Interval system - Revision history

FloraC: FloraC moved page Interval systems to Interval system without leaving a redirect: Oops, forgot about WP:PLURAL

2023-12-02T10:38:02Z

FloraC moved page Interval systems to Interval system without leaving a redirect: Oops, forgot about WP:PLURAL

← Older revision	Revision as of 10:38, 2 December 2023
(No difference)

FloraC: FloraC moved page Musical Interval Systems to Interval systems without leaving a redirect: "Tuning system" doesn't take the "musical" modifier so this need not either; WP:NCCAPS

2023-12-02T10:37:26Z

FloraC moved page Musical Interval Systems to Interval systems without leaving a redirect: "Tuning system" doesn't take the "musical" modifier so this need not either; WP:NCCAPS

← Older revision	Revision as of 10:37, 2 December 2023
(No difference)

FloraC: Redirected page to Tuning system

2023-12-02T10:36:02Z

Redirected page to Tuning system

← Older revision		Revision as of 10:36, 2 December 2023
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	A ''musical interval system'' is the most general term for describing a "range of notes or musical intervals theoretically available to a composer". The qualification "theoretically" is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system.		#redirect [[Tuning system]]

	A musical interval system can be many different things: a scale, a mode of a scale, a temperament, an equal (or otherwise systematic) division of some interval, a prime-, odd-, or integer-limit form of Just Intonation, or any arbitrarily-defined set of musical intervals to be used in composition. Xenharmony in general deals with musical interval systems of all kinds, provided they are distinct from 12-tone equal temperament. There are many gray areas regarding what sort of musical interval systems qualify as "xenharmonic" or not, and no rigorous definition seems to be universally acceptable among xenharmonists.

	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [http://en.wikipedia.org/wiki/MIDI_Tuning_Standard MIDI tuning standard]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it, as in practice only a finite range of 12EDO is used. From a theoretical point of view, 12EDO has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12EDO semitones.

	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.

	Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.

	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[harmonic limit]]; all intervals in a [[~~Just intonation subgroups\|just intonation subgroup]]; and all intervals in a~~ [~~[Regular Temperaments\|regular temperament]].~~

	~~[[Category:~~Tuning]]

Fredg999 category edits: Moving from Category:Overview to Category:Tuning using Cat-a-lot

2023-03-06T22:14:28Z

Moving from Category:Overview to Category:Tuning using Cat-a-lot

← Older revision		Revision as of 22:14, 6 March 2023
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	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[harmonic limit]]; all intervals in a [[Just intonation subgroups\|just intonation subgroup]]; and all intervals in a [[Regular Temperaments\|regular temperament]].		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[harmonic limit]]; all intervals in a [[Just intonation subgroups\|just intonation subgroup]]; and all intervals in a [[Regular Temperaments\|regular temperament]].

	[[Category:~~Overview~~]]		[[Category:Tuning]]

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T04:36:04Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 04:36, 5 March 2023
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	[[Category:Overview]]		[[Category:Overview]]
	~~[[Category:Theory]]~~

Xenwolf: +cat, links simplified

2020-05-31T22:24:12Z

+cat, links simplified

← Older revision		Revision as of 22:24, 31 May 2020
Line 7:		Line 7:
	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.		Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.

	Among open systems, the most important kinds are [[~~Periodic_scale\|~~periodic ~~scales~~]] and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.		Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.

	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[~~Harmonic_Limit\|~~harmonic limit]]; all intervals in a [[~~Just_intonation_subgroups~~\|just intonation subgroup]]; and all intervals in a [[~~Regular_Temperaments~~\|regular temperament]].		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[harmonic limit]]; all intervals in a [[Just intonation subgroups\|just intonation subgroup]]; and all intervals in a [[Regular Temperaments\|regular temperament]].
	[[Category:~~overview~~]]
			[[Category:Overview]]
			[[Category:Theory]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		A ''musical interval system'' is the most general term for describing a "range of notes or musical intervals theoretically available to a composer". The qualification "theoretically" is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system.
	~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~
	~~: This revision was by author [[User:igliashon\|igliashon]] and made on <tt>2014-06-29 21:02:59 UTC</tt>.<br>~~
	~~: The original revision id was <tt>515233968</tt>.<br>~~
	~~: The revision comment was: <tt></tt><br>~~
	~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~
	~~<h4>Original Wikitext content:</h4>~~
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A //musical interval system// is the most general term for describing a "range of notes or musical intervals theoretically available to a composer". The qualification "theoretically" is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system.

	A musical interval system can be many different things: a scale, a mode of a scale, a temperament, an equal (or otherwise systematic) division of some interval, a prime-, odd-, or integer-limit form of Just Intonation, or any arbitrarily-defined set of musical intervals to be used in composition. Xenharmony in general deals with musical interval systems of all kinds, provided they are distinct from 12-tone equal temperament. There are many gray areas regarding what sort of musical interval systems qualify as "xenharmonic" or not, and no rigorous definition seems to be universally acceptable among xenharmonists.		A musical interval system can be many different things: a scale, a mode of a scale, a temperament, an equal (or otherwise systematic) division of some interval, a prime-, odd-, or integer-limit form of Just Intonation, or any arbitrarily-defined set of musical intervals to be used in composition. Xenharmony in general deals with musical interval systems of all kinds, provided they are distinct from 12-tone equal temperament. There are many gray areas regarding what sort of musical interval systems qualify as "xenharmonic" or not, and no rigorous definition seems to be universally acceptable among xenharmonists.

	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [[http://en.wikipedia.org/wiki/MIDI_Tuning_Standard\|MIDI tuning standard]]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it, as in practice only a finite range of 12EDO is used. From a theoretical point of view, 12EDO has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12EDO semitones.		The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [http://en.wikipedia.org/wiki/MIDI_Tuning_Standard MIDI tuning standard]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it, as in practice only a finite range of 12EDO is used. From a theoretical point of view, 12EDO has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12EDO semitones.

	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.		Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.

	Among open systems, the most important kinds are [[~~Periodic scale~~\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [[http://en.wikipedia.org/wiki/Abelian_group\|abelian groups]], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.		Among open systems, the most important kinds are [[Periodic_scale\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.

	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[~~Harmonic Limit~~\|harmonic limit]]; all intervals in a [[~~Just intonation subgroups~~\|just intonation subgroup]]; and all intervals in a [[~~Regular Temperaments~~\|regular temperament]].~~</pre></div>~~		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[Harmonic_Limit\|harmonic limit]]; all intervals in a [[Just_intonation_subgroups\|just intonation subgroup]]; and all intervals in a [[Regular_Temperaments\|regular temperament]].
	~~<h4>Original HTML content~~:~~</h4>~~		[[Category:overview]]
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Musical Interval Systems</title></head><body>A <em>musical interval system</em> is the most general term for describing a &quot;range of notes or musical intervals theoretically available to a composer&quot;. The qualification &quot;theoretically&quot; is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system.<br />
	~~<br />~~
	A musical interval system can be many different things: a scale, a mode of a scale, a temperament, an equal (or otherwise systematic) division of some interval, a prime-, odd-, or integer-limit form of Just Intonation, or any arbitrarily-defined set of musical intervals to be used in composition. Xenharmony in general deals with musical interval systems of all kinds, provided they are distinct from 12-tone equal temperament. There are many gray areas regarding what sort of musical interval systems qualify as &quot;xenharmonic&quot; or not, and no rigorous definition seems to be universally acceptable among xenharmonists.<br />
	~~<br />~~
	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/MIDI_Tuning_Standard" rel="nofollow">MIDI tuning standard</a>. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it, as in practice only a finite range of 12EDO is used. From a theoretical point of view, 12EDO has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12EDO semitones.<br />
	~~<br />~~
	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.<br />
	~~<br />~~
	Among open systems, the most important kinds are <a class="wiki_link" href="/Periodic%20scale">periodic scales</a> and group systems. The latter refers to &quot;groups&quot; in the mathematical sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian groups</a>, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.<br />
	~~<br />~~
	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given <a class="wiki_link" href="/Harmonic%20Limit">harmonic limit</a>; all intervals in a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>; and all intervals in a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>.</body></html></pre></div>

Wikispaces>igliashon: Imported revision 515233968 - Original comment:

2014-06-29T21:02:59Z

**Imported revision 515233968 - Original comment: **

← Older revision		Revision as of 21:02, 29 June 2014
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:~~keenanpepper~~\|~~keenanpepper~~]] and made on <tt>~~2013~~-07-~~26 20~~:10:54 UTC</tt>.<br>		: This revision was by author [[User:igliashon\|igliashon]] and made on <tt>2014-06-29 21:02:59 UTC</tt>.<br>
	: The original revision id was <tt>~~442482880~~</tt>.<br>		: The original revision id was <tt>515233968</tt>.<br>
	: The revision comment was: <tt>~~how is this relevant at all?~~</tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
	<h4>Original Wikitext content:</h4>		<h4>Original Wikitext content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">By //musical interval system// is ~~meant~~ a range of notes or musical intervals theoretically available to a composer. The qualification "theoretically" is important, as such systems ~~usually include notes~~ or ~~intervals which are~~ not actually ~~audible by human beings~~.		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A //musical interval system// is the most general term for describing a "range of notes or musical intervals theoretically available to a composer". The qualification "theoretically" is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system.

	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [[http://en.wikipedia.org/wiki/MIDI_Tuning_Standard\|MIDI tuning standard]]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, ~~even though~~ you can~~'t hear or~~ even ~~physically produce most~~ of ~~them~~.		A musical interval system can be many different things: a scale, a mode of a scale, a temperament, an equal (or otherwise systematic) division of some interval, a prime-, odd-, or integer-limit form of Just Intonation, or any arbitrarily-defined set of musical intervals to be used in composition. Xenharmony in general deals with musical interval systems of all kinds, provided they are distinct from 12-tone equal temperament. There are many gray areas regarding what sort of musical interval systems qualify as "xenharmonic" or not, and no rigorous definition seems to be universally acceptable among xenharmonists.

			The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [[http://en.wikipedia.org/wiki/MIDI_Tuning_Standard\|MIDI tuning standard]]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it, as in practice only a finite range of 12EDO is used. From a theoretical point of view, 12EDO has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12EDO semitones.

			Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.

	Among open systems, the most important kinds are [[Periodic scale\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [[http://en.wikipedia.org/wiki/Abelian_group\|abelian groups]], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.		Among open systems, the most important kinds are [[Periodic scale\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [[http://en.wikipedia.org/wiki/Abelian_group\|abelian groups]], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.

	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given [[Harmonic Limit\|harmonic limit]], all intervals in a [[Just intonation subgroups\|just intonation subgroup]], and all intervals in a [[Regular Temperaments\|regular temperament]].</pre></div>		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[Harmonic Limit\|harmonic limit]]; all intervals in a [[Just intonation subgroups\|just intonation subgroup]]; and all intervals in a [[Regular Temperaments\|regular temperament]].</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Musical Interval Systems</title></head><body>By <em>musical interval system</em> is ~~meant~~ a range of notes or musical intervals theoretically available to a composer. The qualification &quot;theoretically&quot; is important, as such systems ~~usually include~~ notes or intervals ~~which~~ are ~~not actually audible~~ by ~~human beings~~.<br />		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Musical Interval Systems</title></head><body>A <em>musical interval system</em> is the most general term for describing a &quot;range of notes or musical intervals theoretically available to a composer&quot;. The qualification &quot;theoretically&quot; is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system.<br />
			<br />
			A musical interval system can be many different things: a scale, a mode of a scale, a temperament, an equal (or otherwise systematic) division of some interval, a prime-, odd-, or integer-limit form of Just Intonation, or any arbitrarily-defined set of musical intervals to be used in composition. Xenharmony in general deals with musical interval systems of all kinds, provided they are distinct from 12-tone equal temperament. There are many gray areas regarding what sort of musical interval systems qualify as &quot;xenharmonic&quot; or not, and no rigorous definition seems to be universally acceptable among xenharmonists.<br />
			<br />
			The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/MIDI_Tuning_Standard" rel="nofollow">MIDI tuning standard</a>. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it, as in practice only a finite range of 12EDO is used. From a theoretical point of view, 12EDO has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12EDO semitones.<br />
	<br />		<br />
	~~The most basic~~ of ~~distinctions among such systems is between~~ open ~~and closed systems, where a closed~~ system ~~has~~ a finite ~~set of possible musical~~ intervals, ~~and an open system has an infinite set. An example of a closed system would be all 2097151 notes of~~ the ~~<~~a ~~class="wiki_link_ext" href="http://en.wikipedia.org/wiki/MIDI_Tuning_Standard" rel="nofollow">MIDI tuning standard</a>~~. An example of ~~an open system~~ is ~~12EDO~~, ~~which puts no limit on how high or low the range of tones extends. From~~ a ~~practical point of view MTS is vastly more capable of representing~~ musical intervals ~~than 12EDO,~~ and ~~in fact includes it~~. ~~From a theoretical point~~ of ~~view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most~~ of ~~them~~.<br />		Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.<br />
	<br />		<br />
	Among open systems, the most important kinds are <a class="wiki_link" href="/Periodic%20scale">periodic scales</a> and group systems. The latter refers to &quot;groups&quot; in the mathematical sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian groups</a>, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.<br />		Among open systems, the most important kinds are <a class="wiki_link" href="/Periodic%20scale">periodic scales</a> and group systems. The latter refers to &quot;groups&quot; in the mathematical sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian groups</a>, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.<br />
	<br />		<br />
	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given <a class="wiki_link" href="/Harmonic%20Limit">harmonic limit</a>, all intervals in a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>, and all intervals in a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>.</body></html></pre></div>		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given <a class="wiki_link" href="/Harmonic%20Limit">harmonic limit</a>; all intervals in a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>; and all intervals in a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>.</body></html></pre></div>

Wikispaces>keenanpepper: Imported revision 442482880 - Original comment: how is this relevant at all?

2013-07-26T20:10:54Z

**Imported revision 442482880 - Original comment: how is this relevant at all?**

← Older revision		Revision as of 20:10, 26 July 2013
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:~~lobawad~~\|~~lobawad~~]] and made on <tt>~~2011~~-07-~~04 16~~:53:01 UTC</tt>.<br>		: This revision was by author [[User:keenanpepper\|keenanpepper]] and made on <tt>2013-07-26 20:10:54 UTC</tt>.<br>
	: The original revision id was <tt>~~239955287~~</tt>.<br>		: The original revision id was <tt>442482880</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt>how is this relevant at all?</tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
	<h4>Original Wikitext content:</h4>		<h4>Original Wikitext content:</h4>
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	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [[http://en.wikipedia.org/wiki/MIDI_Tuning_Standard\|MIDI tuning standard]]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.		The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [[http://en.wikipedia.org/wiki/MIDI_Tuning_Standard\|MIDI tuning standard]]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.

	Among open systems, the most important kinds are [[Periodic scale\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [[http://en.wikipedia.org/wiki/Abelian_group\|abelian groups]], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.~~The patterns used in 'Change Ringing' of bells are one example of ways one can go through all the permutations of a set of tones by exchanging only one element at a time~~		Among open systems, the most important kinds are [[Periodic scale\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [[http://en.wikipedia.org/wiki/Abelian_group\|abelian groups]], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.

	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given [[Harmonic Limit\|harmonic limit]], all intervals in a [[Just intonation subgroups\|just intonation subgroup]], and all intervals in a [[Regular Temperaments\|regular temperament]].</pre></div>		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given [[Harmonic Limit\|harmonic limit]], all intervals in a [[Just intonation subgroups\|just intonation subgroup]], and all intervals in a [[Regular Temperaments\|regular temperament]].</pre></div>
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	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/MIDI_Tuning_Standard" rel="nofollow">MIDI tuning standard</a>. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.<br />		The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/MIDI_Tuning_Standard" rel="nofollow">MIDI tuning standard</a>. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.<br />
	<br />		<br />
	Among open systems, the most important kinds are <a class="wiki_link" href="/Periodic%20scale">periodic scales</a> and group systems. The latter refers to &quot;groups&quot; in the mathematical sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian groups</a>, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.~~The patterns used in 'Change Ringing' of bells are one example of ways one can go through all the permutations of a set of tones by exchanging only one element at a time~~<br />		Among open systems, the most important kinds are <a class="wiki_link" href="/Periodic%20scale">periodic scales</a> and group systems. The latter refers to &quot;groups&quot; in the mathematical sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian groups</a>, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.<br />
	<br />		<br />
	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given <a class="wiki_link" href="/Harmonic%20Limit">harmonic limit</a>, all intervals in a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>, and all intervals in a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>.</body></html></pre></div>		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given <a class="wiki_link" href="/Harmonic%20Limit">harmonic limit</a>, all intervals in a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>, and all intervals in a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>.</body></html></pre></div>

Wikispaces>lobawad: Imported revision 239955287 - Original comment:

2011-07-04T16:53:01Z

**Imported revision 239955287 - Original comment: **

← Older revision		Revision as of 16:53, 4 July 2011
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:lobawad\|lobawad]] and made on <tt>2011-07-04 16:52:00 UTC</tt>.<br>		: This revision was by author [[User:lobawad\|lobawad]] and made on <tt>2011-07-04 16:53:01 UTC</tt>.<br>
	: The original revision id was <tt>~~239955209~~</tt>.<br>		: The original revision id was <tt>239955287</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
	<h4>Original Wikitext content:</h4>		<h4>Original Wikitext content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">By //musical interval system// is meant a range of notes or musical intervals theoretically available to a composer. The qualification "theoretically" is important, as such systems usually include notes or intervals which are not actually audible by human beings.		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">By //musical interval system// is meant a range of notes or musical intervals theoretically available to a composer. The qualification "theoretically" is important, as such systems usually include notes or intervals which are not actually audible by human beings.

	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [[http://en.wikipedia.org/wiki/MIDI_Tuning_Standard\|MIDI tuning standard]]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.		The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the [[http://en.wikipedia.org/wiki/MIDI_Tuning_Standard\|MIDI tuning standard]]. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.

	Among open systems, the most important kinds are [[Periodic scale\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [[http://en.wikipedia.org/wiki/Abelian_group\|abelian groups]], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.The patterns used in 'Change Ringing' are one example of ways one can go through all the permutations of a set of tones by exchanging only one element at a time		Among open systems, the most important kinds are [[Periodic scale\|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [[http://en.wikipedia.org/wiki/Abelian_group\|abelian groups]], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.The patterns used in 'Change Ringing' of bells are one example of ways one can go through all the permutations of a set of tones by exchanging only one element at a time

	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given [[Harmonic Limit\|harmonic limit]], all intervals in a [[Just intonation subgroups\|just intonation subgroup]], and all intervals in a [[Regular Temperaments\|regular temperament]].</pre></div>		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given [[Harmonic Limit\|harmonic limit]], all intervals in a [[Just intonation subgroups\|just intonation subgroup]], and all intervals in a [[Regular Temperaments\|regular temperament]].</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Musical Interval Systems</title></head><body>By <em>musical interval system</em> is meant a range of notes or musical intervals theoretically available to a composer. The qualification &quot;theoretically&quot; is important, as such systems usually include notes or intervals which are not actually audible by human beings. <br />		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Musical Interval Systems</title></head><body>By <em>musical interval system</em> is meant a range of notes or musical intervals theoretically available to a composer. The qualification &quot;theoretically&quot; is important, as such systems usually include notes or intervals which are not actually audible by human beings.<br />
	<br />		<br />
	The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/MIDI_Tuning_Standard" rel="nofollow">MIDI tuning standard</a>. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.<br />		The most basic of distinctions among such systems is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/MIDI_Tuning_Standard" rel="nofollow">MIDI tuning standard</a>. An example of an open system is 12EDO, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12EDO, and in fact includes it. From a theoretical point of view, 12EDO has an infinite set of available intervals, even though you can't hear or even physically produce most of them.<br />
	<br />		<br />
	Among open systems, the most important kinds are <a class="wiki_link" href="/Periodic%20scale">periodic scales</a> and group systems. The latter refers to &quot;groups&quot; in the mathematical sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian groups</a>, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.The patterns used in 'Change Ringing' are one example of ways one can go through all the permutations of a set of tones by exchanging only one element at a time<br />		Among open systems, the most important kinds are <a class="wiki_link" href="/Periodic%20scale">periodic scales</a> and group systems. The latter refers to &quot;groups&quot; in the mathematical sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian groups</a>, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.The patterns used in 'Change Ringing' of bells are one example of ways one can go through all the permutations of a set of tones by exchanging only one element at a time<br />
	<br />		<br />
	Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given <a class="wiki_link" href="/Harmonic%20Limit">harmonic limit</a>, all intervals in a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>, and all intervals in a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>.</body></html></pre></div>		Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz, all real numbers under addition, regarded as intervals in cents, all positive rational numbers, regarded as intervals from a chosen 1/1, all rational numbers in a given <a class="wiki_link" href="/Harmonic%20Limit">harmonic limit</a>, all intervals in a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>, and all intervals in a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>.</body></html></pre></div>