OS - Revision history

Cmloegcmluin: /* Vs. OS */

2025-09-29T16:51:30Z

Vs. OS

← Older revision		Revision as of 16:51, 29 September 2025
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	Yet another term for this structure is an [[isoharmonic series]].		Yet another term for this structure is an [[isoharmonic series]].

	=== Vs. OS ===		=== Vs. OD ===

	By specifying n, your OS will be equivalent to some [[OD\|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.		By specifying n, your OS will be equivalent to some [[OD\|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

Cmloegcmluin: /* As shifted overtone series */

2024-09-12T01:31:30Z

As shifted overtone series

← Older revision		Revision as of 01:31, 12 September 2024
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	=== As shifted overtone series ===		=== As shifted overtone series ===

	Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is probably irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].		Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is probably irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].

			Yet another term for this structure is an [[isoharmonic series]].

	=== Vs. OS ===		=== Vs. OS ===

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

2023-10-19T20:35:54Z

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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	! 8		! 8
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(4/4)		\|(4/4)
	\|7/4		\|7/4
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	\|28/4		\|28/4
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\|0.81		\|0.81
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	\|2.81		\|2.81
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(1/1)		\|(1/1)
	\|4/7		\|4/7

BudjarnLambeth: Added to category "xenharmonic series"

2023-04-26T05:17:50Z

Added to category "xenharmonic series"

← Older revision		Revision as of 05:17, 26 April 2023
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	[[Category:Harmonic]]		[[Category:Harmonic]]
	[[Category:Harmonic series‏‎]]		[[Category:Harmonic series‏‎]]
			[[Category:Xenharmonic series]]

Cmloegcmluin: add link

2023-03-30T15:30:36Z

add link

← Older revision		Revision as of 15:30, 30 March 2023
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	== Specification ==		== Specification ==

	Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence.		Its full specification is (n-)OSp: (n pitches of an) [[otonal]] sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence.

	== Formula ==		== Formula ==

Cmloegcmluin: /* Relationship to other tunings */ address Flora's recent concerns about rationality vs. irrationality in this system

2023-03-28T21:41:08Z

Relationship to other tunings: 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. AFS ===		=== Vs. AFS ===

	An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-~~AFSp is irrational~~.		An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-OSp must be rational.

	=== As shifted overtone series ===		=== As shifted overtone series ===

	Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].		Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is probably irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].

	=== Vs. OS ===		=== Vs. OS ===

Cmloegcmluin: fix section capitalization

2021-11-11T19:34:48Z

fix section capitalization

← Older revision		Revision as of 19:34, 11 November 2021
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	== Relationship to other tunings ==		== Relationship to other tunings ==

	=== vs. AFS ===		=== Vs. AFS ===

	An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.		An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.
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	Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].		Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].

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

	By specifying n, your OS will be equivalent to some [[OD\|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.		By specifying n, your OS will be equivalent to some [[OD\|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

	=== vs. US ===		=== Vs. US ===

	The analogous undertone equivalent of an OS is a [[US]].		The analogous undertone equivalent of an OS is a [[US]].

Fredg999: Categories

2021-08-30T04:04:13Z

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

2021-03-24T20:38:04Z

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

← Older revision		Revision as of 20:38, 24 March 2021
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	An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings\|arithmetic]] and [[Harmonotonic tunings\|harmonotonic]] tuning.		An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings\|arithmetic]] and [[Harmonotonic tunings\|harmonotonic]] tuning.

	Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.		== Specification ==

	The "n" is optional. If unspecified, you describe an open-ended sequence~~. By specifying n, your sequence will be equivalent to some [[OD\|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7~~.		Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence.

	~~The analogous undertone equivalent of an OS is a [[US]].~~		== Formula ==

	The formula for step <span><math>k</math></span> of an OSp is:		The formula for step <span><math>k</math></span> of an OSp is:
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	</math>		</math>

	OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].		== Tips ==

	The OSp could be read as "1 out of every p harmonics of the harmonic series" (starting with harmonic 1). So OS2 would give the odd harmonics: 1, 3, 5, 7...		The OSp could be read as "1 out of every p harmonics of the harmonic series" (starting with harmonic 1). So OS2 would give the odd harmonics: 1, 3, 5, 7...
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	For an example combining specifying the numerator and denominator: if you say OS3/4, in other words vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <span><math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math></span>, or in other words, a class iii [[isoharmonic_chords\|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.		For an example combining specifying the numerator and denominator: if you say OS3/4, in other words vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <span><math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math></span>, or in other words, a class iii [[isoharmonic_chords\|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.

			== Relationship to other tunings ==

			=== vs. AFS ===

			An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.

			=== As shifted overtone series ===

			Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].

			=== vs. OS ===

			By specifying n, your OS will be equivalent to some [[OD\|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

			=== vs. US ===

			The analogous undertone equivalent of an OS is a [[US]].

			== Examples ==

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

Cmloegcmluin: add formula for mathematician benefit

2021-03-23T22:52:24Z

add formula for mathematician benefit

← Older revision		Revision as of 22:52, 23 March 2021
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	The analogous undertone equivalent of an OS is a [[US]].		The analogous undertone equivalent of an OS is a [[US]].

			The formula for step <span><math>k</math></span> of an OSp is:

			<math>
			f(k) = 1 + k⋅p
			</math>

	OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].		OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation\|derivation]].

← Older revision		Revision as of 04:04, 30 August 2021
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	So we can see that <span><math>\frac 13</math></span> was the right amount to shift by because it is the delta from the starting position <span><math>1</math></span> to <span><math>\frac 43</math></span>, the latter of which is the reciprocal of the target step size <span><math>\frac 34</math></span> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <span><math>1</math></span> when we resize all steps from 1 to the target step size by multiplying everything by it.		So we can see that <span><math>\frac 13</math></span> was the right amount to shift by because it is the delta from the starting position <span><math>1</math></span> to <span><math>\frac 43</math></span>, the latter of which is the reciprocal of the target step size <span><math>\frac 34</math></span> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <span><math>1</math></span> when we resize all steps from 1 to the target step size by multiplying everything by it.

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