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.

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:

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

<math>

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

</math>

If you want to describe overtones 1-9 ~~with~~ 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 ~~his~~ as an [[OS|OS, or overtone sequence]], by simply saying 8-OS.

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-[[EFD]]p (equal frequency division) is that the p for an OD must be rational.

=== 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 23:

Line 53:

! 4

|-

! frequency (f)

! frequency (''f'', ratio)

|(4/4)

|5/4

Line 30:

Line 60:

|8/4

|-

! pitch (~~log₂f~~)

! pitch (log₂''f'', octaves)

|(0)

|0.32

Line 37:

Line 67:

|1

|-

! length (1/f)

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

|(4/4)

|4/5

Line 45:

Line 75:

|}

~~[[Category:Overtone]]~~

~~[[Category:Overtone‏‎ series]]~~

[[Category:Otonality]]

[[Category:Harmonic]]

[[Category:Harmonic series‏‎]]

@@ 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.
+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:
-The equivalent utonal version of an OD is a [[UD]].
+<math>
+f(k) = 1 + (\frac kn)(p-1)
+</math>
-If you want to describe overtones 1-9 with 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 his as an [[OS|OS, or overtone sequence]], by simply saying 8-OS.
+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-[[EFD]]p (equal frequency division) is that the p for an OD must be rational.
+=== 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 23: / Line 53: @@
 ! 4
 |-
-! frequency (f)
+! frequency (''f'', ratio)
 |(4/4)
 |5/4
@@ Line 30: / Line 60: @@
 |8/4
 |-
-! pitch (log₂f)
+! pitch (log₂''f'', octaves)
 |(0)
 |0.32
@@ Line 37: / Line 67: @@
 |1
 |-
-! length (1/f)
+! length (1/''f'', ratio)
 |(4/4)
 |4/5
@@ Line 45: / Line 75: @@
 |}
-[[Category:Overtone]]
-[[Category:Overtone‏‎ series]]
 [[Category:Otonality]]
 [[Category:Harmonic]]
 [[Category:Harmonic series‏‎]]