Keenan Pepper's explanation of vals - Revision history

ArrowHead294 at 02:27, 31 August 2024

2024-08-31T02:27:43Z

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

2023-03-06T21:52:06Z

Removing from Category:Definition using Cat-a-lot

← Older revision		Revision as of 21:52, 6 March 2023
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	Any piece of music can be performed without "comma issues" in any temperament in which all the appropriate commas vanish. It's okay if more vanish unnecessarily, but never less, or else issues arise. Some examples:		Any piece of music can be performed without "comma issues" in any temperament in which all the appropriate commas vanish. It's okay if more vanish unnecessarily, but never less, or else issues arise. Some examples:

	<ul><li>A piece written in 12edo (assuming the patent val because it's the only one that makes musical sense), that actually takes advantage of more than one comma vanishing, cannot be played in any significantly different temperament without arbitrarily modifying the music. The octave stretch can be changed, for example, but it has to be a 12-note circulating temperament, with 12 roughly equal steps, or else some of the puns / comma pumps will not work.</li><li>On the other hand, a piece written in meantone temperament (for example a piece written in conventional notation with everything "spelled" properly and no enharmonic puns) can not only be played in 12edo, but also in 19edo or 26edo or 31edo, or an infinite number of other equal temperaments in which 81/80 vanishes. It could even be performed in 17edo using the 17c val - it will sound quite different, but everything will still work out in a logical way. The same goes for 7edo - major and minor triads both become neutral triads, but that applies in a uniform way to all chord progressions and everything still "works".</li><li>However, a piece written in meantone temperament cannot be performed in a non-meantone temperament like 22edo (using the patent val), or for that matter JI, without serious comma issues arising. It's possible to do it, but you'd basically be writing a new version of the piece from scratch rather than mechanically and faithfully translating it. You could play it in 22edo using a much worse val (<22 35 52\| rather than <22 35 51\|), but the chords would be far from the most accurate approximations that 22edo offers and would sound unnecessarily out-of-tune.</li><li>A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course...</li></ul> ~~[[Category:definition]]~~		<ul><li>A piece written in 12edo (assuming the patent val because it's the only one that makes musical sense), that actually takes advantage of more than one comma vanishing, cannot be played in any significantly different temperament without arbitrarily modifying the music. The octave stretch can be changed, for example, but it has to be a 12-note circulating temperament, with 12 roughly equal steps, or else some of the puns / comma pumps will not work.</li><li>On the other hand, a piece written in meantone temperament (for example a piece written in conventional notation with everything "spelled" properly and no enharmonic puns) can not only be played in 12edo, but also in 19edo or 26edo or 31edo, or an infinite number of other equal temperaments in which 81/80 vanishes. It could even be performed in 17edo using the 17c val - it will sound quite different, but everything will still work out in a logical way. The same goes for 7edo - major and minor triads both become neutral triads, but that applies in a uniform way to all chord progressions and everything still "works".</li><li>However, a piece written in meantone temperament cannot be performed in a non-meantone temperament like 22edo (using the patent val), or for that matter JI, without serious comma issues arising. It's possible to do it, but you'd basically be writing a new version of the piece from scratch rather than mechanically and faithfully translating it. You could play it in 22edo using a much worse val (<22 35 52\| rather than <22 35 51\|), but the chords would be far from the most accurate approximations that 22edo offers and would sound unnecessarily out-of-tune.</li><li>A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course...</li></ul> [[Category:Guides]]
	[[Category:Guides]]
	[[Category:val]]		[[Category:val]]

Fredg999 category edits: Moving from Category:Practical help to Category:Guides using Cat-a-lot

2023-02-21T05:49:23Z

Moving from Category:Practical help to Category:Guides using Cat-a-lot

← Older revision		Revision as of 05:49, 21 February 2023
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	<ul><li>A piece written in 12edo (assuming the patent val because it's the only one that makes musical sense), that actually takes advantage of more than one comma vanishing, cannot be played in any significantly different temperament without arbitrarily modifying the music. The octave stretch can be changed, for example, but it has to be a 12-note circulating temperament, with 12 roughly equal steps, or else some of the puns / comma pumps will not work.</li><li>On the other hand, a piece written in meantone temperament (for example a piece written in conventional notation with everything "spelled" properly and no enharmonic puns) can not only be played in 12edo, but also in 19edo or 26edo or 31edo, or an infinite number of other equal temperaments in which 81/80 vanishes. It could even be performed in 17edo using the 17c val - it will sound quite different, but everything will still work out in a logical way. The same goes for 7edo - major and minor triads both become neutral triads, but that applies in a uniform way to all chord progressions and everything still "works".</li><li>However, a piece written in meantone temperament cannot be performed in a non-meantone temperament like 22edo (using the patent val), or for that matter JI, without serious comma issues arising. It's possible to do it, but you'd basically be writing a new version of the piece from scratch rather than mechanically and faithfully translating it. You could play it in 22edo using a much worse val (<22 35 52\| rather than <22 35 51\|), but the chords would be far from the most accurate approximations that 22edo offers and would sound unnecessarily out-of-tune.</li><li>A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course...</li></ul> [[Category:definition]]		<ul><li>A piece written in 12edo (assuming the patent val because it's the only one that makes musical sense), that actually takes advantage of more than one comma vanishing, cannot be played in any significantly different temperament without arbitrarily modifying the music. The octave stretch can be changed, for example, but it has to be a 12-note circulating temperament, with 12 roughly equal steps, or else some of the puns / comma pumps will not work.</li><li>On the other hand, a piece written in meantone temperament (for example a piece written in conventional notation with everything "spelled" properly and no enharmonic puns) can not only be played in 12edo, but also in 19edo or 26edo or 31edo, or an infinite number of other equal temperaments in which 81/80 vanishes. It could even be performed in 17edo using the 17c val - it will sound quite different, but everything will still work out in a logical way. The same goes for 7edo - major and minor triads both become neutral triads, but that applies in a uniform way to all chord progressions and everything still "works".</li><li>However, a piece written in meantone temperament cannot be performed in a non-meantone temperament like 22edo (using the patent val), or for that matter JI, without serious comma issues arising. It's possible to do it, but you'd basically be writing a new version of the piece from scratch rather than mechanically and faithfully translating it. You could play it in 22edo using a much worse val (<22 35 52\| rather than <22 35 51\|), but the chords would be far from the most accurate approximations that 22edo offers and would sound unnecessarily out-of-tune.</li><li>A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course...</li></ul> [[Category:definition]]
	[[Category:~~practical_help~~]]		[[Category:Guides]]
	[[Category:val]]		[[Category:val]]

Cmloegcmluin: link to new page for Supports

2022-01-25T19:02:45Z

link to new page for Supports

← Older revision		Revision as of 19:02, 25 January 2022
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	A specific equal temperament (as opposed to an EDO) is represented by a single val. There are infinitely many vals, and therefore technically speaking infinitely many equal temperaments, corresponding to each EDO, but usually only a handful are anywhere near good enough to make sense musically. For example, <12 18 27\| is a different 5-limit temperament with 12 equal steps to the octave, which maps 3/2 to 600 cents, and 5/4 to 300 cents, so it's musically useless. But in some cases there are different equal temperaments with the same number of steps that are about equally good. For example, 17edo could be used as two different equal temperaments in the 5-limit: one corresponds to the patent val <17 27 39\|, in which 5/4 is very flat at 353 cents, and one corresponds to the <17 27 40\| val (sometimes called "17c"), in which 5/4 is very sharp at 424 cents. Although these could be realized musically as exactly the same notes (17 equal divisions of 2/1), they are nevertheless different temperaments, because a 4:5:6 chord, for example, would be represented differently.		A specific equal temperament (as opposed to an EDO) is represented by a single val. There are infinitely many vals, and therefore technically speaking infinitely many equal temperaments, corresponding to each EDO, but usually only a handful are anywhere near good enough to make sense musically. For example, <12 18 27\| is a different 5-limit temperament with 12 equal steps to the octave, which maps 3/2 to 600 cents, and 5/4 to 300 cents, so it's musically useless. But in some cases there are different equal temperaments with the same number of steps that are about equally good. For example, 17edo could be used as two different equal temperaments in the 5-limit: one corresponds to the patent val <17 27 39\|, in which 5/4 is very flat at 353 cents, and one corresponds to the <17 27 40\| val (sometimes called "17c"), in which 5/4 is very sharp at 424 cents. Although these could be realized musically as exactly the same notes (17 equal divisions of 2/1), they are nevertheless different temperaments, because a 4:5:6 chord, for example, would be represented differently.

	A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, <17 27 40\| tempers out 81/80 (exercise for the reader), but the patent val for 17edo does not. Thus, if you played a meantone comma pump (which depends on 81/80 vanishing to return to the same pitch) in 17edo using the <17 27 40\| val, it would work, whereas the version using the patent val would not return to the same pitch. We summarize this by saying "<17 27 40\| ~~supports~~ meantone temperament" or "<17 27 40\| is a meantone val".		A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, <17 27 40\| tempers out 81/80 (exercise for the reader), but the patent val for 17edo does not. Thus, if you played a meantone comma pump (which depends on 81/80 vanishing to return to the same pitch) in 17edo using the <17 27 40\| val, it would work, whereas the version using the patent val would not return to the same pitch. We summarize this by saying "<17 27 40\| [[support]]s meantone temperament" or "<17 27 40\| is a meantone val".

	Temperaments other than equal temperaments (that is, rank 2 and higher) can be constructed out of vals. This operation can be denoted "v1 ^ v2" or "v1 & v2". One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals.		Temperaments other than equal temperaments (that is, rank 2 and higher) can be constructed out of vals. This operation can be denoted "v1 ^ v2" or "v1 & v2". One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals.

Mike Battaglia: Mike Battaglia moved page Keenan's explanation of vals to Keenan Pepper's explanation of vals: There are just too many Keenan's

2021-11-17T02:25:39Z

Mike Battaglia moved page Keenan's explanation of vals to Keenan Pepper's explanation of vals: There are just too many Keenan's

← Older revision	Revision as of 02:25, 17 November 2021
(No difference)

Keenan Pepper at 23:37, 28 October 2018

2018-10-28T23:37:00Z

← Older revision		Revision as of 23:37, 28 October 2018
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	<ul><li>A piece written in 12edo (assuming the patent val because it's the only one that makes musical sense), that actually takes advantage of more than one comma vanishing, cannot be played in any significantly different temperament without arbitrarily modifying the music. The octave stretch can be changed, for example, but it has to be a 12-note circulating temperament, with 12 roughly equal steps, or else some of the puns / comma pumps will not work.</li><li>On the other hand, a piece written in meantone temperament (for example a piece written in conventional notation with everything "spelled" properly and no enharmonic puns) can not only be played in 12edo, but also in 19edo or 26edo or 31edo, or an infinite number of other equal temperaments in which 81/80 vanishes. It could even be performed in 17edo using the 17c val - it will sound quite different, but everything will still work out in a logical way. The same goes for 7edo - major and minor triads both become neutral triads, but that applies in a uniform way to all chord progressions and everything still "works".</li><li>However, a piece written in meantone temperament cannot be performed in a non-meantone temperament like 22edo (using the patent val), or for that matter JI, without serious comma issues arising. It's possible to do it, but you'd basically be writing a new version of the piece from scratch rather than mechanically and faithfully translating it. You could play it in 22edo using a much worse val (<22 35 52\| rather than <22 35 51\|), but the chords would be far from the most accurate approximations that 22edo offers and would sound unnecessarily out-of-tune.</li><li>A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course...</li></ul> [[Category:definition]]		<ul><li>A piece written in 12edo (assuming the patent val because it's the only one that makes musical sense), that actually takes advantage of more than one comma vanishing, cannot be played in any significantly different temperament without arbitrarily modifying the music. The octave stretch can be changed, for example, but it has to be a 12-note circulating temperament, with 12 roughly equal steps, or else some of the puns / comma pumps will not work.</li><li>On the other hand, a piece written in meantone temperament (for example a piece written in conventional notation with everything "spelled" properly and no enharmonic puns) can not only be played in 12edo, but also in 19edo or 26edo or 31edo, or an infinite number of other equal temperaments in which 81/80 vanishes. It could even be performed in 17edo using the 17c val - it will sound quite different, but everything will still work out in a logical way. The same goes for 7edo - major and minor triads both become neutral triads, but that applies in a uniform way to all chord progressions and everything still "works".</li><li>However, a piece written in meantone temperament cannot be performed in a non-meantone temperament like 22edo (using the patent val), or for that matter JI, without serious comma issues arising. It's possible to do it, but you'd basically be writing a new version of the piece from scratch rather than mechanically and faithfully translating it. You could play it in 22edo using a much worse val (<22 35 52\| rather than <22 35 51\|), but the chords would be far from the most accurate approximations that 22edo offers and would sound unnecessarily out-of-tune.</li><li>A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course...</li></ul> [[Category:definition]]
	~~[[Category:keenan]]~~
	[[Category:practical_help]]		[[Category:practical_help]]
	[[Category:val]]		[[Category:val]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

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Wikispaces>keenanpepper: Imported revision 288620398 - Original comment:

2011-12-28T00:12:17Z

**Imported revision 288620398 - Original comment: **

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	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:~~clumma~~\|~~clumma~~]] and made on <tt>2011-12-~~27 17~~:54:32 UTC</tt>.<br>		: This revision was by author [[User:keenanpepper\|keenanpepper]] and made on <tt>2011-12-28 00:12:17 UTC</tt>.<br>
	: The original revision id was <tt>~~288601092~~</tt>.<br>		: The original revision id was <tt>288620398</tt>.<br>
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	The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.		The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.

	Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo ~~(denoted "h12")~~ is <12 19 28\|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the ~~h12~~ mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.		Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is <12 19 28\|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.

	==Vals supporting temperaments==		==Vals supporting temperaments==
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	The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.<br />		The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.<br />
	<br />		<br />
	Every EDO has a particular val associated to it called the &quot;patent val&quot;. This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo ~~(denoted &quot;h12&quot;)~~ is &lt;12 19 28\|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the ~~h12~~ mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.<br />		Every EDO has a particular val associated to it called the &quot;patent val&quot;. This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is &lt;12 19 28\|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.<br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Vals supporting temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Vals supporting temperaments</h2>		<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Vals supporting temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Vals supporting temperaments</h2>

Wikispaces>clumma: Imported revision 288601092 - Original comment:

2011-12-27T17:54:32Z

**Imported revision 288601092 - Original comment: **

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Wikispaces>keenanpepper: Imported revision 288537988 - Original comment:

2011-12-26T23:25:39Z

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