OD: Difference between revisions

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An '''OD''', or '''otonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

~~Its full specification is n-ODp: n otonal divisions of rational interval p.~~

== Specification ==

~~The only difference between~~ n-ODp ~~and~~ n~~-EFDp is that~~ the p ~~for an [[EFD]] is irrational~~.

Its full specification is n-ODp: n otonal divisions of the rational interval p.

~~The nth [[Overtone scale|overtone mode, or over-n scale]] is equivalent to n-ODO. So is n-[[ADO]].~~

== Formula ==

~~Your sequence will be equivalent to some [[OS|OS (otonal sequence)]]. E.g. 8-OD7~~ = ~~8-OS3/4, because to get from 1 to 7 you cover 6 overtones, and 6 divided by 8 is 3/4.~~

~~The equivalent utonal version of an OD is a [[UD]].~~

If you want to describe overtones 1-9 as an OD you would need to use 8-OD9, because there are only 8 steps from 1 to 9. You could think of it like 9 is the 8th overtone, so you're really dividing 8 by 8. You're dividing the number of overtones. Alternatively, you could describe this as an [[OS|OS, or overtone sequence]], by simply saying 8-OS.

To find the steps for an n-ODp, begin by recognizing that while the multiplicative interval relating your root position to the end position 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 frequency space that you are dividing up is not actually <math>p</math>, but <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 frequency of step <math>k</math> of an n-ODp is:

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

== Tips ==

If you want to describe overtones 1-9 as an OD you would need to use 8-OD9, because there are only 8 steps from 1 to 9. You could think of it like 9 is the 8th overtone, so you're really dividing 8 by 8. You're dividing the number of overtones. Alternatively, you could describe this as an [[OS|OS, or overtone sequence]], by simply saying 8-OS.

== Relationship to other tunings ==

=== vs. ED ===

It is possible to — instead of equally dividing the octave in 12 equal parts by pitch, or 12-EDO — divide it into 12 equal parts by length. You will have 12-UDO.

=== vs. EFD ===

The only difference between n-ODp and n-EFDp is that the p for an [[EFD|EFD (equal frequency division)]] is irrational, and therefore its pitches and intervals are all irrational too.

=== vs. ADO ===

The nth [[Overtone scale|overtone mode, or over-n scale]] is equivalent to n-ODO. So is n-[[ADO]].

=== vs. OS ===

Any ODO will be equivalent to some [[OS|OS (otonal sequence)]]. E.g. 8-OD7 = 8-OS3/4, because to get from 1 to 7 you cover 6 overtones, and 6 divided by 8 is 3/4.

=== vs. UD ===

The equivalent utonal version of an OD is a [[UD|UD (utonal sequence)]].

== Examples ==

{| class="wikitable"

@@ Line 1: / Line 1: @@
 An '''OD''', or '''otonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Its full specification is n-ODp: n otonal divisions of rational interval p.
+== Specification ==
-The only difference between n-ODp and n-EFDp is that the p for an [[EFD]] is irrational.
+Its full specification is n-ODp: n otonal divisions of the rational interval p.
-The nth [[Overtone scale|overtone mode, or over-n scale]] is equivalent to n-ODO. So is n-[[ADO]].
+== Formula ==
-Your sequence will be equivalent to some [[OS|OS (otonal sequence)]]. E.g. 8-OD7 = 8-OS3/4, because to get from 1 to 7 you cover 6 overtones, and 6 divided by 8 is 3/4.
-The equivalent utonal version of an OD is a [[UD]].
-If you want to describe overtones 1-9 as an OD you would need to use 8-OD9, because there are only 8 steps from 1 to 9. You could think of it like 9 is the 8th overtone, so you're really dividing 8 by 8. You're dividing the number of overtones. Alternatively, you could describe this as an [[OS|OS, or overtone sequence]], by simply saying 8-OS.
 To find the steps for an n-ODp, begin by recognizing that while the multiplicative interval relating your root position to the end position 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 frequency space that you are dividing up is not actually <span><math>p</math></span>, but <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 frequency of step <span><math>k</math></span> of an n-ODp is:
@@ Line 20: / Line 14: @@
 This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>f(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>f(k)</math></span> is simply <span><math>1 + (p-1) = p</math></span>.
+== Tips ==
+If you want to describe overtones 1-9 as an OD you would need to use 8-OD9, because there are only 8 steps from 1 to 9. You could think of it like 9 is the 8th overtone, so you're really dividing 8 by 8. You're dividing the number of overtones. Alternatively, you could describe this as an [[OS|OS, or overtone sequence]], by simply saying 8-OS.
+== Relationship to other tunings ==
+=== vs. ED ===
+It is possible to — instead of equally dividing the octave in 12 equal parts by pitch, or 12-EDO — divide it into 12 equal parts by length. You will have 12-UDO.
+=== vs. EFD ===
+The only difference between n-ODp and n-EFDp is that the p for an [[EFD|EFD (equal frequency division)]] is irrational, and therefore its pitches and intervals are all irrational too.
+=== vs. ADO ===
+The nth [[Overtone scale|overtone mode, or over-n scale]] is equivalent to n-ODO. So is n-[[ADO]].
+=== vs. OS ===
+Any ODO will be equivalent to some [[OS|OS (otonal sequence)]]. E.g. 8-OD7 = 8-OS3/4, because to get from 1 to 7 you cover 6 overtones, and 6 divided by 8 is 3/4.
+=== vs. UD ===
+The equivalent utonal version of an OD is a [[UD|UD (utonal sequence)]].
+== Examples ==
 {| class="wikitable"