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:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:01:01 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:29:33 UTC</tt>.<br>

: The original revision id was <tt>~~250539028~~</tt>.<br>

: The original revision id was <tt>250541350</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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<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">[[toc|flat]]

=Abstract=

A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc~~. Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO~~.

A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.

A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].

Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO. A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].

For example, the 5-limit val <12 19 28| tells us that we'~~re viewing~~ 12 steps of 12-equal as representing 2/1, 19 steps of 12-equal as representing 3/1, and 28 steps of 12-equal as representing 5/1. ~~This means~~ that ~~we view 3/2 as mapping to 19-12=7 steps of~~ 12-equal, that ~~we view 5~~/4 ~~as mapping~~ to ~~28-24=4~~ steps of 12~~-equal, and that we view 81/80 as mapping to 19*4 -~~ 12*4 - 28 = 0 steps of 12~~-equal~~.

For example, the 5-limit val <12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing a tempered 5/1. If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit <12 19 28 34| val.

If you ~~would~~ like to ~~assume~~ the ~~perspective that the 1000 cent interval in 12-equal is a~~ very ~~tempered 7/4, then that decision can be represented by using~~ the ~~7-limit <12 19 28 34| val~~. If you ~~would, for some reason, like to say that~~ the ~~7/4 maps~~ to ~~300 cents instead~~, ~~as absurd as~~ that ~~may be~~, ~~that would be reflected by~~ the ~~<12 19 28 27| val. It's not recommended to use vals like that~~, ~~but~~ the ~~mathematics will allow you to do it if you want,~~ in the ~~same way that a brick will allow you~~ to ~~hit yourself in~~ the ~~face~~ with it.

Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what [[comma pump|comma pumps]] are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as [[meantone]] or [[Porcupine|porcupine]] temperament, and other operations as of yet undiscovered.

~~Vals form the basis~~ for ~~all of regular temperament theory. They are important because they provide~~ a ~~way to mathematically formalize the chosen~~ JI ~~perspective one takes on~~ an EDO. ~~As such~~, ~~they allow us to harness~~ the ~~very powerful realm~~ of ~~mathematics to manipulate our musical intuitions.~~

(Caveat: a more formal definition for a val is that it maps JI intervals onto an EDO, rather than the other way around. This definition, while being perhaps less intuitive, is more applicable in some of the deeper implications of vals in regular temperament theory, which are not dealt with in this abstract.)

~~See also: [[patent val]]~~.

=Definition=

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=Example=

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

The rank-1 [[7-limit]] patent [[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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<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><a href="#Abstract">Abstract</a> | <a href="#Definition">Definition</a> | <a href="#Vals and Monzos">Vals and Monzos</a> | <a href="#Example">Example</a>

<h1 id="toc0"><a name="Abstract"></a>Abstract</h1>

A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc~~. Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO.<br />~~

A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.<br />

~~<br />~~

A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<br />

<br />

~~For example~~, the ~~5-limit val~~ &lt;~~12 19 28~~| ~~tells us that we're viewing 12 steps of 12-equal as representing~~ 2/1, ~~19 steps of 12-equal as representing~~ 3/1, ~~and 28 steps of 12-equal as representing~~ 5/1. ~~This means~~ that ~~we view 3/2 as mapping~~ to ~~19-12~~=~~7 steps of 12-equal, that we view 5~~/~~4 as mapping to 28-24=4 steps of 12-equal, and that we view 81~~/~~80 as mapping to 19*4 - 12*4 - 28 = 0 steps of 12-equal~~.<br />

Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO. A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<br />

<br />

If you would like to assume the perspective that the ~~1000 cent~~ interval in 12-equal is a very tempered 7/4, then that decision can be represented by using the 7-limit &lt;12 19 28 34| val. If you would, for some reason, like to say that the 7/4 maps to 300 cents instead, as absurd as that may be, that would be reflected by the &lt;12 19 28 27| val. It's not recommended to use vals like that, but the mathematics will allow you to do it if you want, in the same way that a brick will allow you to hit yourself in the face with it.<br />

For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing a tempered 5/1. If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit &lt;12 19 28 34| val.<br />

<br />

Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective ~~one takes~~ on an EDO. As such, they allow us to harness the very powerful realm of mathematics to ~~manipulate our~~ musical intuitions.<br />

Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what <a class="wiki_link" href="/comma%20pump">comma pumps</a> are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as <a class="wiki_link" href="/meantone">meantone</a> or <a class="wiki_link" href="/Porcupine">porcupine</a> temperament, and other operations as of yet undiscovered.<br />

<br />

~~See also~~: ~~<~~a ~~class="wiki_link" href="/patent%20val">patent~~ val~~</a>~~.<br />

(Caveat: a more formal definition for a val is that it maps JI intervals onto an EDO, rather than the other way around. This definition, while being perhaps less intuitive, is more applicable in some of the deeper implications of vals in regular temperament theory, which are not dealt with in this abstract.)<br />

<br />

<h1 id="toc1"><a name="Definition"></a>Definition</h1>

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<br />

<h1 id="toc4"><a name="Example"></a>Example</h1>

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 />

The rank-1 <a class="wiki_link" href="/7-limit">7-limit</a> patent <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:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:01:01 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:29:33 UTC</tt>.<br>
-: The original revision id was <tt>250539028</tt>.<br>
+: The original revision id was <tt>250541350</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 8: / Line 8: @@
 <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">[[toc|flat]]
 =Abstract=
-A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc. Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO.
+A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.
-A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].
+Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO. A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].
-For example, the 5-limit val &lt;12 19 28| tells us that we're viewing 12 steps of 12-equal as representing 2/1, 19 steps of 12-equal as representing 3/1, and 28 steps of 12-equal as representing 5/1. This means that we view 3/2 as mapping to 19-12=7 steps of 12-equal, that we view 5/4 as mapping to 28-24=4 steps of 12-equal, and that we view 81/80 as mapping to 19*4 - 12*4 - 28 = 0 steps of 12-equal.
+For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing a tempered 5/1. If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit &lt;12 19 28 34| val.
-If you would like to assume the perspective that the 1000 cent interval in 12-equal is a very tempered 7/4, then that decision can be represented by using the 7-limit &lt;12 19 28 34| val. If you would, for some reason, like to say that the 7/4 maps to 300 cents instead, as absurd as that may be, that would be reflected by the &lt;12 19 28 27| val. It's not recommended to use vals like that, but the mathematics will allow you to do it if you want, in the same way that a brick will allow you to hit yourself in the face with it.
+Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what [[comma pump|comma pumps]] are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as [[meantone]] or [[Porcupine|porcupine]] temperament, and other operations as of yet undiscovered.
-Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective one takes on an EDO. As such, they allow us to harness the very powerful realm of mathematics to manipulate our musical intuitions.
+(Caveat: a more formal definition for a val is that it maps JI intervals onto an EDO, rather than the other way around. This definition, while being perhaps less intuitive, is more applicable in some of the deeper implications of vals in regular temperament theory, which are not dealt with in this abstract.)
-See also: [[patent val]].
 =Definition=
@@ Line 52: / Line 50: @@
 =Example=
-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
+The rank-1 [[7-limit]] patent [[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 64: / Line 62: @@
 <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:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Abstract"&gt;Abstract&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Vals and Monzos"&gt;Vals and Monzos&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Example"&gt;Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Abstract"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Abstract&lt;/h1&gt;
-  A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc. Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO.&lt;br /&gt;
+  A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.&lt;br /&gt;
-&lt;br /&gt;
-A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &amp;lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some &lt;a class="wiki_link" href="/harmonic%20limit"&gt;prime limit&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-For example, the 5-limit val &amp;lt;12 19 28| tells us that we're viewing 12 steps of 12-equal as representing 2/1, 19 steps of 12-equal as representing 3/1, and 28 steps of 12-equal as representing 5/1. This means that we view 3/2 as mapping to 19-12=7 steps of 12-equal, that we view 5/4 as mapping to 28-24=4 steps of 12-equal, and that we view 81/80 as mapping to 19*4 - 12*4 - 28 = 0 steps of 12-equal.&lt;br /&gt;
+Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO. A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &amp;lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some &lt;a class="wiki_link" href="/harmonic%20limit"&gt;prime limit&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-If you would like to assume the perspective that the 1000 cent interval in 12-equal is a very tempered 7/4, then that decision can be represented by using the 7-limit &amp;lt;12 19 28 34| val. If you would, for some reason, like to say that the 7/4 maps to 300 cents instead, as absurd as that may be, that would be reflected by the &amp;lt;12 19 28 27| val. It's not recommended to use vals like that, but the mathematics will allow you to do it if you want, in the same way that a brick will allow you to hit yourself in the face with it.&lt;br /&gt;
+For example, the 5-limit val &amp;lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing a tempered 5/1. If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit &amp;lt;12 19 28 34| val.&lt;br /&gt;
 &lt;br /&gt;
-Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective one takes on an EDO. As such, they allow us to harness the very powerful realm of mathematics to manipulate our musical intuitions.&lt;br /&gt;
+Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pumps&lt;/a&gt; are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; or &lt;a class="wiki_link" href="/Porcupine"&gt;porcupine&lt;/a&gt; temperament, and other operations as of yet undiscovered.&lt;br /&gt;
 &lt;br /&gt;
-See also: &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt;.&lt;br /&gt;
+(Caveat: a more formal definition for a val is that it maps JI intervals onto an EDO, rather than the other way around. This definition, while being perhaps less intuitive, is more applicable in some of the deeper implications of vals in regular temperament theory, which are not dealt with in this abstract.)&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Definition&lt;/h1&gt;
@@ Line 111: / Line 107: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Example&lt;/h1&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;
+  The rank-1 &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; patent &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;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:1: