Vals and tuning space: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-16 17:16:29 UTC</tt>.<br>

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-16 17:21:36 UTC</tt>.<br>

: The original revision id was <tt>~~241605141~~</tt>.<br>

: The original revision id was <tt>241605431</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>

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As an example, consider meantone temperament, where 81/80 vanishes. Meantone can be considered a 5-limit rank-2 temperament, defined by the val [<1 1 0|, <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 a val 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.

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 a val 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. Also, a handy "trick" to simplify things further, whenever one of the generators of a temperament is a 2/1: you can simply ignore the val for that generator by assuming octave equivalence (i.e. 3/1=3/2=6/1 etc.). Doing this would reduce 5-limit meantone to a single val (the one for the 3/2 generator), written <0 1 4|.

==Formal definition==

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<br />

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|].<br />

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 <em>is</em> 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 a val 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.<br />

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 <em>is</em> 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 a val 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. Also, a handy &quot;trick&quot; to simplify things further, whenever one of the generators of a temperament is a 2/1: you can simply ignore the val for that generator by assuming octave equivalence (i.e. 3/1=3/2=6/1 etc.). Doing this would reduce 5-limit meantone to a single val (the one for the 3/2 generator), written &lt;0 1 4|.<br />

<br />

<h2 id="toc1"><a name="x-Formal definition"></a>Formal definition</h2>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-16 17:16:29 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-16 17:21:36 UTC</tt>.<br>
-: The original revision id was <tt>241605141</tt>.<br>
+: The original revision id was <tt>241605431</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>
@@ Line 10: / Line 10: @@
 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 a val 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.
+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 a val 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. Also, a handy "trick" to simplify things further, whenever one of the generators of a temperament is a 2/1: you can simply ignore the val for that generator by assuming octave equivalence (i.e. 3/1=3/2=6/1 etc.). Doing this would reduce 5-limit meantone to a single val (the one for the 3/2 generator), written &lt;0 1 4|.
 ==Formal definition==
@@ Line 51: / Line 51: @@
 &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|].&lt;br /&gt;
-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 a val 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.&lt;br /&gt;
+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 a val 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. Also, a handy &amp;quot;trick&amp;quot; to simplify things further, whenever one of the generators of a temperament is a 2/1: you can simply ignore the val for that generator by assuming octave equivalence (i.e. 3/1=3/2=6/1 etc.). Doing this would reduce 5-limit meantone to a single val (the one for the 3/2 generator), written &amp;lt;0 1 4|.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Formal definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Formal definition&lt;/h2&gt;