Val: 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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 00:06:58 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 00:07:44 UTC</tt>.

: The original revision id was <tt>~~250564722~~</tt>.

: The original revision id was <tt>250564792</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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**What are vals and what are they for?**

A val is a map which sends just intonation intervals to integers. One

A val is a map which sends just intonation intervals to integers. One sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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 tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.

sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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 tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.

A val maps all intervals in some [[harmonic limit]] in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit p]].

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What are vals and what are they for?

A val is a map which sends just intonation intervals to integers. One ~~ ~~

A val is a map which sends just intonation intervals to integers. One sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.

sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.

A val maps all intervals in some <a class="wiki_link" href="/harmonic%20limit">harmonic limit</a> in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit p</a>.

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 00:06:58 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 00:07:44 UTC</tt>.<br>
-: The original revision id was <tt>250564722</tt>.<br>
+: The original revision id was <tt>250564792</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 9: / Line 9: @@
 **What are vals and what are they for?**
-A val is a map which sends just intonation intervals to integers. One
+A val is a map which sends just intonation intervals to integers. One sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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 tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.
-sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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 tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.
 A val maps all intervals in some [[harmonic limit]] in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit p]].
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 &lt;strong&gt;What are vals and what are they for?&lt;/strong&gt; &lt;br /&gt;
 &lt;br /&gt;
-A val is a map which sends just intonation intervals to integers. One &lt;br /&gt;
+A val is a map which sends just intonation intervals to integers. One sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.&lt;br /&gt;
-sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. 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). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.&lt;br /&gt;
 &lt;br /&gt;
 A val maps all intervals in some &lt;a class="wiki_link" href="/harmonic%20limit"&gt;harmonic limit&lt;/a&gt; in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive 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 ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some &lt;a class="wiki_link" href="/harmonic%20limit"&gt;prime limit p&lt;/a&gt;.&lt;br /&gt;