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 22:48:15 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 22:52:27 UTC</tt>.<br>

: The original revision id was <tt>~~250558732~~</tt>.<br>

: The original revision id was <tt>250559034</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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=Abstract=

A val ~~provides~~ a ~~way~~ to ~~map~~ intervals in an equal or well-temperament ~~out~~ as being "tempered" versions of more fundamental ~~just intonation~~ 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 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 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]].

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 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.

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

<br />

A val ~~provides~~ a ~~way~~ to ~~map~~ intervals in an equal or well-temperament ~~out~~ as being ~~&quot;~~tempered~~&quot;~~ versions of more fundamental ~~just intonation~~ 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).<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 />

<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 />

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 />

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 />

@@ 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 22:48:15 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 22:52:27 UTC</tt>.<br>
-: The original revision id was <tt>250558732</tt>.<br>
+: The original revision id was <tt>250559034</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 provides a way to map intervals in an equal or well-temperament out as being "tempered" versions of more fundamental just intonation 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 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 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]].
+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 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.
@@ 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 provides a way to map intervals in an equal or well-temperament out as being &amp;quot;tempered&amp;quot; versions of more fundamental just intonation 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).&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;
 &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;
+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;
 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;