Vals and tuning space: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-17 12:14:49 UTC</tt>.<br>

: The original revision id was <tt>241661281</tt>.<br>

<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">==Introduction==  

A val "maps" just intonation to a certain number of steps; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form &lt;a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest. A rank r temperament will have r generators, and thus will be defined by r vals. In the usual coordinates for the [[Harmonic limit|p-limit]], the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments will be defined by a val &lt;a b c|, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments will have two vals: [&lt;a1 b1 c1|, &lt;a2 b2 c2|] Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be located by (b1, b2) and (c1, c2) respectively.

As an example, consider meantone temperament, where 81/80 vanishes. Meantone can be considered a 5-limit rank-2 temperament, defined by the val [&lt;1 1 0|, &lt;0 1 4|]. This tells us just about everything we need to know about how the 5-limit is mapped in meantone: since 2/1 is mapped (1, 0), that tells us that the first generator //is// a 2/1, and since 3/1 is mapped to (1,1), that tells us that the 2nd generator is a 3/2; then, since 5/1 is mapped to (0,4), aka four 3/2s up, that tells us that 81/64 (which is (3/2)^4) equals 5/1 (which is 80/64). Since 81/64 is equated with 80/64 here, that tells us that 81/80 is tempered out! Thus it is possible to derive from the mapping the approximate size of the two generators, the commas that are tempered out, and roughly the complexity of the temperament (the number of notes of the temperament we need to reach all the prime harmonics in the p-limit). This makes the val an extremely compact and useful bit of notation for describing regular temperaments, since we can discern almost everything we need to know about the temperament essentially at a glance. Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming octave equivalence (i.e. 3/1=3/2=6/1 etc). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written &lt;0 1 4|.

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Vals and Tuning Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Introduction&lt;/h2&gt;

  A val &amp;quot;maps&amp;quot; just intonation to a certain number of steps; by putting vals together we can define the mapping of a &lt;a class="wiki_link" href="/regular%20temperament"&gt;regular temperament&lt;/a&gt; and thereby define the temperament. A val is written in the form &amp;lt;a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a &lt;a class="wiki_link" href="/just%20intonation%20subgroup"&gt;just intonation subgroup&lt;/a&gt; is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest. A rank r temperament will have r generators, and thus will be defined by r vals. In the usual coordinates for the &lt;a class="wiki_link" href="/Harmonic%20limit"&gt;p-limit&lt;/a&gt;, the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments will be defined by a val &amp;lt;a b c|, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments will have two vals: [&amp;lt;a1 b1 c1|, &amp;lt;a2 b2 c2|] Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be located by (b1, b2) and (c1, c2) respectively.&lt;br /&gt;

As an example, consider meantone temperament, where 81/80 vanishes. Meantone can be considered a 5-limit rank-2 temperament, defined by the val [&amp;lt;1 1 0|, &amp;lt;0 1 4|]. This tells us just about everything we need to know about how the 5-limit is mapped in meantone: since 2/1 is mapped (1, 0), that tells us that the first generator &lt;em&gt;is&lt;/em&gt; a 2/1, and since 3/1 is mapped to (1,1), that tells us that the 2nd generator is a 3/2; then, since 5/1 is mapped to (0,4), aka four 3/2s up, that tells us that 81/64 (which is (3/2)^4) equals 5/1 (which is 80/64). Since 81/64 is equated with 80/64 here, that tells us that 81/80 is tempered out! Thus it is possible to derive from the mapping the approximate size of the two generators, the commas that are tempered out, and roughly the complexity of the temperament (the number of notes of the temperament we need to reach all the prime harmonics in the p-limit). This makes the val an extremely compact and useful bit of notation for describing regular temperaments, since we can discern almost everything we need to know about the temperament essentially at a glance. Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming octave equivalence (i.e. 3/1=3/2=6/1 etc). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written &amp;lt;0 1 4|.&lt;br /&gt;