Keenan Pepper's explanation of vals: Difference between revisions
Wikispaces>keenanpepper **Imported revision 288395084 - Original comment: ** |
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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>2011-12- | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-26 23:25:39 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>288537988</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">A val is a function that assigns a whole number to a JI interval in a **consistent** way. Here's an example of what "consistent" means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you **multiply** their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is | <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 val is a function that assigns a whole number to a JI interval in a **consistent** way. Here's an example of what "consistent" means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you **multiply** their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is "homomorphism". | ||
Because of the consistency property, if you want to write down a specific val, you don't need to give its value on every possible JI interval you might want to apply it to. Instead, you only need to give the value for primes, because every other interval can be decomposed into primes. For example, if I tell you that v(2/1) = 12, v(3/1) = 19, and v(5/1) = 28, then you can figure out the value of v applied to any other interval of 5-limit JI. For example, 25/24 = 5^2/(2^3*3), so v(25/24) = 2*v(5) - 3*v(2) - v(3) = 2*28 - 3*12 - 19 = 1, so 25/24 must be mapped to 1 by this val. The standard way to write this val in symbols is | Because of the consistency property, if you want to write down a specific val, you don't need to give its value on every possible JI interval you might want to apply it to. Instead, you only need to give the value for primes, because every other interval can be decomposed into primes. For example, if I tell you that v(2/1) = 12, v(3/1) = 19, and v(5/1) = 28, then you can figure out the value of v applied to any other interval of 5-limit JI. For example, 25/24 = 5^2/(2^3*3), so v(25/24) = 2*v(5) - 3*v(2) - v(3) = 2*28 - 3*12 - 19 = 1, so 25/24 must be mapped to 1 by this val. The standard way to write this val in symbols is | ||
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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 is <12 19 28|. This is **not** the same thing as the function you'd get by rounding every interval independently to the nearest number of steps - that kind of rounding is not consistent, so it isn't a val at all. 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 that's farther away from the JI interval 625/512. | 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 "12p") is <12 19 28|. This is **not** the same thing as the function you'd get by rounding every interval independently to the nearest number of steps - that kind of rounding is not consistent, so it isn't a val at all. 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 that's farther away from the JI interval 625/512. | ||
In other words, the result of tempering using a patent val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for non-prime intervals. | In other words, the result of tempering using a patent val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for non-prime intervals. | ||
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In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, http://x31eq.com/temper/net.html , type "12p 19p" into the equal temperaments field, and type "5" into the limit field. It tells you that the resulting temperament is called "meantone", it has 81/80 as its only "unison vector" (aka comma), and other information you might find useful. | In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, http://x31eq.com/temper/net.html , type "12p 19p" into the equal temperaments field, and type "5" into the limit field. It tells you that the resulting temperament is called "meantone", it has 81/80 as its only "unison vector" (aka comma), and other information you might find useful. | ||
===Practical consequences=== | ===Practical consequences=== | ||
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: | ||
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* 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". | * 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". | ||
* 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. | * 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. | ||
* A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course... | * A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course...</pre></div> | ||
</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>Keenan's explanation of vals</title></head><body>A val is a function that assigns a whole number to a JI interval in a <strong>consistent</strong> way. Here's an example of what &quot;consistent&quot; means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you <strong>multiply</strong> their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is | <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>Keenan's explanation of vals</title></head><body>A val is a function that assigns a whole number to a JI interval in a <strong>consistent</strong> way. Here's an example of what &quot;consistent&quot; means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you <strong>multiply</strong> their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is &quot;homomorphism&quot;.<br /> | ||
<br /> | <br /> | ||
Because of the consistency property, if you want to write down a specific val, you don't need to give its value on every possible JI interval you might want to apply it to. Instead, you only need to give the value for primes, because every other interval can be decomposed into primes. For example, if I tell you that v(2/1) = 12, v(3/1) = 19, and v(5/1) = 28, then you can figure out the value of v applied to any other interval of 5-limit JI. For example, 25/24 = 5^2/(2^3*3), so v(25/24) = 2*v(5) - 3*v(2) - v(3) = 2*28 - 3*12 - 19 = 1, so 25/24 must be mapped to 1 by this val. The standard way to write this val in symbols is<br /> | Because of the consistency property, if you want to write down a specific val, you don't need to give its value on every possible JI interval you might want to apply it to. Instead, you only need to give the value for primes, because every other interval can be decomposed into primes. For example, if I tell you that v(2/1) = 12, v(3/1) = 19, and v(5/1) = 28, then you can figure out the value of v applied to any other interval of 5-limit JI. For example, 25/24 = 5^2/(2^3*3), so v(25/24) = 2*v(5) - 3*v(2) - v(3) = 2*28 - 3*12 - 19 = 1, so 25/24 must be mapped to 1 by this val. The standard way to write this val in symbols is<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.<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 is &lt;12 19 28|. This is <strong>not</strong> the same thing as the function you'd get by rounding every interval independently to the nearest number of steps - that kind of rounding is not consistent, so it isn't a val at all. 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 that's farther away from the JI interval 625/512.<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;12p&quot;) is &lt;12 19 28|. This is <strong>not</strong> the same thing as the function you'd get by rounding every interval independently to the nearest number of steps - that kind of rounding is not consistent, so it isn't a val at all. 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 that's farther away from the JI interval 625/512.<br /> | ||
<br /> | <br /> | ||
In other words, the result of tempering using a patent val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for non-prime intervals.<br /> | In other words, the result of tempering using a patent val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for non-prime intervals.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Vals supporting temperaments-Practical consequences"></a><!-- ws:end:WikiTextHeadingRule:2 -->Practical consequences</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Vals supporting temperaments-Practical consequences"></a><!-- ws:end:WikiTextHeadingRule:2 -->Practical consequences</h3> | ||
<br /> | <br /> | ||
Any piece of music can be performed without &quot;comma issues&quot; 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:<br /> | Any piece of music can be performed without &quot;comma issues&quot; 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:<br /> | ||
<br /> | <br /> | ||
<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 &quot;spelled&quot; 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 &quot;works&quot;.</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 (&lt;22 35 52| rather than &lt;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></body></html></pre></div> | <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 &quot;spelled&quot; 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 &quot;works&quot;.</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 (&lt;22 35 52| rather than &lt;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></body></html></pre></div> | ||