Keenan Pepper's explanation of vals: Difference between revisions
Wikispaces>keenanpepper **Imported revision 288537988 - Original comment: ** |
Wikispaces>clumma **Imported revision 288601092 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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: | : This revision was by author [[User:clumma|clumma]] and made on <tt>2011-12-27 17:54:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>288601092</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> | ||
| Line 14: | Line 14: | ||
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 " | 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. | ||
In other words, the result of tempering | |||
==Vals supporting temperaments== | ==Vals supporting temperaments== | ||
| Line 24: | Line 22: | ||
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| supports meantone temperament" or "<17 27 40| is a meantone val". | ||
Temperaments other than equal temperaments (that is, rank | 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. | ||
For example, consider the statement "5-limit meantone is | For example, consider the statement "5-limit meantone is 12 & 19". Here's a list of the simplest commas tempered out of those two 5-limit equal temperaments: | ||
* | * 12: 81/80, 128/125, 648/625, 2048/2025, 6561/6400... | ||
* | * 19: 81/80, 3125/3072, 6561/6400, 15625/15552... | ||
In the | In the 12-equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19, all of the commas in the second list vanish. In the temperament you get from combining them, "12 & 19", only the commas common to both lists are tempered out. These are 81/80, 6561/6400 = (81/80)^2, (81/80)^3, (81/80)^4... In other words, it works out that 81/80 is the only basic, fundamental comma that vanishes in both 12 and 19 - all the other commas are powers of 81/80, so they automatically vanish if 81/80 vanishes. So we say that the 5-limit temperament "12 & 19" is the same thing as "the 81/80 temperament" or "meantone". | ||
In practice, the easy way to find information about temperaments like this is to go to Graham Breed's | In practice, the easy way to find information about temperaments like this is to go to Graham Breed's [[http://x31eq.com/temper/net.html|temperament finder]], type "12 19" 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=== | ||
| Line 52: | Line 50: | ||
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; | 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 /> | ||
In other words, the result of tempering | |||
<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> | ||
| Line 62: | Line 58: | ||
A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, &lt;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 &lt;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 &quot;&lt;17 27 40| supports meantone temperament&quot; or &quot;&lt;17 27 40| is a meantone val&quot;.<br /> | A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, &lt;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 &lt;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 &quot;&lt;17 27 40| supports meantone temperament&quot; or &quot;&lt;17 27 40| is a meantone val&quot;.<br /> | ||
<br /> | <br /> | ||
Temperaments other than equal temperaments (that is, rank | Temperaments other than equal temperaments (that is, rank 2 and higher) can be constructed out of vals. This operation can be denoted &quot;v1 ^ v2&quot; or &quot;v1 &amp; v2&quot;. 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.<br /> | ||
<br /> | <br /> | ||
For example, consider the statement &quot;5-limit meantone is | For example, consider the statement &quot;5-limit meantone is 12 &amp; 19&quot;. Here's a list of the simplest commas tempered out of those two 5-limit equal temperaments:<br /> | ||
<br /> | <br /> | ||
<ul><li> | <ul><li>12: 81/80, 128/125, 648/625, 2048/2025, 6561/6400...</li><li>19: 81/80, 3125/3072, 6561/6400, 15625/15552...</li></ul><br /> | ||
In the | In the 12-equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19, all of the commas in the second list vanish. In the temperament you get from combining them, &quot;12 &amp; 19&quot;, only the commas common to both lists are tempered out. These are 81/80, 6561/6400 = (81/80)^2, (81/80)^3, (81/80)^4... In other words, it works out that 81/80 is the only basic, fundamental comma that vanishes in both 12 and 19 - all the other commas are powers of 81/80, so they automatically vanish if 81/80 vanishes. So we say that the 5-limit temperament &quot;12 &amp; 19&quot; is the same thing as &quot;the 81/80 temperament&quot; or &quot;meantone&quot;.<br /> | ||
<br /> | <br /> | ||
In practice, the easy way to find information about temperaments like this is to go to Graham Breed's | In practice, the easy way to find information about temperaments like this is to go to Graham Breed's <a class="wiki_link_ext" href="http://x31eq.com/temper/net.html" rel="nofollow">temperament finder</a>, type &quot;12 19&quot; into the equal temperaments field, and type &quot;5&quot; into the limit field. It tells you that the resulting temperament is called &quot;meantone&quot;, it has 81/80 as its only &quot;unison vector&quot; (aka comma), and other information you might find useful.<br /> | ||
<br /> | <br /> | ||
<!-- 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> | ||