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'''ED5''' means '''Division of the Fifth Harmonic ([[5/1]], pentave) into n equal parts'''.
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.  


== Theory ==
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 fifth harmonic is particularly wide as far as equivalences go. There are (at absolute most) ~4.8 pentaves within the human hearing range; imagine if that were the case with octaves. If one does indeed deal with pentave equivalence, this range restriction is a crucial consideration. Pentave equivalence itself may have a basis in Western music seeing as minor chords have an octave of 5 in their root (i.e. 10:12:15).


One way to treat 5/1 as an equivalence is by eliminating the primes 2 and 3. The most fundamental chord in this paradigm is 5:7:11. This chord can be approximated in a 5.7.11 (or "no-twos-or-threes [[11-limit]]") subgroup [[regular temperament]] by eliminating the comma 859375/823543, equating a stack of 7 [[7/5]] generators with [[11/5]]. Other equivalences that could be used for such "no-two-or-threes" music include [[Ed11/5|equal divisions of 11/5]] and [[Ed11/7|equal divisions of 11/7]].
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 pentave based tuning is hyperpyth (see [[17ed5|17ED5]]). However, perhaps the more common reason to use these scales is in approximation with lower harmonic factors than 5. This approach is highlighted by Hieronymus ([[20ed5|20ED5]]) which itself is a zeta peak tuning (not "no-fives", full on zeta). Other reasons for taking the nth root of 5 include finding temperaments like orwell, meantone, and thuja. This approach can of course be used indiscriminately.
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


Some equal divisions of the pentave are known by alternate names or have special interest:
* [[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


* [[3ed5|3ED5]] [[orwell]] generator
== Individual pages for ed5's ==
* [[4ed5|4ED5]] [[meantone]] generator
* [[5ed5|5ED5]] [[2L_7s|thuja]] generator
* [[6ed5|6ED5]] [[Trienstonic clan #Uncle|uncle]] generator
* [[8ed5|8ED5]] [[mohajira]] generator
* [[Hyperpyth]] tuning (e.g. [[17ed5|17ED5]])
* [[20ed5|20ED5]] Hieronymus Tuning
* [[25ed5|25ED5]] (Stockhausen, McLaren)
 
== Individual pages for ED5s ==
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…99
|+ style=white-space:nowrap | 0…99
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| [[99ed5|99]]
| [[99ed5|99]]
|}
|}
; 100 and beyond
* [[116ed5|116]], [[139ed5|139]], [[175ed5|175]], [[256ed5|256]]
<!-- Uncomment this when there are more pages
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | 100…199
|+ style=white-space:nowrap | 100…199
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| [[299ed5|299]]
| [[299ed5|299]]
|}
|}
-->


==ED5-EDO correspondence==
== Ed5–edo correspondence ==
Following ED5s (up to 339) contain good correspondences to EDO tunings<ref>EDO with relative error of 5th harmonic below 1/3</ref>.
Following ed5's (up to 339) contain good correspondences to edo tunings<ref>Edo with relative error of 5th harmonic below 1/3</ref>.


{| class="wikitable"
{| class="wikitable center-1 center-2"
|-
|-
! | ED5
! Ed5
! |EDO
! Edo
! | Comments
! Comments
|-
|-
| |[[7ed5|7ED5]]
| [[7ed5]]
| |[[3edo|3EDO]]
| [[3edo]]
| | 7ED5 is 3EDO with ~5.9 cent compressed octaves. <br>Equivalently, 3edo is 7ED5 with pentaves stretched by ~13.7 cents. <br>Patent vals match through the 67-limit.
| 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|9ED5]]
| [[9ed5]]
| |[[4edo|4EDO]]
| [[4edo]]
| |Very rough correspondence (~38 cent octave stretch), but patent vals agree through the 7-limit.
| Very rough correspondence (~38 cent octave stretch), but patent vals agree through the 7-limit.
|-
|-
| |[[14ed5|14ED5]]
| [[14ed5]]
| |[[6edo|6EDO]]
| [[6edo]]
| |Same 5.9 cent octave compression as 7ED5~3EDO. Patent vals agree through the 17-limit.
| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
|-
|-
| |[[16ed5|16ED5]]
| [[16ed5]]
| |[[7edo|7EDO]]
| [[7edo]]
| | Only rough correspondence (~19 cent octave stretch). Patent vals differ in the 7-limit.
| Only rough correspondence (~19 cent octave stretch). Patent vals differ in the 7-limit.
|-
|-
| |[[21ed5|21ED5]]
| [[21ed5]]
| |[[9edo|9EDO]]
| [[9edo]]
| |Same 5.9 cent octave compression as 7ED5~3EDO. Patent vals agree through the 59-limit.
| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 59-limit.
|-
|-
| |[[23ed5|23ED5]]
| [[23ed5]]
| |[[10edo|10EDO]]
| [[10edo]]
| | 23ED5 is 10EDO with ~11 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 11 and 17.
| 23ed5 is 10edo with ~11 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 11 and 17.
|-
|-
| |[[28ed5|28ED5]]
| [[28ed5]]
| |[[12edo|12EDO]]
| [[12edo]]
| |Same 5.9 cent octave compression as 7ED5~3EDO. Patent vals match through the 19-limit, with the exception of 13 (as expected).
| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 19-limit, with the exception of 13 (as expected).
|-
|-
| |[[30ed5|30ED5]]
| [[30ed5]]
| |[[13edo|13EDO]]
| [[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).
| 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|35ED5]]
| [[35ed5]]
| |[[15edo|15EDO]]
| [[15edo]]
| |Same 5.9 cent octave compression as 7ED5~3EDO. Patent vals agree through the 13-limit.
| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 13-limit.
|-
|-
| |[[37ed5|37ED5]]
| [[37ed5]]
| |[[16edo|16EDO]]
| [[16edo]]
| |37ED5 is 16EDO with ~4.9 cent stretched octaves. Patent vals match through the 23-limit.
| 37ed5 is 16edo with ~4.9 cent stretched octaves. Patent vals match through the 23-limit.
|-
|-
| |[[42ed5|42ED5]]
| [[42ed5]]
| |[[18edo|18EDO]]
| [[18edo]]
| |Same 5.9 cent octave compression as 7ED5~3EDO. Patent vals match through the 17-limit, with the exception of 11.
| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 17-limit, with the exception of 11.
|-
|-
| |[[44ed5|44ED5]]
| [[44ed5]]
| |[[19edo|19EDO]]
| [[19edo]]
| |44ED5 is 19EDO with ~3.2 cent stretched octaves. Patent vals match through the 13-limit.
| 44ed5 is 19edo with ~3.2 cent stretched octaves. Patent vals match through the 13-limit.
|-
|-
| |[[49ed5|49ED5]]
| [[49ed5]]
| |[[21edo|21EDO]]
| [[21edo]]
| |Same 5.9 cent octave compression as 7ED5~3EDO. Patent vals agree through the 17-limit.
| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
|-
|-
| |[[51ed5|51ED5]]
| [[51ed5]]
| |[[22edo|22EDO]]
| [[22edo]]
| |51ED5 is 22EDO with ~1.9 cent stretched octaves. Patent vals match through the 19-limit.
| 51ed5 is 22edo with ~1.9 cent stretched octaves. Patent vals match through the 19-limit.
|-
|-
| | [[56ed5|56ED5]]
| [[56ed5]]
| | [[24edo|24EDO]]
| [[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.
| 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|58ED5]]
| [[58ed5]]
| |[[25edo|25EDO]]
| [[25edo]]
| |58ED5 is 25EDO with ~1.0 cent stretched octaves. Patent vals match through the 73-limit, with the exception of 13.
| 58ed5 is 25edo with ~1.0 cent stretched octaves. Patent vals match through the 73-limit, with the exception of 13.
|-
|-
| | [[63ed5|63ED5]]
| [[63ed5]]
| | [[27edo|27EDO]]
| [[27edo]]
| | This is a rough correspondence, but the (7n)ED5 ~ (3n)EDO sequence is momentarily fixed. Patent vals match through the 13-limit.
| This is a rough correspondence, but the (7n)ed5 ~ (3n)edo sequence is momentarily fixed. Patent vals match through the 13-limit.
|-
|-
| |[[65ed5|65ED5]]
| [[65ed5]]
| |[[28edo|28EDO]]
| [[28edo]]
| |65ED5 is 28EDO with ~0.26 cent stretched octaves. Patent vals match through the 43-limit.
| 65ed5 is 28edo with ~0.26 cent stretched octaves. Patent vals match through the 43-limit.
|-
|-
| |
| [[72ed5]]
[[72ed5|72ED5]]
| [[31edo]]
| |
| 72ed5 is 31edo with ~0.34 cent compressed octaves. Patent vals match through the 31-limit.
[[31edo|31EDO]]
| |72ED5 is 31EDO with ~0.34 cent compressed octaves. Patent vals match through the 31-limit.
|-
|-
| |[[74ed5|74ED5]]
| [[74ed5]]
| |[[32edo|32EDO]]
| [[32edo]]
| |Same 4.9 cent octave stretch as 37ED5~16EDO. Patent vals match through the 13-limit with the exception of 7 and 11.
| Same 4.9 cent octave stretch as 37ed5~16edo. Patent vals match through the 13-limit with the exception of 7 and 11.
|-
|-
| |[[79ed5|79ED5]]
| [[79ed5]]
| |[[34edo|34EDO]]
| [[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).
| 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]]
[[81ed5|81ED5]]
| [[35edo]]
| |[[35edo|35EDO]]
| 81ed5 is 35edo with ~4.0 cent stretched octaves. Patent vals match through the 11-limit.
| |81ED5 is 35EDO with ~4.0 cent stretched octaves. Patent vals match through the 11-limit.
|-
|-
| |[[84ed5|84ED5]]
| [[84ed5]]
| |[[36edo|36EDO]]
| [[36edo]]
| |Same 5.9 cent octave compression as 7ED5~3EDO. Patent vals match through the 11-limit, with the exception of 7.
| Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 11-limit, with the exception of 7.
|-
|-
| |[[86ed5|86ED5]]
| [[86ed5]]
| |[[37edo|37EDO]]
| [[37edo]]
| |86ED5 is 37EDO with ~1.2 cent compressed octaves. Patent vals match through the 19-limit.
| 86ed5 is 37edo with ~1.2 cent compressed octaves. Patent vals match through the 19-limit.
|-
|-
| |[[88ed5|88ED5]]
| [[88ed5]]
| |[[38edo|38EDO]]
| [[38edo]]
| |Same 3.2 cent octave stretch as 44ED5~19EDO. Patent vals differ in the 7-limit.
| Same 3.2 cent octave stretch as 44ed5~19edo. Patent vals differ in the 7-limit.
|-
|-
| |[[93ed5|93ED5]]
| [[93ed5]]
| | [[40edo|40EDO]]
| [[40edo]]
| |93ED5 is 40EDO with ~1.6 cent compressed octaves. Patent vals match through the 7-limit.
| 93ed5 is 40edo with ~1.6 cent compressed octaves. Patent vals match through the 7-limit.
|-
|-
| |[[95ed5|95ED5]]
| [[95ed5]]
| |[[41edo|41EDO]]
| [[41edo]]
| |95ED5 is 41EDO with ~2.5 cent stretched octaves. Patent vals match through the 11-limit.
| 95ed5 is 41edo with ~2.5 cent stretched octaves. Patent vals match through the 11-limit.
|-
|-
| |
| [[100ed5]]
[[100ed5|100ED5]]
| [[43edo]]
| |[[43edo|43EDO]]
| 100ed5 is 43edo with ~1.9 cent compressed octaves. Patent vals match through the 37-limit.
| |100ED5 is 43EDO with ~1.9 cent compressed octaves. Patent vals match through the 37-limit.
|-
|-
| |
| [[102ed5]]
[[102ed5|102ED5]]
| [[44edo]]
| |[[44edo|44EDO]]
| Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 23-limit, with the exception of 7.
| |Same 1.9 cent octave stretch as 51ED5~22EDO. Patent vals match through the 23-limit, with the exception of 7.
|-
|-
| |
| [[107ed5]]
[[107ed5|107ED5]]
| [[46edo]]
| | [[46edo|46EDO]]
| 107ed5 is 46edo with ~2.1 cent compressed octaves. Patent vals match through the 11-limit.
| |107ED5 is 46EDO with ~2.1 cent compressed octaves. Patent vals match through the 11-limit.
|-
|-
| |
| [[109ed5]]
[[109ed5|109ED5]]
| [[47edo]]
| |[[47edo|47EDO]]
| 109ed5 is 47edo with ~1.4 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 11.
| |109ED5 is 47EDO with ~1.4 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 11.
|-
|-
| |
| [[114ed5]]
[[114ed5|114ED5]]
| [[49edo]]
| |[[49edo|49EDO]]
| 114ed5 is 49edo with ~2.4 cent compressed octaves. Patent vals match through the 11-limit.
| |114ED5 is 49EDO with ~2.4 cent compressed octaves. Patent vals match through the 11-limit.
|-
|-
| |
| [[116ed5]]
[[116ed5|116ED5]]
| [[50edo]]
| |[[50edo|50EDO]]
| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals agree through the 43-limit.
| |Same 1.0 cent octave stretch as 58ED5~25EDO. Patent vals agree through the 43-limit.
|-
|-
| |
| [[121ed5]]
[[121ed5|121ED5]]
| [[52edo]]
| |[[52edo|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).
| |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]]
[[123ed5|123ED5]]
| [[53edo]]
| | [[53edo|53EDO]]
| 123ed5 is 53edo with ~0.61 cent stretched octaves. Patent vals match through the 29-limit.
| |123ED5 is 53EDO with ~0.61 cent stretched octaves. Patent vals match through the 29-limit.
|-
|-
| |
| [[128ed5]]
[[128ed5|128ED5]]
| [[55edo]]
| |[[55edo|55EDO]]
| 128ed5 is 55edo with ~2.8 cent compressed octaves, but the correspondence is rough. Patent vals differ in the 7-limit.
| |128ED5 is 55EDO with ~2.8 cent compressed octaves, but the correspondence is rough. Patent vals differ in the 7-limit.
|-
|-
| |
| [[130ed5]]
[[130ed5|130ED5]]
| [[56edo]]
| |[[56edo|56EDO]]
| Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 103-limit.
| |Same 0.26 cent octave stretch as 65ED5~28EDO. Patent vals agree through the 103-limit.
|-
|-
| |
| [[135ed5]]
[[135ed5|135ED5]]
| [[58edo]]
| |[[58edo|58EDO]]
| 135ed5 is 58edo with ~2.9 cent compressed octaves. Patent vals match through the 13-limit.
| |135ED5 is 58EDO with ~2.9 cent compressed octaves. Patent vals match through the 13-limit.
|-
|-
| |
| [[137ed5]]
[[137ed5|137ED5]]
| [[59edo]]
| |[[59edo|59EDO]]
| 137ed5 is 59edo with ~0.05 cent compressed octaves. Patent vals match through the 179-limit.
| | 137ED5 is 59EDO with ~0.05 cent compressed octaves. Patent vals match through the 179-limit.
|-
|-
| |
| [[139ed5]]
[[139ed5|139ED5]]
| [[60edo]]
| |[[60edo|60EDO]]
| 139ed5 is 60edo with ~2.7 cent stretched octaves. Pantent vals match through the 31-limit with the exception of 11.
| | 139ED5 is 60EDO with ~2.7 cent stretched octaves. Pantent vals match through the 31-limit with the exception of 11.
|-
|-
| |
| [[144ed5]]
[[144ed5|144ED5]]
| [[62edo]]
| | [[62edo|62EDO]]
| Same 0.34 cent octave compression as 72ed5~31edo. Patent vals match through the 19-limit, with the exception of 11.
| |Same 0.34 cent octave compression as 72ED5~31EDO. Patent vals match through the 19-limit, with the exception of 11.
|-
|-
| |
| [[146ed5]]
[[146ed5|146ED5]]
| [[63edo]]
| |[[63edo|63EDO]]
| 146ed5 is 63edo with ~2.3 cent stretched octaves. Patent vals match through the 13-limit.
| |146ED5 is 63EDO with ~2.3 cent stretched octaves. Patent vals match through the 13-limit.
|-
|-
| |
| [[151ed5]]
[[151ed5|151ED5]]
| [[65edo]]
| |[[65edo|65EDO]]
| 151ed5 is 65edo with ~0.59 cent compressed octaves. Patent vals match through the 53-limit, with the exception of 7 (as expected).
| |151ED5 is 65EDO with ~0.59 cent compressed octaves. Patent vals match through the 53-limit, with the exception of 7 (as expected).
|-
|-
| |
| [[153ed5]]
[[153ed5|153ED5]]
| [[66edo]]
| |[[66edo|66EDO]]
| Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 13-limit, with the exception of 3.
| |Same 1.9 cent octave stretch as 51ED5~22EDO. Patent vals match through the 13-limit, with the exception of 3.
|-
|-
| |
| [[158ed5]]
[[158ed5|158ED5]]
| [[68edo]]
| |[[68edo|68EDO]]
| Same 0.83 cent octave compression as 79ed5~34edo. Patent vals agree through the 23-limit.
| |Same 0.83 cent octave compression as 79ED5~34EDO. Patent vals agree through the 23-limit.
|-
|-
| |
| [[160ed5]]
[[160ed5|160ED5]]
| [[69edo]]
| |[[69edo|69EDO]]
| 160ed5 is 69edo with ~1.6 cent stretched octaves. Patent vals differ in the 7-limit.
| |160ED5 is 69EDO with ~1.6 cent stretched octaves. Patent vals differ in the 7-limit.
|-
|-
| |
| [[165ed5]]
[[165ed5|165ED5]]
| [[71edo]]
| |[[71edo|71EDO]]
| 165ed5 is 71edo with ~1.0 cent compressed octaves. Patent vals match through the 37-limit.
| |165ED5 is 71EDO with ~1.0 cent compressed octaves. Patent vals match through the 37-limit.
|-
|-
| |
| [[167ed5]]
[[167ed5|167ED5]]
| [[72edo]]
| |[[72edo|72EDO]]
| 167ed5 is 72edo with ~1.3 cent stretched octaves. Patent vals agree through the 19-limit.
| |167ED5 is 72EDO with ~1.3 cent stretched octaves. Patent vals agree through the 19-limit.
|-
|-
| |
| [[172ed5]]
[[172ed5|172ED5]]
| [[74edo]]
| |[[74edo|74EDO]]
| Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 13-limit.
| |Same 1.2 cent octave compression as 86ED5~37EDO. Patent vals agree through the 13-limit.
|-
|-
| |
| [[174ed5]]
[[174ed5|174ED5]]
| [[75edo]]
| |[[75edo|75EDO]]
| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals differ in the 7-limit.
| | Same 1.0 cent octave stretch as 58ED5~25EDO. Patent vals differ in the 7-limit.
|-
|-
| |
| [[179ed5]]
[[179ed5|179ED5]]
| [[77edo]]
| | [[77edo|77EDO]]
| 179ed5 is 77edo with ~1.4 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 11.
| |179ED5 is 77EDO with ~1.4 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 11.
|-
|-
| |
| [[181ed5]]
[[181ed5|181ED5]]
| [[78edo]]
| |[[78edo|78EDO]]
| 181ed5 is 78edo with ~0.73 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 13.
| |181ED5 is 78EDO with ~0.73 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 13.
|-
|-
| |
| [[186ed5]]
[[186ed5|186ED5]]
| [[80edo]]
| | [[80edo|80EDO]]
| Same 1.6 cent octave compression as 93ed5~40edo. Patent vals agree through the 29-limit.
| |Same 1.6 cent octave compression as 93ED5~40EDO. Patent vals agree through the 29-limit.
|-
|-
| |
| [[188ed5]]
[[188ed5|188ED5]]
| [[81edo]]
| |[[81edo|81EDO]]
| 188ed5 is 81edo with ~0.49 cent stretched octaves. Patent vals match through the 41-limit.
| |188ED5 is 81EDO with ~0.49 cent stretched octaves. Patent vals match through the 41-limit.
|-
|-
| |
| [[193ed5]]
[[193ed5|193ED5]]
| [[83edo]]
| |
| 193ed5 is 83edo with ~1.7 cent compressed octaves. Patent vals match through the 7-limit.
[[83edo|83EDO]]
| | 193ED5 is 83EDO with ~1.7 cent compressed octaves. Patent vals match through the 7-limit.
|-
|-
| |
| [[195ed5]]
[[195ed5|195ED5]]
| [[84edo]]
| |[[84edo|84EDO]]
| Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 31-limit.
| | Same 0.26 cent octave stretch as 65ED5~28EDO. Patent vals agree through the 31-limit.
|-
|-
| |
| [[200ed5]]
[[200ed5|200ED5]]
| [[86edo]]
| |[[86edo|86EDO]]
| Same 1.9 cent octeave compression as 100ed5~43edo. Patent vals match alternate primes through the 17-limit.
| | Same 1.9 cent octeave compression as 100ED5~43EDO. Patent vals match alternate primes through the 17-limit.
|-
|-
| |
| [[202ed5]]
[[202ed5|202ED5]]
| [[87edo]]
| |[[87edo|87EDO]]
| 202ed5 is 87edo with ~0.05 cent stretched octaves. Patent vals match through the 71-limit.
| |202ED5 is 87EDO with ~0.05 cent stretched octaves. Patent vals match through the 71-limit.
|-
|-
| |
| [[204ed5]]
[[204ed5|204ED5]]
| [[88edo]]
| |[[88edo|88EDO]]
| Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 7-limit.
| |Same 1.9 cent octave stretch as 51ED5~22EDO. Patent vals match through the 7-limit.
|-
|-
| |
| [[209ed5]]
[[209ed5|209ED5]]
| [[90edo]]
| |[[90edo|90EDO]]
| 209ed5 is 90edo with ~0.15 cent compressed octaves. Patent vals match through the 53-limit.
| |209ED5 is 90EDO with ~0.15 cent compressed octaves. Patent vals match through the 53-limit.
|-
|-
| |
| [[211ed5]]
[[211ed5|211ED5]]
| [[91edo]]
| |[[91edo|91EDO]]
| 211ed5 is 91edo with ~1.7 cent stretched octaves. Patent vals match through the 7-limit.
| | 211ED5 is 91EDO with ~1.7 cent stretched octaves. Patent vals match through the 7-limit.
|-
|-
| |
| [[216ed5]]
[[216ed5|216ED5]]
| [[93edo]]
| |[[93edo|93EDO]]
| Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 31-limit.
| |Same 0.34 cent octave compression as 72ED5~31EDO. Patent vals agree through the 31-limit.
|-
|-
| |
| [[218ed5]]
[[218ed5|218ED5]]
| [[94edo]]
| |[[94edo|94EDO]]
| Same 1.4 cent octave stretch as 109ed5~47edo. Patent vals match through the 23-limit, with the exception of 13.
| |Same 1.4 cent octave stretch as 109ED5~47EDO. Patent vals match through the 23-limit, with the exception of 13.
|-
|-
| |
| [[223ed5]]
[[223ed5|223ED5]]
| [[96edo]]
| |[[96edo|96EDO]]
| 223ed5 is 96edo with ~0.51 cent compressed octaves. Patent vals match through the 13-limit.
| | 223ED5 is 96EDO with ~0.51 cent compressed octaves. Patent vals match through the 13-limit.
|-
|-
| |
| [[225ed5]]
[[225ed5|225ED5]]
| [[97edo]]
| |[[97edo|97EDO]]
| 225ed5 is 97edo with ~1.2 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 11.
| |225ED5 is 97EDO with ~1.2 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 11.
|-
|-
| |
| [[230ed5]]
[[230ed5|230ED5]]
| [[99edo]]
| |[[99edo|99EDO]]
| 230ed5 is 99edo with ~0.67 cent compressed octaves. Patent vals match through the 7-limit.
| |230ED5 is 99EDO with ~0.67 cent compressed octaves. Patent vals match through the 7-limit.
|-
|-
| |
| [[232ed5]]
[[232ed5|232ED5]]
| [[100edo]]
| |
| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 13-limit.
[[100edo|100EDO]]
| |Same 1.0 cent octave stretch as 58ED5~25EDO. Patent vals match through the 13-limit.
|-
|-
| |
| [[237ed5]]
[[237ed5|237ED5]]
| [[102edo]]
| |
| Same 0.83 cent octave compression as 79ed5~34edo. Patent vals differ in the 7-limit.
[[102edo|102EDO]]
| |Same 0.83 cent octave compression as 79ED5~34EDO. Patent vals differ in the 7-limit.
|-
|-
| |
| [[239ed5]]
[[239ed5|239ED5]]
| [[103edo]]
| |
| 239ed5 is 103edo with ~0.80 cent stretched octaves. Patent vals match through the 17-limit.
[[103edo|103EDO]]
| |239ED5 is 103EDO with ~0.80 cent stretched octaves. Patent vals match through the 17-limit.
|-
|-
| |
| [[244ed5]]
[[244ed5|244ED5]]
| [[105edo]]
| |
| 244ed5 is 105edo with ~0.97 cent compressed octaves. Patent vals disagree in the 5-limit and higher prime limit.
[[105edo|105EDO]]
| |244ED5 is 105EDO with ~0.97 cent compressed octaves. Patent vals disagree in the 5-limit and higher prime limit.
|-
|-
| |
| [[246ed5]]
[[246ed5|246ED5]]
| [[106edo]]
| |
| Same 0.61 cent octave stretch as 123ed5~53edo. Patent vals match through the 43-limit, with the exception of 7.
[[106edo|106EDO]]
| | Same 0.61 cent octave stretch as 123ED5~53EDO. Patent vals match through the 43-limit, with the exception of 7.
|-
|-
| |
| [[251ed5]]
[[251ed5|251ED5]]
| [[108edo]]
| |
| 251ed5 is 108edo with ~1.1 cent compressed octaves. Patent vals match through the 13-limit.
[[108edo|108EDO]]
| |251ED5 is 108EDO with ~1.1 cent compressed octaves. Patent vals match through the 13-limit.
|-
|-
| |
| [[253ed5]]
[[253ed5|253ED5]]
| [[109edo]]
| |
| 253ed5 is 109edo with ~0.43 cent stretched octaves. Patent vals match through the 13-limit.
[[109edo|109EDO]]
| | 253ED5 is 109EDO with ~0.43 cent stretched octaves. Patent vals match through the 13-limit.
|-
|-
| |
| [[258ed5]]
[[258ed5|258ED5]]
| [[111edo]]
| |
| Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 19-limit.
[[111edo|111EDO]]
| |Same 1.2 cent octave compression as 86ED5~37EDO. Patent vals agree through the 19-limit.
|-
|-
| |
| [[260ed5]]
[[260ed5|260ED5]]
| [[112edo]]
| |
| Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals match through the 47-limit, with the exception of 3.
[[112edo|112EDO]]
| |Same 0.26 cent octave stretch as 65ED5~28EDO. Patent vals match through the 47-limit, with the exception of 3.
|-
|-
| |
| [[265ed5]]
[[265ed5|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.
[[114edo|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]]
[[267ed5|267ED5]]
| [[115edo]]
| |
| 267ed5 is 115edo with ~0.10 cent stretched octaves. Patent vals match through the 17-limit.
[[115edo|115EDO]]
| |267ED5 is 115EDO with ~0.10 cent stretched octaves. Patent vals match through the 17-limit.
|-
|-
| |
| [[274ed5]]
[[274ed5|274ED5]]
| [[118edo]]
| |
| Same 0.05 cent octave compression as 137ed5~59edo. Patent vals agree through the 103-limit.
[[118edo|118EDO]]
| |Same 0.05 cent octave compression as 137ED5~59EDO. Patent vals agree through the 103-limit.
|-
|-
| |
| [[276ed5]]
[[276ed5|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.
[[119edo|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]]
[[281ed5|281ED5]]
| [[121edo]]
| |
| 281ed5 is 121edo with ~0.20 cent compressed octaves. Patent vals match through the 29-limit.
[[121edo|121EDO]]
| |281ED5 is 121EDO with ~0.20 cent compressed octaves. Patent vals match through the 29-limit.
|-
|-
| |
| [[283ed5]]
[[283ed5|283ED5]]
| [[122edo]]
| |
| 283ed5 is 122edo with ~1.2 cent stretched octaves. Patent vals match through the 13-limit.
[[122edo|122EDO]]
| |283ED5 is 122EDO with ~1.2 cent stretched octaves. Patent vals match through the 13-limit.
|-
|-
| |
| [[288ed5]]
[[288ed5|288ED5]]
| [[124edo]]
| |
| Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 23-limit.
[[124edo|124EDO]]
| |Same 0.34 cent octave compression as 72ED5~31EDO. Patent vals agree through the 23-limit.
|-
|-
| |
| [[290ed5]]
[[290ed5|290ED5]]
| [[125edo]]
| |
| Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 37-limit, with the exception of 13.
[[125edo|125EDO]]
| |Same 1.0 cent octave stretch as 58ED5~25EDO. Patent vals match through the 37-limit, with the exception of 13.
|-
|-
| |
| [[295ed5]]
[[295ed5|295ED5]]
| [[127edo]]
| |
| 295ed5 is 127edo with ~0.47 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 11.
[[127edo|127EDO]]
| |295ED5 is 127EDO with ~0.47 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 11.
|-
|-
| |
| [[297ed5]]
[[297ed5|297ED5]]
| [[128edo]]
| |
| 297ed5 is 128edo with ~0.84 cent stretched octaves. Patent vals match through the 7-limit.
[[128edo|128EDO]]
| |297ED5 is 128EDO with ~0.84 cent stretched octaves. Patent vals match through the 7-limit.
|-
|-
| |
| [[302ed5]]
[[302ed5|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.
[[130edo|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]]
[[304ed5|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.
[[131edo|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]]
[[309ed5|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.
[[133edo|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]]
[[311ed5|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.
[[134edo|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]]
[[316ed5|316ED5]]
| [[136edo]]
| |
| Same 0.83 cent octave compression as 79ed5~34edo. Patent vals match through the 41-limit, with the exception of 13 and 23.
[[136edo|136EDO]]
| |Same 0.83 cent octave compression as 79ED5~34EDO. Patent vals match through the 41-limit, with the exception of 13 and 23.
|-
|-
| |
| [[318ed5]]
[[318ed5|318ED5]]
| [[137edo]]
| |
| 318ed5 is 137edo with ~0.39 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 7.
[[137edo|137EDO]]
| |318ED5 is 137EDO with ~0.39 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 7.
|-
|-
| |
| [[323ed5]]
[[323ed5|323ED5]]
| [[139edo]]
| |
| 323ed5 is 139edo with ~0.94 cent compressed compressed octaves. Patent vals match alternating primes through the 11-limit.
[[139edo|139EDO]]
| |323ED5 is 139EDO with ~0.94 cent compressed compressed octaves. Patent vals match alternating primes through the 11-limit.
|-
|-
| |
| [[325ed5]]
[[325ed5|325ED5]]
| [[140edo]]
| |
| Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals match through the 73-limit, with the exception of 31 and 67.
[[140edo|140EDO]]
| | Same 0.26 cent octave stretch as 65ED5~28EDO. Patent vals match through the 73-limit, with the exception of 31 and 67.
|-
|-
| |
| [[330ed5]]
[[330ed5|330ED5]]
| [[142edo]]
| |
| Same 1.0 cent octave compression as 165ed5~71edo. Patent vals match through the 13-limit, with the exception of 11.
[[142edo|142EDO]]
| |Same 1.0 cent octave compression as 165ED5~71EDO. Patent vals match through the 13-limit, with the exception of 11.
|-
|-
| |
| [[332ed5]]
[[332ed5|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.
[[143edo|143EDO]]
| | 332ED5 is 143EDO with ~0.13 cent stretched octaves. Patent vals match through the 29-limit, with the exception of 7, 17 and 19.
|-
|-
| |
| [[337ed5]]
[[337ed5|337ED5]]
| [[145edo]]
| |
| 337ed5 is 145edo with ~1.1 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.
[[145edo|145EDO]]
| |337ED5 is 145EDO with ~1.1 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.
|-
|-
| |[[339ed5|339ED5]]
| [[339ed5]]
| |[[146edo|146EDO]]
| [[146edo]]
| |339ED5 is practically identical to 146EDO, with a slight octave stretch (~0.0053 cents). Patent vals match through the 827-limit.
| 339ed5 is practically identical to 146edo, with a slight octave stretch (~0.0053 cents). Patent vals match through the 827-limit.
|}
|}
<references />
<references />


==See also==
== See also ==
*[[Pentave Reduced Harmonics]]
* [[Pentave Reduced Harmonics]]
*[[Pentave Reduced Subharmonics]]
* [[Pentave Reduced Subharmonics]]
*[[Relative errors of small ED5s]]
* [[Relative errors of small ED5s]]
*http://www.nonoctave.com/tuning/fifth_harmonic.html
 
== External links ==
* [http://www.nonoctave.com/tuning/fifth_harmonic.html| Nonoctave.com: tuning: equal division of the fifth harmonic]


[[Category:Equal-step tuning]]
[[Category:Ed5's| ]]
[[Category:Ed5| ]] <!-- main article -->
<!-- main article -->
[[category:Nonoctave]]
[[Category:Lists of scales]]
[[Category:Pentave]]
[[Category:Pentave]]


{{Todo|add sound example}}
{{Todo|add sound example}}
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