Vals and tuning space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 250559912 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 250560438 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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 23: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 23:09:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>250560438</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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A val is a map representing 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 is a map representing 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]]. | 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 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]]. | ||
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. | 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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A val is a map representing 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 /> | A val is a map representing 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 /> | <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 /> | 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 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</a>.<br /> | ||
<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 /> | 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 /> | ||