Keenan Pepper's explanation of vals: Difference between revisions
Wikispaces>keenanpepper **Imported revision 288394060 - Original comment: ** |
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>2011-12-24 | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-24 13:06:16 UTC</tt>.<br> | ||
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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> | ||
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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=== | |||
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: | |||
* 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. | |||
* 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. | |||
* A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course... | |||
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In the 12p equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19p, all of the commas in the second list vanish. In the temperament you get from combining them, &quot;12p &amp; 19p&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 12p and 19p - 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;12p &amp; 19p&quot; is the same thing as &quot;the 81/80 temperament&quot; or &quot;meantone&quot;.<br /> | In the 12p equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19p, all of the commas in the second list vanish. In the temperament you get from combining them, &quot;12p &amp; 19p&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 12p and 19p - 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;12p &amp; 19p&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 temperament finder, <!-- ws:start:WikiTextUrlRule: | In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, <!-- ws:start:WikiTextUrlRule:49:http://x31eq.com/temper/net.html --><a class="wiki_link_ext" href="http://x31eq.com/temper/net.html" rel="nofollow">http://x31eq.com/temper/net.html</a><!-- ws:end:WikiTextUrlRule:49 --> , type &quot;12p 19p&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 /> | |||
<!-- 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 /> | |||
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 /> | |||
<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> | |||