Val
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**What are vals and what are they for?** =Definition= A val is a map representing how to view the intervals in a single [[periods and generators|chain of generators]] as the 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. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths). A val accomplishes the goal of mapping all intervals in some [[harmonic limit|harmonic limit ]]by simply notating how many steps in the chain it takes to get to each of the limit's primes. Since every positive rational number can be described as a product of primes, any mapping for the primes hence implies a mapping for all of the positive rational numbers within the prime limit. By mapping the primes and letting the composite rationals fall where they may, a val tells us which interval in the chain represents the tempered 3/2, which interval represents the tempered 5/4, and so forth. Vals are usually written in the notation <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]]. Vals are important because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to 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. =**Example EDO**= Consider the 5-limit val <12 19 28]. This val tells us that you should view 12 generator steps as mapping to the octave 2/1. Since the temperament which maps 12 generator steps to the octave is 12-EDO, this means you're describing 12-EDO. The val <12 19 28], in addition to saying that 12 steps of 12-equal represents 2/1, also states explicitly that 19 steps of 12-equal represents a tempered 3/1, and 28 steps of 12-equal represents a tempered 5/1. Now assume you'd like to extend 12-EDO into the 7-limit. 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 <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 <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 See also: [[Monzos and Interval Space]], [[Patent val]], [[Vals and Tuning Space]]
Original HTML content:
<html><head><title>Vals</title></head><body><strong>What are vals and what are they for?</strong><br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> A val is a map representing how to view the intervals in a single <a class="wiki_link" href="/periods%20and%20generators">chain of generators</a> as the 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. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths).<br /> <br /> A val accomplishes the goal of mapping all intervals in some <a class="wiki_link" href="/harmonic%20limit">harmonic limit </a>by simply notating how many steps in the chain it takes to get to each of the limit's primes. Since every positive rational number can be described as a product of primes, any mapping for the primes hence implies a mapping for all of the positive rational numbers within the prime limit. By mapping the primes and letting the composite rationals fall where they may, a val tells us which interval in the chain represents the tempered 3/2, which interval represents the tempered 5/4, and so forth.<br /> <br /> Vals are usually written in the notation <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 <a class="wiki_link" href="/harmonic%20limit">prime limit p</a>.<br /> <br /> Vals are important because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to figure out what commas your temperament eliminates, what <a class="wiki_link" href="/comma%20pump">comma pumps</a> 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Example EDO"></a><!-- ws:end:WikiTextHeadingRule:2 --><strong>Example EDO</strong></h1> Consider the 5-limit val <12 19 28]. This val tells us that you should view 12 generator steps as mapping to the octave 2/1. Since the temperament which maps 12 generator steps to the octave is 12-EDO, this means you're describing 12-EDO.<br /> <br /> The val <12 19 28], in addition to saying that 12 steps of 12-equal represents 2/1, also states explicitly that 19 steps of 12-equal represents a tempered 3/1, and 28 steps of 12-equal represents a tempered 5/1.<br /> <br /> Now assume you'd like to extend 12-EDO into the 7-limit. 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 <12 19 28 34] val.<br /> <br /> If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <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<br /> <br /> See also: <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>, <a class="wiki_link" href="/Patent%20val">Patent val</a>, <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a></body></html>