Val
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[[toc|flat]] =Abstract= 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 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. 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. 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. See also: [[Monzos and Interval Space]], [[Patent val]]
Original HTML content:
<html><head><title>Vals</title></head><body><!-- ws:start:WikiTextTocRule:2:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:2 --><!-- ws:start:WikiTextTocRule:3: --><a href="#Abstract">Abstract</a><!-- ws:end:WikiTextTocRule:3 --><!-- ws:start:WikiTextTocRule:4: --> <!-- ws:end:WikiTextTocRule:4 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Abstract"></a><!-- ws:end:WikiTextHeadingRule:0 -->Abstract</h1> 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.<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 <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 /> 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.<br /> <br /> 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.<br /> <br /> 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 /> 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></body></html>