ELD: Difference between revisions

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An '''ELD''', or '''~~equal length~~ division''', is ~~a kind of~~ [[Arithmetic tunings|arithmetic]] and [[~~Harmonotonic tunings|harmonotonic~~]] tuning.

An '''ELD''' ('''equal length division'''), '''ALD''' ('''arithmetic length division'''), or '''IFD''' ('''inverse-arithmetic frequency division'''), is an [[Arithmetic tunings|arithmetic]] and [[period]]ic [[tuning]] in which each period is divided to a number of steps of equal length difference.

== Specification ==

Its full specification is n-~~ELDp:~~ n equal length divisions of ~~the irrational interval~~ p.

Its full specification is ''n''-ELD-''p'' (''n'' equal length divisions of ''p''), or ''n''-ALD-''p'' (''n'' arithmetic length divisions of ''p''), or ''n''-IFD-''p'' (''n'' inverse-arithmetic frequency division of ''p'').

== Formula ==

To find the steps for an n-~~ELDp~~, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <math>p</math> (or <math>\frac p1</math>), if you are going to move arithmetically (by repeated addition) from <math>1</math> to <math>p</math>, then the difference in string length that you need to cover is not actually <math>p</math>, but only <math>p - 1</math>. And because you are dividing it into <math>n</math> parts, each step will have a size of <math>\frac{p-1}{n}</math>. So, the formula for the length of step <math>k</math> of an n-ELDp is:

To find the steps for an ''n''-ELD-''p'', begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <math>p</math> (or <math>\frac p1</math>), if you are going to move arithmetically (by repeated addition) from <math>1</math> to <math>p</math>, then the difference in string length that you need to cover is not actually <math>p</math>, but only <math>p - 1</math>. And because you are dividing it into <math>n</math> parts, each step will have a size of <math>\frac{p-1}{n}</math>. So, the formula for the length of step <math>k</math> of an n-ELDp is:

<math>

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=== Vs. UD ===

~~The only difference between an n-ELDp and an~~ [[UD|n-~~UDp~~ (or utonal division)]] is that the p ~~for a utonal division is~~ rational.

An [[UD|''n''-UD-''p'' (or utonal division)]] is equivalent to an ''n''-ELD-''p'' except that the period ''p'' of the UD must be rational.

=== Vs. EFD ===

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=== Vs. ALS ===

An ELD will be equivalent to some [[ALS|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-~~ALSp~~.

One period of an ELD will be equivalent to some [[ALS|ALS (arithmetic length sequence)]]; specifically ''n''-ELD((''p'' - 1)/''n'') = ''n''-ALS-''p''.

=== Vs. EDL ===

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! 4

|-

! frequency (f)

! frequency (''f'', ratio)

|(1)

|1.11

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|φ

|-

! pitch (~~log₂f~~)

! pitch (log₂''f'', octaves)

|(0)

|0.14

Line 68:

|0.69

|-

! length (1/f)

! length (1/''f'', ratio)

|(1)

|0.90

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! 4

|-

! frequency (f)

! frequency (''f'', ratio)

|(1)

|0.87

Line 93:

|1/φ

|-

! pitch (~~log₂f~~)

! pitch (log₂''f'', octaves)

|(0)

| -0.21

Line 100:

| -0.69

|-

! length (1/f)

! length (1/''f'', ratio)

|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1

|1+(1/4)(φ-1) = (1φ + 3)/4

@@ Line 1: / Line 1: @@
-An '''ELD''', or '''equal length division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
+An '''ELD''' ('''equal length division'''), '''ALD''' ('''arithmetic length division'''), or '''IFD''' ('''inverse-arithmetic frequency division'''), is an [[Arithmetic tunings|arithmetic]] and [[period]]ic [[tuning]] in which each period is divided to a number of steps of equal length difference.
 == Specification ==
-Its full specification is n-ELDp: n equal length divisions of the irrational interval p.
+Its full specification is ''n''-ELD-''p'' (''n'' equal length divisions of ''p''), or ''n''-ALD-''p'' (''n'' arithmetic length divisions of ''p''), or ''n''-IFD-''p'' (''n'' inverse-arithmetic frequency division of ''p'').
 == Formula ==
-To find the steps for an n-ELDp, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <span><math>p</math></span> (or <span><math>\frac p1</math></span>), if you are going to move arithmetically (by repeated addition) from <span><math>1</math></span> to <span><math>p</math></span>, then the difference in string length that you need to cover is not actually <span><math>p</math></span>, but only <span><math>p - 1</math></span>. And because you are dividing it into <span><math>n</math></span> parts, each step will have a size of <span><math>\frac{p-1}{n}</math></span>. So, the formula for the length of step <span><math>k</math></span> of an n-ELDp is:
+To find the steps for an ''n''-ELD-''p'', begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <span><math>p</math></span> (or <span><math>\frac p1</math></span>), if you are going to move arithmetically (by repeated addition) from <span><math>1</math></span> to <span><math>p</math></span>, then the difference in string length that you need to cover is not actually <span><math>p</math></span>, but only <span><math>p - 1</math></span>. And because you are dividing it into <span><math>n</math></span> parts, each step will have a size of <span><math>\frac{p-1}{n}</math></span>. So, the formula for the length of step <span><math>k</math></span> of an n-ELDp is:
 <math>
@@ Line 27: / Line 27: @@
 === Vs. UD ===
-The only difference between an n-ELDp and an [[UD|n-UDp (or utonal division)]] is that the p for a utonal division is rational.
+An [[UD|''n''-UD-''p'' (or utonal division)]] is equivalent to an ''n''-ELD-''p'' except that the period ''p'' of the UD must be rational.
 === Vs. EFD ===
@@ Line 35: / Line 35: @@
 === Vs. ALS ===
-An ELD will be equivalent to some [[ALS|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.
+One period of an ELD will be equivalent to some [[ALS|ALS (arithmetic length sequence)]]; specifically ''n''-ELD((''p'' - 1)/''n'') = ''n''-ALS-''p''.
 === Vs. EDL ===
@@ Line 54: / Line 54: @@
 ! 4
 |-
-! frequency (f)
+! frequency (''f'', ratio)
 |(1)
 |1.11
@@ Line 61: / Line 61: @@
 |φ
 |-
-! pitch (log₂f)
+! pitch (log₂''f'', octaves)
 |(0)
 |0.14
@@ Line 68: / Line 68: @@
 |0.69
 |-
-! length (1/f)
+! length (1/''f'', ratio)
 |(1)
 |0.90
@@ Line 86: / Line 86: @@
 ! 4
 |-
-! frequency (f)
+! frequency (''f'', ratio)
 |(1)
 |0.87
@@ Line 93: / Line 93: @@
 |1/φ
 |-
-! pitch (log₂f)
+! pitch (log₂''f'', octaves)
 |(0)
 | -0.21
@@ Line 100: / Line 100: @@
 | -0.69
 |-
-! length (1/f)
+! length (1/''f'', ratio)
 |(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1
 |1+(1/4)(φ-1) = (1φ + 3)/4