UD: Difference between revisions

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A '''UD''', or '''utonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

~~Its full specification is n-UDp: n utonal divisions of rational interval p. An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps|undertone mode, or under-n scale]].~~

== Specification ==

~~The only difference between~~ n-UDp ~~and [[ELD|~~n~~-ELDp (equal length division)]] is that the p for UD is~~ rational~~, while the~~ p ~~for ELD is irrational.~~

Its full specification is n-UDp: n utonal divisions of rational interval p.

~~Your sequence will be equivalent to some [[US|US (utonal sequence)]]. E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.~~

~~An [[EDL|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n)~~.

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.

== Formula ==

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.

To find the steps for an n-UDp, 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-UDp is:

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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 ED ===

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.

=== vs under-n series ===

An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps|undertone mode, or under-n scale]].

=== vs ELD ===

The only difference between n-UDp and [[ELD|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.

=== vs US ===

A UD will be equivalent to some [[US|US (utonal sequence)]]. E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.

=== vs EDL ===

An [[EDL|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).

== Examples ==

{| class="wikitable"

@@ Line 1: / Line 1: @@
 A '''UD''', or '''utonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Its full specification is n-UDp: n utonal divisions of rational interval p. An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps|undertone mode, or under-n scale]].
+== Specification ==
-The only difference between n-UDp and [[ELD|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.
+Its full specification is n-UDp: n utonal divisions of rational interval p.
-Your sequence will be equivalent to some [[US|US (utonal sequence)]]. E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.
-An [[EDL|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).
-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.
+== Formula ==
-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.
 To find the steps for an n-UDp, 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-UDp is:
@@ Line 20: / Line 14: @@
 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 ED ===
+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.
+=== vs under-n series ===
+An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps|undertone mode, or under-n scale]].
+=== vs ELD ===
+The only difference between n-UDp and [[ELD|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.
+=== vs US ===
+A UD will be equivalent to some [[US|US (utonal sequence)]]. E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.
+=== vs EDL ===
+An [[EDL|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).
+== Examples ==
 {| class="wikitable"