Keenan Pepper's explanation of vals: Difference between revisions
Wikispaces>keenanpepper **Imported revision 288320296 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 288348732 - 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>2011-12-23 | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-23 22:13:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>288348732</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 a " | <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 a "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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For example, consider the statement "5-limit meantone is 12p & 19p"...</pre></div> | For example, consider the statement "5-limit meantone is 12p & 19p"...</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 a &quot; | <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 a &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 /> | ||