Arithmetic tuning - Revision history

Cmloegcmluin: /* See also */ remove link to AID because Shaahin's pages are below the level of pedagogical clarity and conceptual coherency that this page should link to; and the link to the arithmetic mean is outdated because I have since realized the inherent link between means and progressions

2023-04-19T16:23:11Z

See also: remove link to AID because Shaahin's pages are below the level of pedagogical clarity and conceptual coherency that this page should link to; and the link to the arithmetic mean is outdated because I have since realized the inherent link between means and progressions

← Older revision		Revision as of 16:23, 19 April 2023
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	It is convenient that the three basic types of divisions — OD, ED, and UD — all begin with vowels. As do the three rational types of sequences — OS, AS, and US.		It is convenient that the three basic types of divisions — OD, ED, and UD — all begin with vowels. As do the three rational types of sequences — OS, AS, and US.

	~~== See also ==~~

	* [[Arithmetic mean]]: uses the word "arithmetic" in a somewhat different sense than it is used in "arithmetic sequence", though in both cases it refers to addition
	* [[AID\|AID (Arithmetic irrational divisions]])

	[[Category:Otonality and utonality]]		[[Category:Otonality and utonality]]

BudjarnLambeth: /* See also */

2023-04-19T08:18:16Z

BudjarnLambeth: /* See also */

2023-04-19T08:09:22Z

Cmloegcmluin: /* Types */ the link should use the same terminology as the rest of the bullet point (and page as whole); if others prefer a link to ODO in addition to the existing link to EDO, then in order to achieve parallelism we should also have a link to UDO

2023-04-18T15:24:49Z

Types: the link should use the same terminology as the rest of the bullet point (and page as whole); if others prefer a link to ODO in addition to the existing link to EDO, then in order to achieve parallelism we should also have a link to UDO

← Older revision		Revision as of 15:24, 18 April 2023
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	Equal frequency steps:		Equal frequency steps:
	* [[OD\|OD, otonal division]] (such as an [[~~AFDO~~]])		* [[OD\|OD, otonal division]] (such as an [[ODO]])
	* [[EFD\|EFD, equal frequency division]]		* [[EFD\|EFD, equal frequency division]]
	* [[OS\|OS, otonal sequence]]		* [[OS\|OS, otonal sequence]]
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	Equal length steps:		Equal length steps:
	* [[UD\|UD, utonal division]]		* [[UD\|UD, utonal division]] (such as a [[UDO]])
	* [[ELD\|ELD, equal length division]]		* [[ELD\|ELD, equal length division]]
	* [[US\|US, utonal sequence]]		* [[US\|US, utonal sequence]]

BudjarnLambeth: Added the words "(such as an AFDO)"

2023-04-18T09:11:31Z

Added the words "(such as an AFDO)"

← Older revision		Revision as of 09:11, 18 April 2023
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	Equal frequency steps:		Equal frequency steps:
	* [[OD\|OD, otonal division]]		* [[OD\|OD, otonal division]] (such as an [[AFDO]])
	* [[EFD\|EFD, equal frequency division]]		* [[EFD\|EFD, equal frequency division]]
	* [[OS\|OS, otonal sequence]]		* [[OS\|OS, otonal sequence]]

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

2023-03-28T21:42:11Z

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

← Older revision		Revision as of 21:42, 28 March 2023
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	Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, 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.		Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, 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.

	If the new step size is irrational, the tuning is no longer JI, so we ~~use~~ a ~~different~~ acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ.		If the new step size is irrational, the tuning is no longer JI, so we fall back to a more general acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ. Because different acronyms are used to distinguish rational (JI) tunings from general tunings which include irrational (non-JI) tunings, while the acronyms used for general tunings technically include the JI tunings, these general acronyms are more useful when reserved for non-JI tunings, and this is what is typically done. So when "irrational" is used on this page, it more accurately means "probably irrational".

	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 of OS]].		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 of OS]].

Cmloegcmluin: /* Types */ reorder, for better parallelism within lists

2023-03-28T21:26:12Z

Types: reorder, for better parallelism within lists

← Older revision		Revision as of 21:26, 28 March 2023
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	Equal pitch steps:		Equal pitch steps:
	* [[EPD\|EPD, or ED, equal (pitch) division]] (such as an [[EDO]])		* [[EPD\|EPD, or ED, equal (pitch) division]] (such as an [[EDO]])
			* [[AS\|AS, ambitonal sequence]]
	* [[APS\|APS, arithmetic pitch sequence]]		* [[APS\|APS, arithmetic pitch sequence]]
	* [[AS\|AS, ambitonal sequence]]

	Equal length steps:		Equal length steps:
	* [[UD\|UD, utonal division]]		* [[UD\|UD, utonal division]]
	* [[ELD\|ELD, equal length division]]		* [[ELD\|ELD, equal length division]]
			* [[US\|US, utonal sequence]]
	* [[ALS\|ALS, arithmetic length sequence]]		* [[ALS\|ALS, arithmetic length sequence]]
	* [[US\|US, utonal sequence]]

	== Basic examples ==		== Basic examples ==

Cmloegcmluin: the typical reader will not share this special confusion with arithmetic mean, so I have moved that to a see also at the end, and instead clarified what arithmetic means in this context

2023-03-03T16:34:25Z

the typical reader will not share this special confusion with arithmetic mean, so I have moved that to a see also at the end, and instead clarified what arithmetic means in this context

← Older revision		Revision as of 16:34, 3 March 2023
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	An '''arithmetic tuning''' is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound). This ''arithmetic~~'' does not refer to measurements in~~ the ~~arithmetic scale, as in~~ [[arithmetic ~~mean~~]], ~~but any arithmetic-like scale~~.		An '''arithmetic tuning''' is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound).

			This sense of the word arithmetic comes from the mathematical concept of an [[Wikipedia:Arithmetic sequence\|arithmetic sequence]], which means a sequence with a constant difference between successive terms.

	All arithmetic tunings are [[harmonotonic tunings]].		All arithmetic tunings are [[harmonotonic tunings]].
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	It is convenient that the three basic types of divisions — OD, ED, and UD — all begin with vowels. As do the three rational types of sequences — OS, AS, and US.		It is convenient that the three basic types of divisions — OD, ED, and UD — all begin with vowels. As do the three rational types of sequences — OS, AS, and US.

			== See also ==

			* [[Arithmetic mean]]: uses the word "arithmetic" in a somewhat different sense than it is used in "arithmetic sequence", though in both cases it refers to addition

	[[Category:Otonality and utonality]]		[[Category:Otonality and utonality]]

FloraC: Clarify the term; heading levels and other styles

2023-03-03T11:40:37Z

Clarify the term; heading levels and other styles

← Older revision		Revision as of 11:40, 3 March 2023
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	An arithmetic tuning is one which has equal step sizes ''of any kind of quantity'', whether that be ~~'''~~pitch~~'''~~, ~~'''~~frequency~~'''~~, or ~~'''~~length~~'''~~ (of the resonating entity producing the sound).		An '''arithmetic tuning''' is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound). This ''arithmetic'' does not refer to measurements in the arithmetic scale, as in [[arithmetic mean]], but any arithmetic-like scale.

	All arithmetic tunings are [[harmonotonic tunings]].		All arithmetic tunings are [[harmonotonic tunings]].

	=== Types ===		== Types ==

	Equal frequency steps:		Equal frequency steps:
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	* [[US\|US, utonal sequence]]		* [[US\|US, utonal sequence]]

	=== Basic examples ===		== Basic examples ==

	Basic examples of arithmetic tunings:		Basic examples of arithmetic tunings:

	# the '''overtone''' series has equal steps of '''frequency''' (1, 2, 3, 4, etc.; adding 1 each step)		# the ''overtone'' series has equal steps of ''frequency'' (1, 2, 3, 4, etc.; adding 1 each step)
	# any ''~~'EDO'~~'' has equal steps of '''pitch''' (~~12-EDO~~ goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)		# any ''edo'' has equal steps of ''pitch'' (12edo goes 0\12, 1\12, 2\12, 3\12, etc.; adding 1\12 each step)
	# the '''undertone''' series has equal steps of '''length''' (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)		# the ''undertone'' series has equal steps of ''length'' (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)

	=== Sequences ===		== Sequences ==
	Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, 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.		Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, 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.

	If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ.		If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ.
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	Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.		Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

	=== Divisions ===		== Divisions ==
	So far we~~'ve~~ looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.		So far we have looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.

	The most common example of this type of tuning is ~~12-EDO~~, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this ~~12-EPDO~~, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why ~~12-EDO~~ is the better name).		The most common example of this type of tuning is 12edo, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12epdo, for 12 equal ''pitch'' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12edo is the better name).

	But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have ~~12-EFDO~~, and in the latter case, you will have ~~12-ELDO~~. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as ~~12-EFDφ~~. So it would be more appropriate to name these two tunings ~~12-ODO~~ and ~~12-UDO~~, for otonal divisions of the octave and utonal divisions of the octave, respectively.		But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by ''frequency'', or ''length''. In the former case, you will have 12efdo, and in the latter case, you will have 12eldo. However, that is not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12efdφ. So it would be more appropriate to name these two tunings 12odo and 12udo, for otonal divisions of the octave and utonal divisions of the octave, respectively.

	=== Comparing arithmetic tunings ===		== Comparing arithmetic tunings ==

	We can state a few helpful analogies:		We can state a few helpful analogies:
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	[[Category:Subharmonic series‏‎]]		[[Category:Subharmonic series‏‎]]
	[[Category:Equal-step tuning‏‎]]		[[Category:Equal-step tuning‏‎]]
	~~[[Category:Equal divisions of the octave‏‎]]~~

FloraC: FloraC moved page Arithmetic tunings to Arithmetic tuning: WP:PLURAL

2023-02-14T19:40:55Z

FloraC moved page Arithmetic tunings to Arithmetic tuning: WP:PLURAL

← Older revision	Revision as of 19:40, 14 February 2023
(No difference)

← Older revision		Revision as of 15:24, 18 April 2023
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	Equal frequency steps:		Equal frequency steps:
	* [[OD\|OD, otonal division]] (such as an [[~~AFDO~~]])		* [[OD\|OD, otonal division]] (such as an [[ODO]])
	* [[EFD\|EFD, equal frequency division]]		* [[EFD\|EFD, equal frequency division]]
	* [[OS\|OS, otonal sequence]]		* [[OS\|OS, otonal sequence]]
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	Equal length steps:		Equal length steps:
	* [[UD\|UD, utonal division]]		* [[UD\|UD, utonal division]] (such as a [[UDO]])
	* [[ELD\|ELD, equal length division]]		* [[ELD\|ELD, equal length division]]
	* [[US\|US, utonal sequence]]		* [[US\|US, utonal sequence]]

← Older revision		Revision as of 21:26, 28 March 2023
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	Equal pitch steps:		Equal pitch steps:
	* [[EPD\|EPD, or ED, equal (pitch) division]] (such as an [[EDO]])		* [[EPD\|EPD, or ED, equal (pitch) division]] (such as an [[EDO]])
			* [[AS\|AS, ambitonal sequence]]
	* [[APS\|APS, arithmetic pitch sequence]]		* [[APS\|APS, arithmetic pitch sequence]]
	* [[AS\|AS, ambitonal sequence]]

	Equal length steps:		Equal length steps:
	* [[UD\|UD, utonal division]]		* [[UD\|UD, utonal division]]
	* [[ELD\|ELD, equal length division]]		* [[ELD\|ELD, equal length division]]
			* [[US\|US, utonal sequence]]
	* [[ALS\|ALS, arithmetic length sequence]]		* [[ALS\|ALS, arithmetic length sequence]]
	* [[US\|US, utonal sequence]]

	== Basic examples ==		== Basic examples ==

← Older revision		Revision as of 08:18, 19 April 2023
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	* [[Arithmetic mean]]: uses the word "arithmetic" in a somewhat different sense than it is used in "arithmetic sequence", though in both cases it refers to addition		* [[Arithmetic mean]]: uses the word "arithmetic" in a somewhat different sense than it is used in "arithmetic sequence", though in both cases it refers to addition
	* [[AID]]		* [[AID\|AID (Arithmetic irrational divisions]])

	[[Category:Otonality and utonality]]		[[Category:Otonality and utonality]]

← Older revision		Revision as of 08:09, 19 April 2023
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	* [[Arithmetic mean]]: uses the word "arithmetic" in a somewhat different sense than it is used in "arithmetic sequence", though in both cases it refers to addition		* [[Arithmetic mean]]: uses the word "arithmetic" in a somewhat different sense than it is used in "arithmetic sequence", though in both cases it refers to addition
			* [[AID]]

	[[Category:Otonality and utonality]]		[[Category:Otonality and utonality]]