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) ==

One of the key advantages of dividing the perfect fifth (3/2) into equal parts is that it creates scales where the interval between the unison (1/1) and the mapped minor third (6/5) is the same as the interval between the mapped major third (5/4) and the perfect fifth (3/2). This symmetry arises because the product of (6/5) and (5/4) equals (3/2). Consequently, the errors in approximating the minor third and the major third are of equal magnitude but in opposite directions. Similarly, when dividing the perfect fourth (4/3) into equal parts, the interval between the unison (1/1) and the mapped septimal major second (8/7) matches the interval between the mapped septimal minor third (7/6) and the perfect fourth (4/3), as (8/7) multiplied by (7/6) equals (4/3). Thus, the errors in approximating the septimal major second and the septimal minor third are also equal in size but opposite in direction. In essence, equal divisions of the perfect fourth (4/3) relate to 7-limit intervals in the same way that equal divisions of the perfect fifth (3/2) relate to 5-limit intervals.

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, 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) ==
+One of the key advantages of dividing the perfect fifth (3/2) into equal parts is that it creates scales where the interval between the unison (1/1) and the mapped minor third (6/5) is the same as the interval between the mapped major third (5/4) and the perfect fifth (3/2). This symmetry arises because the product of (6/5) and (5/4) equals (3/2). Consequently, the errors in approximating the minor third and the major third are of equal magnitude but in opposite directions. Similarly, when dividing the perfect fourth (4/3) into equal parts, the interval between the unison (1/1) and the mapped septimal major second (8/7) matches the interval between the mapped septimal minor third (7/6) and the perfect fourth (4/3), as (8/7) multiplied by (7/6) equals (4/3). Thus, the errors in approximating the septimal major second and the septimal minor third are also equal in size but opposite in direction. In essence, equal divisions of the perfect fourth (4/3) relate to 7-limit intervals in the same way that equal divisions of the perfect fifth (3/2) relate to 5-limit intervals.
+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, 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}}