Ed4/3: Difference between revisions

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An '''equal division of the fourth''' ('''ed4/3''') is an [[equal-step tuning]] in which the perfect fourth ([[4/3]]) is [[Just intonation|justly tuned]] and is divided in a given number of equal steps~~. The fourth can be treated as an [[equave]], but it is not necessary and, more importantly, it is not well known whether most listeners can hear it as such~~.

An '''equal division of the fourth''' ('''ed4/3''') is an [[equal-step tuning]] in which the perfect fourth ([[4/3]]) is [[Just intonation|justly tuned]] and is divided in a given number of equal steps.

The expression ''equal division of the fourth'' could be interpreted as applying to other [[interval]]s in the region of the fourth (see [[:Category: Fourth]]), such as [[15/11]]. However, these should be named more specifically and be treated on other pages to avoid any confusion.

The utility of the fourth as ~~a base~~ is apparent by being used at the base of so much Neo-Medieval harmony. Many, though not all, ~~of these~~ scales have a ~~pseudo (~~false) octave, with various degrees of accuracy~~, but which context(s), if any, it is very perceptually important in is as yet an open question~~.

The utility of the fourth as structural scaffolding is apparent by being used at the base of so much Neo-Medieval harmony (see [[tetrachord]]). Division of 4/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed4/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.

~~Incidentally, one way~~ to ~~treat 4~~/3 ~~as an equivalence~~ is the use of the 12:13:14:(16) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]]. Whereas in meantone it takes (an octave-reduced stack of) four [[3/2]] to get to [[5/4]], here it takes (a fourth-reduced stack of) eight [[7/6]] to get to [[13/12]] (tempering out the comma [[5764801/5750784]]). So, doing this yields 13-, 15-, and 28-note [[mos scale]]s for ed4/3's. While the notes are rather closer together, the scheme is uncannily similar to meantone.

One approach to some ed4/3 tunings is the use of the 12:13:14:(16) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]]. Whereas in meantone it takes (an octave-reduced stack of) four [[3/2]] to get to [[5/4]], here it takes (a fourth-reduced stack of) eight [[7/6]] to get to [[13/12]] (tempering out the comma [[5764801/5750784]]). So, doing this yields 13-, 15-, and 28-note [[mos scale]]s for ed4/3's. While the notes are rather closer together, the scheme is uncannily similar to meantone.

== 7-limit, analogy with equal divisions of (3/2) ==

Line 12:

ED4/3 tuning systems that accurately represent the intervals 8/7 and 7/6 include: [[13ed4/3]] (1.31 cent error), [[15ed4/3]] (1.25 cent error), and [[28ed4/3]] (0.06 cent error).

In this sense, [[13ed4/3]], [[15ed4/3]], and [[28ed4/3]] are to the division of the fourth what [[9edf|9ed3/2]], [[11edf|11ed3/2,]] and [[20edf|20ed3/2]] are to the division of the fifth.

In this sense, [[13ed4/3]], [[15ed4/3]], and [[28ed4/3]] are to the division of the fourth what [[9edf|9ed3/2]], [[11edf|11ed3/2]], and [[20edf|20ed3/2]] are to the division of the fifth, and what [[5edo]], [[7edo]], and [[12edo]] are to the division of the octave.

== Individual pages for ed4/3s ==

* [[~~Cube Root of P4|~~3ed4/3]] ~~(aka Cube Root~~ of P4)

* [[4ed4/3]]

{| class="wikitable center-all"

* [[~~5ed4/3|~~5ed4/3]] ~~(aka~~ Quintilipyth scale{{citation needed|date=December 2021|reason=Who used that term?}})

|+ 0…9

* [[6ed4/3]] ~~(aka~~ Sextilipyth scale{{citation needed|date=December 2021|reason=Who used that term?}})

|-

* [[7ed4/3]]

! Standard name

* [[8ed4/3]]

! Common name

* [[~~Noleta|~~9ed4/3]] ~~(aka~~ Noleta scale)

|-

| [[3ed4/3]]

| ED cube root of P4

|-

| [[4ed4/3]]

|

|-

| [[5ed4/3]]

| Quintilipyth scale <br>{{citation needed|date=December 2021|reason=Who used that term?}}

|-

| [[6ed4/3]]

| Sextilipyth scale <br>{{citation needed|date=December 2021|reason=Who used that term?}}

|-

| [[7ed4/3]]

|

|-

| [[8ed4/3]]

|

|-

| [[9ed4/3]]

| Noleta scale

|}

{| class="wikitable center-all"

|+ style=white-space:nowrap | 10…49

|-

| [[10ed4/3|10]]

| [[11ed4/3|11]]

| [[12ed4/3|12]]

| [[13ed4/3|13]]

| [[14ed4/3|14]]

| [[15ed4/3|15]]

| [[16ed4/3|16]]

| [[17ed4/3|17]]

| [[18ed4/3|18]]

| [[19ed4/3|19]]

|-

| [[20ed4/3|20]]

| [[21ed4/3|21]]

| [[22ed4/3|22]]

| [[23ed4/3|23]]

| [[24ed4/3|24]]

| [[25ed4/3|25]]

| [[26ed4/3|26]]

| [[27ed4/3|27]]

| [[28ed4/3|28]]

| [[29ed4/3|29]]

|-

| [[30ed4/3|30]]

| [[31ed4/3|31]]

| [[32ed4/3|32]]

| [[33ed4/3|33]]

| [[34ed4/3|34]]

| [[35ed4/3|35]]

| [[36ed4/3|36]]

| [[37ed4/3|37]]

| [[38ed4/3|38]]

| [[39ed4/3|39]]

|-

| [[40ed4/3|40]]

| [[41ed4/3|41]]

| [[42ed4/3|42]]

| [[43ed4/3|43]]

| [[44ed4/3|44]]

| [[45ed4/3|45]]

| [[46ed4/3|46]]

| [[47ed4/3|47]]

| [[48ed4/3|48]]

| [[49ed4/3|49]]

|}

== See also ==

* [[Square root of 13 over 10]] (previously listed here as an "edIV")

[[Category:Ed4/3| ]]

[[Category:Ed4/3's| ]]

~~[[Category:Edonoi]]~~

[[Category:Lists of scales]]

@@ Line 1: / Line 1: @@
-An '''equal division of the fourth''' ('''ed4/3''') is an [[equal-step tuning]] in which the perfect fourth ([[4/3]]) is [[Just intonation|justly tuned]] and is divided in a given number of equal steps. The fourth can be treated as an [[equave]], but it is not necessary and, more importantly, it is not well known whether most listeners can hear it as such.
+An '''equal division of the fourth''' ('''ed4/3''') is an [[equal-step tuning]] in which the perfect fourth ([[4/3]]) is [[Just intonation|justly tuned]] and is divided in a given number of equal steps.
 The expression ''equal division of the fourth'' could be interpreted as applying to other [[interval]]s in the region of the fourth (see [[:Category: Fourth]]), such as [[15/11]]. However, these should be named more specifically and be treated on other pages to avoid any confusion.
-The utility of the fourth as a base is apparent by being used at the base of so much Neo-Medieval harmony. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy, but which context(s), if any, it is very perceptually important in is as yet an open question.
+The utility of the fourth as structural scaffolding is apparent by being used at the base of so much Neo-Medieval harmony (see [[tetrachord]]). Division of 4/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed4/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.
-Incidentally, one way to treat 4/3 as an equivalence is the use of the 12:13:14:(16) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]]. Whereas in meantone it takes (an octave-reduced stack of) four [[3/2]] to get to [[5/4]], here it takes (a fourth-reduced stack of) eight [[7/6]] to get to [[13/12]] (tempering out the comma [[5764801/5750784]]). So, doing this yields 13-, 15-, and 28-note [[mos scale]]s for ed4/3's. While the notes are rather closer together, the scheme is uncannily similar to meantone.
+One approach to some ed4/3 tunings is the use of the 12:13:14:(16) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]]. Whereas in meantone it takes (an octave-reduced stack of) four [[3/2]] to get to [[5/4]], here it takes (a fourth-reduced stack of) eight [[7/6]] to get to [[13/12]] (tempering out the comma [[5764801/5750784]]). So, doing this yields 13-, 15-, and 28-note [[mos scale]]s for ed4/3's. While the notes are rather closer together, the scheme is uncannily similar to meantone.
 == 7-limit, analogy with equal divisions of (3/2) ==
@@ Line 12: / Line 12: @@
 ED4/3 tuning systems that accurately represent the intervals 8/7 and 7/6 include: [[13ed4/3]] (1.31 cent error), [[15ed4/3]] (1.25 cent error), and [[28ed4/3]] (0.06 cent error).
-In this sense, [[13ed4/3]], [[15ed4/3]], and [[28ed4/3]] are to the division of the fourth what [[9edf|9ed3/2]], [[11edf|11ed3/2,]] and [[20edf|20ed3/2]] are to the division of the fifth.
+In this sense, [[13ed4/3]], [[15ed4/3]], and [[28ed4/3]] are to the division of the fourth what [[9edf|9ed3/2]], [[11edf|11ed3/2]], and [[20edf|20ed3/2]] are to the division of the fifth, and what [[5edo]], [[7edo]], and [[12edo]] are to the division of the octave.
 == Individual pages for ed4/3s ==
-* [[Cube Root of P4|3ed4/3]] (aka Cube Root of P4)
-* [[4ed4/3]]
+{| class="wikitable center-all"
-* [[5ed4/3|5ed4/3]] (aka Quintilipyth scale{{citation needed|date=December 2021|reason=Who used that term?}})
+|+ 0…9
-* [[6ed4/3]] (aka Sextilipyth scale{{citation needed|date=December 2021|reason=Who used that term?}})
+|-
-* [[7ed4/3]]
+! Standard name
-* [[8ed4/3]]
+! Common name
-* [[Noleta|9ed4/3]] (aka Noleta scale)
+|-
+| [[3ed4/3]]
+| ED cube root of P4
+|-
+| [[4ed4/3]]
+|
+|-
+| [[5ed4/3]]
+| Quintilipyth scale <br>{{citation needed|date=December 2021|reason=Who used that term?}}
+|-
+| [[6ed4/3]]
+| Sextilipyth scale <br>{{citation needed|date=December 2021|reason=Who used that term?}}
+|-
+| [[7ed4/3]]
+|
+|-
+| [[8ed4/3]]
+|
+|-
+| [[9ed4/3]]
+| Noleta scale
+|}
+{| class="wikitable center-all"
+|+ style=white-space:nowrap | 10…49
+|-
+| [[10ed4/3|10]]
+| [[11ed4/3|11]]
+| [[12ed4/3|12]]
+| [[13ed4/3|13]]
+| [[14ed4/3|14]]
+| [[15ed4/3|15]]
+| [[16ed4/3|16]]
+| [[17ed4/3|17]]
+| [[18ed4/3|18]]
+| [[19ed4/3|19]]
+|-
+| [[20ed4/3|20]]
+| [[21ed4/3|21]]
+| [[22ed4/3|22]]
+| [[23ed4/3|23]]
+| [[24ed4/3|24]]
+| [[25ed4/3|25]]
+| [[26ed4/3|26]]
+| [[27ed4/3|27]]
+| [[28ed4/3|28]]
+| [[29ed4/3|29]]
+|-
+| [[30ed4/3|30]]
+| [[31ed4/3|31]]
+| [[32ed4/3|32]]
+| [[33ed4/3|33]]
+| [[34ed4/3|34]]
+| [[35ed4/3|35]]
+| [[36ed4/3|36]]
+| [[37ed4/3|37]]
+| [[38ed4/3|38]]
+| [[39ed4/3|39]]
+|-
+| [[40ed4/3|40]]
+| [[41ed4/3|41]]
+| [[42ed4/3|42]]
+| [[43ed4/3|43]]
+| [[44ed4/3|44]]
+| [[45ed4/3|45]]
+| [[46ed4/3|46]]
+| [[47ed4/3|47]]
+| [[48ed4/3|48]]
+| [[49ed4/3|49]]
+|}
 == See also ==
 * [[Square root of 13 over 10]] (previously listed here as an "edIV")
-[[Category:Ed4/3| ]] <!-- main article -->
+[[Category:Ed4/3's| ]]
-[[Category:Edonoi]]
+<!-- main article -->
 [[Category:Lists of scales]]
+{{todo|inline=1|cleanup|improve layout}}