ED5: Difference between revisions

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~~The~~ '''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.

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 5th harmonic is particularly wide as far as [[equivalence]]s 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]]). 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]]) which itself is a zeta peak tuning (not "no-~~fives~~", full on zeta)~~. Other reasons 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 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).

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

== 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

* [[3ed5]] – [[orwell]] generator

* [[4ed5]] [[meantone]] generator

* [[4ed5]] – [[meantone]] generator

* [[5ed5]] [[~~2L_7s|~~thuja]] generator

* [[5ed5]] – [[thuja]] generator

* [[6ed5]] [[~~Trienstonic clan #Uncle|~~uncle]] generator

* [[6ed5]] – [[uncle]] generator

* [[8ed5]] [[mohajira]] generator

* [[8ed5]] – [[mohajira]] generator

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

* [[20ed5]] Hieronymus Tuning

* [[20ed5]] – Hieronymus Tuning

* [[25ed5]] (Stockhausen, McLaren)

* [[25ed5]] – Stockhausen, McLaren

== Individual pages for ed5's ==

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| [[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"

|+ style=white-space:nowrap | 100…199

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| [[299ed5|299]]

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== ~~Ed5-edo~~ correspondence ==

== 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>.

{| class="wikitable"

{| class="wikitable center-1 center-2"

|-

! Ed5

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| [[7ed5]]

| [[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]]

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* [[Pentave Reduced Subharmonics]]

* [[Relative errors of small ED5s]]

* http://www.nonoctave.com/tuning/fifth_harmonic.html

[~~[Category~~:~~Equal-step~~ tuning]]

== External links ==

[[Category:Ed5| ]]

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

[[~~category~~:~~Nonoctave~~]]

[[Category:Ed5's| ]]

[[Category:Lists of scales]]

[[Category:Pentave]]

@@ Line 1: / Line 1: @@
-The '''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.
+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 5th harmonic is particularly wide as far as [[equivalence]]s 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]]). 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]]) which itself is a zeta peak tuning (not "no-fives", full on zeta). Other reasons 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 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).
-Some equal divisions of the pentave are known by alternate names or have special interest:
+== 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
+* [[3ed5]] – [[orwell]] generator
-* [[4ed5]] [[meantone]] generator
+* [[4ed5]] – [[meantone]] generator
-* [[5ed5]] [[2L_7s|thuja]] generator
+* [[5ed5]] – [[thuja]] generator
-* [[6ed5]] [[Trienstonic clan #Uncle|uncle]] generator
+* [[6ed5]] – [[uncle]] generator
-* [[8ed5]] [[mohajira]] generator
+* [[8ed5]] – [[mohajira]] generator
 * [[Hyperpyth]] tuning (e.g. [[17ed5]])
-* [[20ed5]] Hieronymus Tuning
+* [[20ed5]] – Hieronymus Tuning
-* [[25ed5]] (Stockhausen, McLaren)
+* [[25ed5]] – Stockhausen, McLaren
 == Individual pages for ed5's ==
@@ Line 132: / Line 132: @@
 | [[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"
 |+ style=white-space:nowrap | 100…199
@@ Line 356: / Line 361: @@
 | [[299ed5|299]]
 |}
+-->
-== Ed5-edo correspondence ==
+== 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>.
-{| class="wikitable"
+{| class="wikitable center-1 center-2"
 |-
 ! Ed5
@@ Line 368: / Line 374: @@
 | [[7ed5]]
 | [[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]]
@@ Line 760: / Line 766: @@
 * [[Pentave Reduced Subharmonics]]
 * [[Relative errors of small ED5s]]
-* http://www.nonoctave.com/tuning/fifth_harmonic.html
-[[Category:Equal-step tuning]]
+== External links ==
-[[Category:Ed5| ]] <!-- main article -->
+* [http://www.nonoctave.com/tuning/fifth_harmonic.html| Nonoctave.com: tuning: equal division of the fifth harmonic]
-[[category:Nonoctave]]
+[[Category:Ed5's| ]]
+<!-- main article -->
+[[Category:Lists of scales]]
 [[Category:Pentave]]
 {{Todo|add sound example}}