Vals and tuning space: Difference between revisions

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 22:52:27 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 22:52:58 UTC</tt>.<br>

: The original revision id was <tt>~~250559034~~</tt>.<br>

: The original revision id was <tt>250559076</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:

=Abstract=

A val is a structure that 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. 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 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 is a structure that 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. 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 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, 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]].

Line 66:

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

<br />

A val is a structure that 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. 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). A val 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 is a structure that 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. 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). 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, etc.<br />

<br />

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 />

@@ 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 22:52:27 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 22:52:58 UTC</tt>.<br>
-: The original revision id was <tt>250559034</tt>.<br>
+: The original revision id was <tt>250559076</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: @@
 =Abstract=
-A val is a structure that 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. 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 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 is a structure that 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. 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 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, 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]].
@@ Line 66: / Line 66: @@
 &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;
   &lt;br /&gt;
-A val is a structure that 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. 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). A val 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 is a structure that 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. 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). 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, etc.&lt;br /&gt;
 &lt;br /&gt;
 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;