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 20:02:11 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 22:47:43 UTC</tt>.<br>

: The original revision id was <tt>~~250544004~~</tt>.<br>

: The original revision id was <tt>250558690</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 provides a way to map intervals in just intonation to a certain number of steps; that is, to an integer. In many cases of interest the val is associated to an EDO, and the val maps to steps of the EDO. For any number "n" of steps in the EDO, a set of JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is the "kernel" of the val, meaning the intervals mapped to 0 ("tempered out".) Hence the val maps from JI to the integers, and also from integers back to sets of just intervals.

A ~~12-EDO~~ val ~~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 provides a way to map intervals in an equal or well-temperament back to just intonation. More specifically, it provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator "steps," of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type).

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]].

A val tells us, when we look at such a temperament, 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 all intervals in this way by simply mapping each of the primes, hence indirectly mapping 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 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 for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.

For example, the 5-limit val <12 19 28| tells us that you'd like to view 12 generator steps as mapping to 2/1, which hence means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a tempered 2/1, it also states that you'd like to view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being 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 for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how 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.

(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain integer number of "steps," which may or may not reflect an EDO in its intuitive sense at all, but rather steps of a much larger generator. 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: [[Monzos and Interval Space]], [[Patent val]]

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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 provides a way to map intervals in just intonation to a ~~certain number of steps; that is,~~ to ~~an integer. In many cases of interest the val is associated~~ to ~~an EDO,~~ and ~~the val maps to steps~~ of ~~the EDO. For any number~~ &quot;n&quot; of ~~steps in the~~ EDO~~, a set~~ of ~~JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is~~ the ~~&quot;kernel&quot;~~ of ~~the val~~, ~~meaning the intervals mapped to 0 (&quot;tempered out&quot;.~~) ~~Hence the val maps from JI to the integers, and also from integers back to sets of just intervals~~.<br />

<br />

A val provides a way to map intervals in an equal or well-temperament back to just intonation. More specifically, it provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator &quot;steps,&quot; of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type).<br />

<br />

A ~~12-EDO~~ val 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 />

A val tells us, when we look at such a temperament, 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 all intervals in this way by simply mapping each of the primes, hence indirectly mapping 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 />

~~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 />

For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 generator steps as mapping to 2/1, which hence means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a tempered 2/1, it also states that you'd like to view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being a tempered 5/1.<br />

<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. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.<br />

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 for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.<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 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 />~~

(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain integer number of &quot;steps,&quot; which may or may not reflect an EDO in its intuitive sense at all, but rather steps of a much larger generator. 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 />

See also: <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>, <a class="wiki_link" href="/Patent%20val">Patent val</a><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:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 20:02:11 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 22:47:43 UTC</tt>.<br>
-: The original revision id was <tt>250544004</tt>.<br>
+: The original revision id was <tt>250558690</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 provides a way to map intervals in just intonation to a certain number of steps; that is, to an integer. In many cases of interest the val is associated to an EDO, and the val maps to steps of the EDO. For any number "n" of steps in the EDO, a set of JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is the "kernel" of the val, meaning the intervals mapped to 0 ("tempered out".) Hence the val maps from JI to the integers, and also from integers back to sets of just intervals.
-A 12-EDO val 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 provides a way to map intervals in an equal or well-temperament back to just intonation. More specifically, it provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator "steps," of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type).
-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]].
+A val tells us, when we look at such a temperament, 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 all intervals in this way by simply mapping each of the primes, hence indirectly mapping 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 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 for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.
+For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 generator steps as mapping to 2/1, which hence means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a tempered 2/1, it also states that you'd like to view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being 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 for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how 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.
-(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain integer number of "steps," which may or may not reflect an EDO in its intuitive sense at all, but rather steps of a much larger generator. 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: [[Monzos and Interval Space]], [[Patent val]]
@@ Line 66: / Line 65: @@
 <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 provides a way to map intervals in just intonation to a certain number of steps; that is, to an integer. In many cases of interest the val is associated to an EDO, and the val maps to steps of the EDO. For any number &amp;quot;n&amp;quot; of steps in the EDO, a set of JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is the &amp;quot;kernel&amp;quot; of the val, meaning the intervals mapped to 0 (&amp;quot;tempered out&amp;quot;.) Hence the val maps from JI to the integers, and also from integers back to sets of just intervals.&lt;br /&gt;
+  &lt;br /&gt;
+A val provides a way to map intervals in an equal or well-temperament back to just intonation. More specifically, it provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator &amp;quot;steps,&amp;quot; of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type).&lt;br /&gt;
 &lt;br /&gt;
-A 12-EDO val 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;
+A val tells us, when we look at such a temperament, 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 all intervals in this way by simply mapping each of the primes, hence indirectly mapping 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;
-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;
+For example, the 5-limit val &amp;lt;12 19 28| tells us that you'd like to view 12 generator steps as mapping to 2/1, which hence means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a tempered 2/1, it also states that you'd like to view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being a tempered 5/1.&lt;br /&gt;
 &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. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &amp;lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.&lt;br /&gt;
+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. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &amp;lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.&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 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;
-(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain integer number of &amp;quot;steps,&amp;quot; which may or may not reflect an EDO in its intuitive sense at all, but rather steps of a much larger generator. 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;
 See also: &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;Monzos and Interval Space&lt;/a&gt;, &lt;a class="wiki_link" href="/Patent%20val"&gt;Patent val&lt;/a&gt;&lt;br /&gt;