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.

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

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

~~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~~.

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

~~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~~).

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

~~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.

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 otonal divisions of the octave and utonal divisions of the octave, respectively.

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 An '''ELD''', or '''equal length division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-A UD is a specific (rational) type of ELD.
+Its full specification is n-ELDp: n equal length divisions of the irrational interval p. 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.
-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.
+The analogous otonal equivalent of an ELD is an [[EFD|EFD (equal frequency division)]].
-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).
+An ELD will be equivalent to some [[ALS|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.
-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.
+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 otonal divisions of the octave and utonal divisions of the octave, respectively.
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