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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 ==
== 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 ==
== 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>
<math>
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== Relationship to other tunings ==
== Relationship to other tunings ==


=== vs. EPD ===
=== Vs. EPD ===


It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''length'''. You will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so ELD are typically reserved for irrational tunings, such as 12-ELDφ. So it would be more appropriate to name this tuning 12-UDO, for utonal divisions of the octave.
It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''length'''. You will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so ELD are typically reserved for irrational tunings, such as 12-ELDφ. So it would be more appropriate to name this tuning 12-UDO, for utonal divisions of the octave.


=== vs. UD ===
=== 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 ===
=== Vs. EFD ===


The analogous otonal equivalent of an ELD is an [[EFD|EFD (equal frequency division)]].
The analogous otonal equivalent of an ELD is an [[EFD|EFD (equal frequency division)]].


=== vs. ALS ===
=== 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 ===
=== Vs. EDL ===


An ELD is not to be confused with [[EDL|EDL, equal division of length]]. The latter term does not take an interval parameter because it is assumed to be the length of an entire string, and then only an octave subset of that is taken.
An ELD is not to be confused with [[EDL|EDL, equal division of length]]. The latter term does not take an interval parameter because it is assumed to be the length of an entire string, and then only an octave subset of that is taken.
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! 4
! 4
|-
|-
! frequency (f)
! frequency (''f'', ratio)
|(1)
|(1)
|1.11
|1.11
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|φ
|φ
|-
|-
! pitch (log₂f)
! pitch (log₂''f'', octaves)
|(0)
|(0)
|0.14
|0.14
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|0.69
|0.69
|-
|-
! length (1/f)
! length (1/''f'', ratio)
|(1)
|(1)
|0.90
|0.90
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! 4
! 4
|-
|-
! frequency (f)
! frequency (''f'', ratio)
|(1)
|(1)
|0.87
|0.87
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|1/φ
|1/φ
|-
|-
! pitch (log₂f)
! pitch (log₂''f'', octaves)
|(0)
|(0)
| -0.21
| -0.21
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| -0.69
| -0.69
|-
|-
! length (1/f)
! length (1/''f'', ratio)
|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1
|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1
|1+(1/4)(φ-1) = (1φ + 3)/4
|1+(1/4)(φ-1) = (1φ + 3)/4
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|}
|}


[[Category:Undertone]]
[[Category:Undertone series]]
[[Category:Utonality]]
[[Category:Utonality]]
[[Category:Subharmonic]]
[[Category:Subharmonic]]
[[Category:Subharmonic series‏‎]]
[[Category:Subharmonic series‏‎]]
Retrieved from "https://en.xen.wiki/w/ELD"