OD - Revision history

Cmloegcmluin: /* Examples */ update row headers per agreement at https://en.xen.wiki/w/Talk:APS

2023-10-19T20:35:44Z

Examples: update row headers per agreement at https://en.xen.wiki/w/Talk:APS

← Older revision		Revision as of 20:35, 19 October 2023
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	! 4		! 4
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(4/4)		\|(4/4)
	\|5/4		\|5/4
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	\|8/4		\|8/4
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\|0.32		\|0.32
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	\|1		\|1
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(4/4)		\|(4/4)
	\|4/5		\|4/5

Cmloegcmluin: /* Vs. EFD */ address Flora's recent concerns about rationality vs. irrationality in this system

2023-03-28T21:41:06Z

Vs. EFD: address Flora's recent concerns about rationality vs. irrationality in this system

← Older revision		Revision as of 21:41, 28 March 2023
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	=== Vs. EFD ===		=== 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~~.		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 ===		=== Vs. ADO ===

Cmloegcmluin: fix section capitalization

2021-11-11T19:38:28Z

fix section capitalization

← Older revision		Revision as of 19:38, 11 November 2021
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	== Relationship to other tunings ==		== Relationship to other tunings ==

	=== vs. ED ===		=== 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.		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 ===		=== 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.		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 ===		=== Vs. ADO ===

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

	=== vs. OS ===		=== 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.		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 ===		=== Vs. UD ===

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

Fredg999: Categories

2021-08-30T04:04:01Z

Cmloegcmluin: break up wall of information into helpful sections that are consistent across all arithmetic tuning pages

2021-03-24T20:37:54Z

break up wall of information into helpful sections that are consistent across all arithmetic tuning pages

@@ 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 ==
 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>.
 {| class="wikitable"

Cmloegcmluin: fix typos

2021-03-24T19:12:07Z

fix typos

← Older revision		Revision as of 19:12, 24 March 2021
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	The equivalent utonal version of an OD is a [[UD]].		The equivalent utonal version of an OD is a [[UD]].

	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.		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:		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:

Cmloegcmluin: add formula for mathematician benefit

2021-03-23T22:53:44Z

add formula for mathematician benefit

← Older revision		Revision as of 22:53, 23 March 2021
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	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.		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.

			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>.

	{\| class="wikitable"		{\| class="wikitable"

Cmloegcmluin at 17:57, 23 March 2021

2021-03-23T17:57:07Z

← Older revision		Revision as of 17:57, 23 March 2021
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	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.		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 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.		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.

Cmloegcmluin at 15:26, 23 March 2021

2021-03-23T15:26:31Z

← Older revision		Revision as of 15:26, 23 March 2021
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	The nth [[Overtone scale\|overtone mode, or over-n scale]] is equivalent to n-ODO. So is n-[[ADO]].		The nth [[Overtone scale\|overtone mode, or over-n scale]] is equivalent to n-ODO. So is n-[[ADO]].

			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.

	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.		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.

Cmloegcmluin at 23:40, 22 March 2021

2021-03-22T23:40:28Z

← Older revision		Revision as of 23:40, 22 March 2021
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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 rational interval p.		Its full specification is n-ODp: n otonal divisions of rational interval p.

← Older revision		Revision as of 04:04, 30 August 2021
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	\|}		\|}

	~~[[Category:Overtone]]~~
	~~[[Category:Overtone‏‎ series]]~~
	[[Category:Otonality]]		[[Category:Otonality]]
	[[Category:Harmonic]]		[[Category:Harmonic]]
	[[Category:Harmonic series‏‎]]		[[Category:Harmonic series‏‎]]