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. | An '''OD''', or '''otonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | |||
Its full specification is n-ODp: n otonal divisions of the rational interval p. | |||
== Formula == | |||
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: | 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: | ||
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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>. | 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" | {| class="wikitable" | ||
Revision as of 20:37, 24 March 2021
An OD, or otonal division, is a kind of arithmetic and harmonotonic tuning.
Specification
Its full specification is n-ODp: n otonal divisions of the rational interval p.
Formula
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]\displaystyle{ p }[/math] (or [math]\displaystyle{ \frac p1 }[/math]), if you are going to move arithmetically (by repeated addition) from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ p }[/math], then the difference in frequency space that you are dividing up is not actually [math]\displaystyle{ p }[/math], but [math]\displaystyle{ p - 1 }[/math]. And because you are dividing it into [math]\displaystyle{ n }[/math] parts, each step will have a size of [math]\displaystyle{ \frac{p-1}{n} }[/math]. So, the formula for the frequency of step [math]\displaystyle{ k }[/math] of an n-ODp is:
[math]\displaystyle{ f(k) = 1 + (\frac kn)(p-1) }[/math]
This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ f(k) }[/math] is simply [math]\displaystyle{ 1 }[/math]. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ f(k) }[/math] is simply [math]\displaystyle{ 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, 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 (equal frequency division) is irrational, and therefore its pitches and intervals are all irrational too.
vs. ADO
The nth 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 (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 (utonal sequence).
Examples
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f) | (4/4) | 5/4 | 6/4 | 7/4 | 8/4 |
| pitch (log₂f) | (0) | 0.32 | 0.58 | 0.81 | 1 |
| length (1/f) | (4/4) | 4/5 | 4/6 | 4/7 | 4/8 |