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 17:09:18 UTC</tt>.<br>

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

: The original revision id was <tt>~~241604729~~</tt>.<br>

: The original revision id was <tt>241605141</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>

<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">==Simple definition==

A val is a numerical representation of the way a regular temperament "maps" to Just intonation, and as such can be said to "define" the temperament. A val is written in the form <a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of "coordinates" along the r dimensions of the temperament. By convention, 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 <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:

A val is a numerical representation of the way a regular temperament "maps" to Just intonation, and as such can be said to "define" the temperament. A val is written in the form <a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of "coordinates" along the r dimensions of the temperament. By convention, 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 <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: [<a1 b1 c1|, <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.

<a1 b1 c1|

<a2 b2 c2|

~~They are usually written on top of each other.~~ 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 ~~following~~ val:

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|].

<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.

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<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>Vals and Tuning Space</title></head><body><h2 id="toc0"><a name="x-Simple definition"></a>Simple definition</h2>

A val is a numerical representation of the way a regular temperament &quot;maps&quot; to Just intonation, and as such can be said to &quot;define&quot; the temperament. A val is written in the form &lt;a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of &quot;coordinates&quot; along the r dimensions of the temperament. By convention, 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: ~~<br />~~

A val is a numerical representation of the way a regular temperament &quot;maps&quot; to Just intonation, and as such can be said to &quot;define&quot; the temperament. A val is written in the form &lt;a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of &quot;coordinates&quot; along the r dimensions of the temperament. By convention, 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.<br />

&lt;a1 b1 c1|~~<br />~~

&lt;a2 b2 c2|~~<br />~~

~~They are usually written on top of each other.~~ 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. <br />

<br />

As an example, consider meantone temperament, where 81/80 vanishes. Meantone can be considered a 5-limit rank-2 temperament, defined by the ~~following~~ val~~:<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 />

&lt;1 1 0|~~<br />~~

&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 />

<br />

@@ 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 17:09:18 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-16 17:16:29 UTC</tt>.<br>
-: The original revision id was <tt>241604729</tt>.<br>
+: The original revision id was <tt>241605141</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>
 <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">==Simple definition==
-A val is a numerical representation of the way a regular temperament "maps" to Just intonation, and as such can be said to "define" the temperament. A val is written in the form &lt;a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of "coordinates" along the r dimensions of the temperament. By convention, 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:
+A val is a numerical representation of the way a regular temperament "maps" to Just intonation, and as such can be said to "define" the temperament. A val is written in the form &lt;a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of "coordinates" along the r dimensions of the temperament. By convention, 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.
-&lt;a1 b1 c1|
-&lt;a2 b2 c2|
-They are usually written on top of each other. 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 following val:
+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|].
-&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.
@@ Line 53: / Line 48: @@
 <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">&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-Simple definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Simple definition&lt;/h2&gt;
-  A val is a numerical representation of the way a regular temperament &amp;quot;maps&amp;quot; to Just intonation, and as such can be said to &amp;quot;define&amp;quot; the temperament. A val is written in the form &amp;lt;a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of &amp;quot;coordinates&amp;quot; along the r dimensions of the temperament. By convention, 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: &lt;br /&gt;
+  A val is a numerical representation of the way a regular temperament &amp;quot;maps&amp;quot; to Just intonation, and as such can be said to &amp;quot;define&amp;quot; the temperament. A val is written in the form &amp;lt;a b c ... x|, where the numbers a b c (and so on) are numbers of generators. A rank r temperament will have r generators, and thus will have r vals. In a p-limit rank-r temperament, all rational numbers that can be expressed in the p-prime-limit are defined by a set of &amp;quot;coordinates&amp;quot; along the r dimensions of the temperament. By convention, 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;
-&amp;lt;a1 b1 c1|&lt;br /&gt;
-&amp;lt;a2 b2 c2|&lt;br /&gt;
-They are usually written on top of each other. 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;
 &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 following val:&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;
-&amp;lt;1 1 0|&lt;br /&gt;
-&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;
 &lt;br /&gt;