ELD: Difference between revisions

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

~~A UD is a specific (rational) type of ELD.~~

== Specification ==

~~So far we~~'~~ve looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having~~ equal ~~step sizes~~ of ~~some kind of quantity (frequency, pitch~~, or length), ~~then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind~~ of ~~quantity~~.

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'').

The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).

== Formula ==

~~But~~ it is ~~also~~ possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by ~~'''frequency''', or~~ '''length'''. ~~In the former case, you will have 12-EFDO, and in the latter case, 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 ~~EFD and~~ ELD are typically reserved for irrational tunings, such as 12-~~EFDφ~~. So it would be more appropriate to name ~~these two tunings 12-ODO and~~ 12-UDO, for ~~otonal~~ divisions of the octave ~~and~~ utonal ~~divisions~~ of the octave~~, respectively~~.

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>

L(k) = 1 + (\frac kn)(p-1)

</math>

This way, when <math>k</math> is <math>0</math>, <math>L(k)</math> is simply <math>1</math>. And when <math>k</math> is <math>n</math>, <math>L(k)</math> is simply <math>1 + (p-1) = p</math>.

== Tip about tunings based on length ==

Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

== Relationship to other tunings ==

=== 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.

=== Vs. UD ===

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 ===

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

=== Vs. ALS ===

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 ===

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.

== Examples ==

{| class="wikitable"

|+example: 4-ELDφ

''(arranged so that the pitches are in ascending order and still begin on 1/1)''

|-

! quantity

Line 19:

Line 54:

! 4

|-

! frequency (f)

! frequency (''f'', ratio)

|(1)

|1.11

Line 26:

Line 61:

|φ

|-

! pitch (~~log₂f~~)

! pitch (log₂''f'', octaves)

|(0)

|0.14

Line 33:

Line 68:

|0.69

|-

! length (1/f)

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

|(1)

|0.90

Line 39:

Line 74:

|0.71

|1/φ

|}

{| class="wikitable"

|+example: 4-ELDφ

''(descending pitches)''

|-

! quantity

! (0)

! 1

! 2

! 3

! 4

|-

! frequency (''f'', ratio)

|(1)

|0.87

|0.76

|0.68

|1/φ

|-

! pitch (log₂''f'', octaves)

|(0)

| -0.21

| -0.39

| -0.55

| -0.69

|-

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

|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1

|1+(1/4)(φ-1) = (1φ + 3)/4

|1+(2/4)(φ-1) = (2φ + 2)/4

|1+(3/4)(φ-1) = (3φ + 1)/4

|1+(4/4)(φ-1) = (4φ + 0)/4 = φ

|}

~~== 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.

~~[[Category:Undertone]]~~

~~[[Category:Undertone series]]~~

[[Category:Utonality]]

[[Category:Subharmonic]]

[[Category:Subharmonic series‏‎]]

@@ Line 1: / Line 1: @@
-An '''ELD''', or '''equal length division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] 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.
-A UD is a specific (rational) type of ELD.
+== Specification ==
-So far we've looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.
+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'').
-The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
+== Formula ==
-But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, 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 EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.
+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>
+L(k) = 1 + (\frac kn)(p-1)
+</math>
+This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>L(k)</math></span> is simply <span><math>1</math></span>. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>L(k)</math></span> is simply <span><math>1 + (p-1) = p</math></span>.
+== Tip about tunings based on length ==
+Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.
+== Relationship to other tunings ==
+=== 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.
+=== Vs. UD ===
+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 ===
+The analogous otonal equivalent of an ELD is an [[EFD|EFD (equal frequency division)]].
+=== Vs. ALS ===
+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 ===
+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.
+== Examples ==
 {| class="wikitable"
 |+example: 4-ELDφ
+''(arranged so that the pitches are in ascending order and still begin on 1/1)''
 |-
 ! quantity
@@ Line 19: / Line 54: @@
 ! 4
 |-
-! frequency (f)
+! frequency (''f'', ratio)
 |(1)
 |1.11
@@ Line 26: / Line 61: @@
 |φ
 |-
-! pitch (log₂f)
+! pitch (log₂''f'', octaves)
 |(0)
 |0.14
@@ Line 33: / Line 68: @@
 |0.69
 |-
-! length (1/f)
+! length (1/''f'', ratio)
 |(1)
 |0.90
@@ Line 39: / Line 74: @@
 |0.71
 |1/φ
+|}
+{| class="wikitable"
+|+example: 4-ELDφ
+''(descending pitches)''
+|-
+! quantity
+! (0)
+! 1
+! 2
+! 3
+! 4
+|-
+! frequency (''f'', ratio)
+|(1)
+|0.87
+|0.76
+|0.68
+|1/φ
+|-
+! pitch (log₂''f'', octaves)
+|(0)
+| -0.21
+| -0.39
+| -0.55
+| -0.69
+|-
+! length (1/''f'', ratio)
+|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1
+|1+(1/4)(φ-1) = (1φ + 3)/4
+|1+(2/4)(φ-1) = (2φ + 2)/4
+|1+(3/4)(φ-1) = (3φ + 1)/4
+|1+(4/4)(φ-1) = (4φ + 0)/4 = φ
 |}
-== 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.
-[[Category:Undertone]]
-[[Category:Undertone series]]
 [[Category:Utonality]]
 [[Category:Subharmonic]]
 [[Category:Subharmonic series‏‎]]