4edt: Difference between revisions

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~~'''4EDT''' is the [[Edt|equal division of the third harmonic]] into four parts of 475.4888 [[cent]]s each, corresponding to 2.5237 [[edo]].~~

~~The 4th root~~ of 3, ~~might~~ be viewed ~~alternately~~ as a ~~degenerate form or a fundamental building block~~ of Bohlen-Pierce ~~harmony~~, analogous to how 5edo ~~relates to~~ diatonic ~~music. The situation is different however, as in this case both 5 and 7 are relatively well represented (opposed to just 3 in 5edo)~~. While the ~~approximations~~ may seem excessively vague, ~~and~~ some might say ~~impossible~~, they are nevertheless categorically important to the perception of the scale~~, and, may even be heard as those harmonies given~~ the width of the "scale". It is doubtful ~~however,~~ that this scale could ~~recieve~~ much melodic treatment, and is more useful as a harmonic entity, either to ~~demonstrate~~ BP harmony, or as a ~~component~~ of scales like [[8edt]].

== Theory ==

4edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until further multiples of 4edt until these intervals see practical use.

4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].

=== Odd harmonics ===

=== Approximation of intervals ===

{| class="wikitable right-all left-4 left-5"

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! Steps

! Cents

! ~~Hekts~~

! [[Hekt]]s

! Corresponding <br>JI intervals

! Comments

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| 0

| [[1/1]]

|

| perfect unison

|-

| 1

| 475.4888

| 325

| [[17/13]], [[21/16]], 25/19, 33/25

| [[17/13]], [[21/16]], [[25/19]], 33/25

|

|-

| 2

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Line 38:

| 650

| [[19/11]], 45/26, [[26/15]], (85/49), 33/19

|

| one step of [[2edt]]

|-

| 3

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''4EDT''' is the [[Edt|equal division of the third harmonic]] into four parts of 475.4888 [[cent]]s each, corresponding to 2.5237 [[edo]].
+{{EDO intro}}
-The 4th root of 3, might be viewed alternately as a degenerate form or a fundamental building block of Bohlen-Pierce harmony, analogous to how 5edo relates to diatonic music. The situation is different however, as in this case both 5 and 7 are relatively well represented (opposed to just 3 in 5edo). While the approximations may seem excessively vague, and some might say impossible, they are nevertheless categorically important to the perception of the scale, and, may even be heard as those harmonies given the width of the "scale". It is doubtful however, that this scale could recieve much melodic treatment, and is more useful as a harmonic entity, either to demonstrate BP harmony, or as a component of scales like [[8edt]].
+== Theory ==
+edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until further multiples of 4edt until these intervals see practical use.
+edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale.  While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].
+=== Odd harmonics ===
+{{Harmonics in equal|4|3|1|intervals=odd}}
+=== Approximation of intervals ===
 {| class="wikitable right-all left-4 left-5"
@@ Line 8: / Line 18: @@
 ! Steps
 ! Cents
-! Hekts
+! [[Hekt]]s
 ! Corresponding <br>JI intervals
 ! Comments
@@ Line 16: / Line 26: @@
 | 0
 | [[1/1]]
-|
+| perfect unison
 |-
 | 1
 | 475.4888
 | 325
-| [[17/13]], [[21/16]], 25/19, 33/25
+| [[17/13]], [[21/16]], [[25/19]], 33/25
 |
 |-
 | 2
@@ Line 28: / Line 38: @@
 | 650
 | [[19/11]], 45/26, [[26/15]], (85/49), 33/19
-|
+| one step of [[2edt]]
 |-
 | 3