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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2011-06-17 07:05:10 UTC</tt>.<br>

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-16 17:09:18 UTC</tt>.<br>

: The original revision id was <tt>~~237277875~~</tt>.<br>

: The original revision id was <tt>241604729</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">The p-limit [[Monzos and Interval Space|monzos]] M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The [[http://planetmath.org/encyclopedia/DualModule.html|dual Z-module]] M* is [[http://en.wikipedia.org/wiki/Group_isomorphism|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of **vals**, is also a free abelian group of rank pi(p). Just as monzos are often written as [[http://mathworld.wolfram.com/Ket.html|kets]], vals are typically written as [[http://mathworld.wolfram.com/Bra.html|bras]].

<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:

<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:

<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.

==Formal definition==

The p-limit [[Monzos and Interval Space|monzos]] M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The [[http://planetmath.org/encyclopedia/DualModule.html|dual Z-module]] M* is [[http://en.wikipedia.org/wiki/Group_isomorphism|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of **vals**, is also a free abelian group of rank pi(p). Just as monzos are often written as [[http://mathworld.wolfram.com/Ket.html|kets]], vals are typically written as [[http://mathworld.wolfram.com/Bra.html|bras]].

If V is a val and M is a monzo of the same rank, then the [[http://mathworld.wolfram.com/AngleBracket.html|angle bracket]] <V|M>, which can also be written V(M), is the result of applying the [[http://en.wikipedia.org/wiki/Group_homomorphism|homomorphism]] V to M. For example, if V = <12 19 28 34| and M = |-5 2 2 -1> then <V|M> equals 12*(-5) + 19*2 + 28*2 - 34 = 0

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==Example==

The [[7-limit]] [[val]] corresponding to [[31edo]] is <31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes

The rank-1 [[7-limit]] [[val]] corresponding to [[31edo]] is <31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes

[[math]]

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which is approximately <31.000 30.916 31.009 30.990|. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt(3838.694), or 61.957. To use the RMS we divide that by sqrt(4)=2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31.</pre></div>

<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>The p-limit <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/DualModule.html" rel="nofollow">dual Z-module</a> M* is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_isomorphism" rel="nofollow">isomorphic</a> to M, but not in a canonical way. Hence it, the group (Z-module) of <strong>vals</strong>, is also a free abelian group of rank pi(p). Just as monzos are often written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">kets</a>, vals are typically written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Bra.html" rel="nofollow">bras</a>.<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;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 />

&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 />

&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 />

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

<br />

The p-limit <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/DualModule.html" rel="nofollow">dual Z-module</a> M* is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_isomorphism" rel="nofollow">isomorphic</a> to M, but not in a canonical way. Hence it, the group (Z-module) of <strong>vals</strong>, is also a free abelian group of rank pi(p). Just as monzos are often written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">kets</a>, vals are typically written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Bra.html" rel="nofollow">bras</a>.<br />

<br />

If V is a val and M is a monzo of the same rank, then the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/AngleBracket.html" rel="nofollow">angle bracket</a> &lt;V|M&gt;, which can also be written V(M), is the result of applying the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow">homomorphism</a> V to M. For example, if V = &lt;12 19 28 34| and M = |-5 2 2 -1&gt; then &lt;V|M&gt; equals 12*(-5) + 19*2 + 28*2 - 34 = 0<br />

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It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is &lt;1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then &lt;JIP|M&gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = &lt;1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.<br />

<br />

<h2 id="~~toc0~~"><a name="x-Example"></a>Example</h2>

<h2 id="toc2"><a name="x-Example"></a>Example</h2>

The <a class="wiki_link" href="/7-limit">7-limit</a> <a class="wiki_link" href="/val">val</a> corresponding to <a class="wiki_link" href="/31edo">31edo</a> is &lt;31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes<br />

The rank-1 <a class="wiki_link" href="/7-limit">7-limit</a> <a class="wiki_link" href="/val">val</a> corresponding to <a class="wiki_link" href="/31edo">31edo</a> is &lt;31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes<br />

<br />

<!-- ws:start:WikiTextMathRule:1:

@@ 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:xenwolf|xenwolf]] and made on <tt>2011-06-17 07:05:10 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-16 17:09:18 UTC</tt>.<br>
-: The original revision id was <tt>237277875</tt>.<br>
+: The original revision id was <tt>241604729</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">The p-limit [[Monzos and Interval Space|monzos]] M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The [[http://planetmath.org/encyclopedia/DualModule.html|dual Z-module]] M* is [[http://en.wikipedia.org/wiki/Group_isomorphism|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of **vals**, is also a free abelian group of rank pi(p). Just as monzos are often written as [[http://mathworld.wolfram.com/Ket.html|kets]], vals are typically written as [[http://mathworld.wolfram.com/Bra.html|bras]].
+<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:
+&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:
+&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.
+==Formal definition==
+The p-limit [[Monzos and Interval Space|monzos]] M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The [[http://planetmath.org/encyclopedia/DualModule.html|dual Z-module]] M* is [[http://en.wikipedia.org/wiki/Group_isomorphism|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of **vals**, is also a free abelian group of rank pi(p). Just as monzos are often written as [[http://mathworld.wolfram.com/Ket.html|kets]], vals are typically written as [[http://mathworld.wolfram.com/Bra.html|bras]].
 If V is a val and M is a monzo of the same rank, then the [[http://mathworld.wolfram.com/AngleBracket.html|angle bracket]] &lt;V|M&gt;, which can also be written V(M), is the result of applying the [[http://en.wikipedia.org/wiki/Group_homomorphism|homomorphism]] V to M. For example, if V = &lt;12 19 28 34| and M = |-5 2 2 -1&gt; then &lt;V|M&gt; equals 12*(-5) + 19*2 + 28*2 - 34 = 0
@@ Line 29: / Line 42: @@
 ==Example==
-The [[7-limit]] [[val]] corresponding to [[31edo]] is &lt;31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes
+The rank-1 [[7-limit]] [[val]] corresponding to [[31edo]] is &lt;31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes
 [[math]]
@@ Line 39: / Line 52: @@
 which is approximately &lt;31.000 30.916 31.009 30.990|. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt(3838.694), or 61.957. To use the RMS we divide that by sqrt(4)=2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31.</pre></div>
 <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;The p-limit &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt; M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The &lt;a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/DualModule.html" rel="nofollow"&gt;dual Z-module&lt;/a&gt; M* is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_isomorphism" rel="nofollow"&gt;isomorphic&lt;/a&gt; to M, but not in a canonical way. Hence it, the group (Z-module) of &lt;strong&gt;vals&lt;/strong&gt;, is also a free abelian group of rank pi(p). Just as monzos are often written as &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow"&gt;kets&lt;/a&gt;, vals are typically written as &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Bra.html" rel="nofollow"&gt;bras&lt;/a&gt;.&lt;br /&gt;
+<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;
+&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;
+&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;
+&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;
+ &lt;br /&gt;
+The p-limit &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt; M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The &lt;a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/DualModule.html" rel="nofollow"&gt;dual Z-module&lt;/a&gt; M* is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_isomorphism" rel="nofollow"&gt;isomorphic&lt;/a&gt; to M, but not in a canonical way. Hence it, the group (Z-module) of &lt;strong&gt;vals&lt;/strong&gt;, is also a free abelian group of rank pi(p). Just as monzos are often written as &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow"&gt;kets&lt;/a&gt;, vals are typically written as &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Bra.html" rel="nofollow"&gt;bras&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 If V is a val and M is a monzo of the same rank, then the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/AngleBracket.html" rel="nofollow"&gt;angle bracket&lt;/a&gt; &amp;lt;V|M&amp;gt;, which can also be written V(M), is the result of applying the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow"&gt;homomorphism&lt;/a&gt; V to M. For example, if V = &amp;lt;12 19 28 34| and M = |-5 2 2 -1&amp;gt; then &amp;lt;V|M&amp;gt; equals 12*(-5) + 19*2 + 28*2 - 34 = 0&lt;br /&gt;
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 It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is &amp;lt;1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then &amp;lt;JIP|M&amp;gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = &amp;lt;1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.&lt;br /&gt;
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-  The &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; corresponding to &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; is &amp;lt;31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes&lt;br /&gt;
+  The rank-1 &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; corresponding to &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; is &amp;lt;31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes&lt;br /&gt;
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