OD: Difference between revisions

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-An '''OD''', or '''otonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
+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 interval p.
+== Specification ==
-The nth [[Overtone scale|overtone mode, or over-n scale]] is equivalent to n-ODO. So is n-[[ADO]].
+Its full specification is n-ODp: n otonal divisions of the rational interval p.
-An OD is a specific (rational) type of [[EFD|EFD, or equal frequency division]].
+== Formula ==
-note there's a kinda tricky aspect which is that if you just want overtones 1-9 you need 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.
+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:
+<math>
+f(k) = 1 + (\frac kn)(p-1)
+</math>
+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 13: / Line 47: @@
 |-
 ! quantity
+! (0)
 ! 1
 ! 2
 ! 3
 ! 4
-! (5)
 |-
-! frequency (f)
+! frequency (''f'', ratio)
-|4/4
+|(4/4)
 |5/4
 |6/4
 |7/4
-|(8/4)
+|8/4
 |-
-! pitch (log₂f)
+! pitch (log₂''f'', octaves)
-|0.00
+|(0)
 |0.32
 |0.58
 |0.81
-|(1.00)
+|1
 |-
-! length (1/f)
+! length (1/''f'', ratio)
-|4/4
+|(4/4)
 |4/5
 |4/6
 |4/7
-|(4/8)
+|4/8
 |}
-[[Category:Overtone]]
-[[Category:Overtone‏‎ series]]
 [[Category:Otonality]]
 [[Category:Harmonic]]
 [[Category:Harmonic series‏‎]]