]]

| [[124ed5|124]]

| [[125ed5|125]]

| [[126ed5|126]]

| [[130ed5|130]]

| [[132ed5|132]]

| [[133ed5|133]]

| [[134ed5|134]]

| [[135ed5|135]]

| [[136ed5|136]]

| [[140ed5|140]]

| [[142ed5|142]]

| [[143ed5|143]]

| [[144ed5|144]]

| [[145ed5|145]]

| [[146ed5|146]]

| [[150ed5|150]]

| [[152ed5|152]]

| [[153ed5|153]]

| [[154ed5|154]]

| [[155ed5|155]]

| [[156ed5|156]]

| [[160ed5|160]]

| [[162ed5|162]]

| [[163ed5|163]]

| [[164ed5|164]]

| [[165ed5|165]]

| [[166ed5|166]]

| [[170ed5|170]]

| [[172ed5|172]]

| [[173ed5|173]]

| [[174ed5|174]]

| [[175ed5|175]]

| [[176ed5|176]]

| [[180ed5|180]]

| [[182ed5|182]]

| [[183ed5|183]]

| [[184ed5|184]]

| [[185ed5|185]]

| [[186ed5|186]]

| [[190ed5|190]]

| [[195ed5|195]]

| [[196ed5|196]]

| [[197ed5|197]]

| [[198ed5|198]]

|+ style=white-space:nowrap | 200…299

| [[210ed5|210]]

| [[212ed5|212]]

| [[213ed5|213]]

| [[214ed5|214]]

| [[215ed5|215]]

| [[216ed5|216]]

| [[220ed5|220]]

| [[222ed5|222]]

| [[223ed5|223]]

| [[224ed5|224]]

| [[225ed5|225]]

| [[226ed5|226]]

| [[230ed5|230]]

| [[232ed5|232]]

| [[233ed5|233]]

| [[234ed5|234]]

| [[235ed5|235]]

| [[236ed5|236]]

| [[240ed5|240]]

| [[242ed5|242]]

| [[243ed5|243]]

| [[244ed5|244]]

| [[245ed5|245]]

| [[246ed5|246]]

| [[250ed5|250]]

| [[252ed5|252]]

| [[253ed5|253]]

| [[254ed5|254]]

| [[255ed5|255]]

| [[256ed5|256]]

| [[260ed5|260]]

| [[262ed5|262]]

| [[263ed5|263]]

| [[264ed5|264]]

| [[265ed5|265]]

| [[266ed5|266]]

| [[270ed5|270]]

| [[272ed5|272]]

| [[273ed5|273]]

| [[274ed5|274]]

| [[275ed5|275]]

| [[276ed5|276]]

| [[280ed5|280]]

| [[282ed5|282]]

| [[283ed5|283]]

| [[284ed5|284]]

| [[285ed5|285]]

| [[286ed5|286]]

| [[290ed5|290]]

| [[295ed5|295]]

| [[296ed5|296]]

| [[297ed5|297]]

| [[298ed5|298]]

| [[299ed5|299]]

 

== Ed5–edo correspondence ==

Following ed5's (up to 339) contain good correspondences to edo tunings<ref>Edo with relative error of 5th harmonic below 1/3</ref>.

 

| 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.

| [[9ed5]]

| [[4edo]]

| 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]]

| [[332ed5]]

| [[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.

| [[145edo]]

| [[339ed5]]

| [[146edo]]

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

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

 

[[Category:Ed5's| ]]

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