Val: Difference between revisions

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-04 19:50:58 UTC</tt>.<br>

: The original revision id was <tt>250696188</tt>.<br>

A val is a map representing how the intervals in a single chain of [[periods and generators|generators]] - commonly an equal temperament - are viewed as tempered versions of intervals in just intonation. They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory.

A val tells us which interval in the equal temperament 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 as a number of "steps" in the equal temperament. By mapping the primes, we hence indirectly map all of the positive rational numbers within 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]].

For example, the 5-limit val &lt;12 19 28| tells us that you should view 12 generator steps as mapping to the octave 2/1, which means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a (possibly tempered) 2/1, it also states that you should view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being a tempered 5/1.

If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit &lt;12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.

Vals are important because they provide a way to mathematically formalize which JI intervals you'd like to view the intervals in some temperament as representing. Once you've figured out how the JI perspective you've chosen to take on a temperament can be represented in val form (sometimes requiring more than one val), you can figure out what commas your temperament eliminates, what [[comma pump|comma pumps]] are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered.

=Beyond EDOs=

Although the obvious use for a val is to relate an EDO to JI, it can also be used to relate more abstract types of "equal temperaments" to JI as well. Most directly, vals provide a way to describe temperaments by mapping JI intervals onto a stack of tempered [[periods and generators|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. For example, a stack of meantone fifths can have JI intervals mapped onto it just like an EDO can, and vals describe these sorts of stacks as well.

 

[to be continued, describe how we get &lt;1 1 0| &lt;0 1 4| for meantone]

  A val is a map representing how the intervals in a single chain of &lt;a class="wiki_link" href="/periods%20and%20generators"&gt;generators&lt;/a&gt; - commonly an equal temperament - are viewed as tempered versions of intervals in just intonation. They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory.&lt;br /&gt;

A val tells us which interval in the equal temperament 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;

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 as a number of &amp;quot;steps&amp;quot; in the equal temperament. By mapping the primes, we hence indirectly map all of the positive rational numbers within 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;

For example, the 5-limit val &amp;lt;12 19 28| tells us that you should view 12 generator steps as mapping to the octave 2/1, which means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a (possibly tempered) 2/1, it also states that you should 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;

If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit &amp;lt;12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &amp;lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.&lt;br /&gt;

Vals are important because they provide a way to mathematically formalize which JI intervals you'd like to view the intervals in some temperament as representing. Once you've figured out how the JI perspective you've chosen to take on a temperament can be represented in val form (sometimes requiring more than one val), you can figure out what commas your temperament eliminates, what &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pumps&lt;/a&gt; are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered.&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Beyond EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Beyond EDOs&lt;/h1&gt;

&lt;br /&gt;

Although the obvious use for a val is to relate an EDO to JI, it can also be used to relate more abstract types of &amp;quot;equal temperaments&amp;quot; to JI as well. Most directly, vals provide a way to describe temperaments by mapping JI intervals onto a stack of tempered &lt;a class="wiki_link" href="/periods%20and%20generators"&gt;generator&lt;/a&gt; &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. For example, a stack of meantone fifths can have JI intervals mapped onto it just like an EDO can, and vals describe these sorts of stacks as well.&lt;br /&gt;

[to be continued, describe how we get &amp;lt;1 1 0| &amp;lt;0 1 4| for meantone]&lt;br /&gt;