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 19:32:54 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:56:44 UTC</tt>.<br>

: The original revision id was <tt>~~250541612~~</tt>.<br>

: The original revision id was <tt>250543568</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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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 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.)

(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain number of equally tempered "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.)

=Definition=

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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 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 />

(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain number of equally tempered &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 />

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

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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 19:32:54 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:56:44 UTC</tt>.<br>
-: The original revision id was <tt>250541612</tt>.<br>
+: The original revision id was <tt>250543568</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 16: / Line 16: @@
 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 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.)
+(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain number of equally tempered "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.)
 =Definition=
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 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 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;
+(Caveat: a more formal definition for a val is that it maps JI intervals onto a certain number of equally tempered &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;
 &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;