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Inthar (talk | contribs)
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Mike Battaglia (talk | contribs)
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IIL: you have very good ideas and we have already written some wiki pages on this. There is also already a notation we have been using so I put that there instead. See the talk page for more details
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== Definition ==
== Definition ==


A val is a set of assignments representing how to view the intervals in a [[temperament]], such as an edo, as approximate versions of intervals in [[just intonation]] (JI). 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 [[Periods and generators|generators]] to JI as well (such as a stack of meantone fifths).
A val is a linear map representing how to view the intervals in a [[temperament]], such as an edo, as approximate versions of intervals in [[just intonation]] (JI). 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 [[Periods and generators|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]] by simply notating how many steps in the chain it takes to get to each of the primes within the limit. 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.
A val accomplishes the goal of mapping all intervals in some [[harmonic limit]] by simply notating how many steps in the chain it takes to get to each of the primes within the limit. 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.
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== Vals in JI subgroups ==
== Vals in JI subgroups ==
We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]].
We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. Gene Smith called these "[Sval|svals]" for short.


Proposed notation: To write a JI ratio as a monzo in a JI subgroup, we choose a [[basis]] for the subgroup and factor an interval into the basis elements as we factor an interval in the ''p''-limit into primes at most ''p''. Then we write the monzo so as to explicitly state what basis elements we factor the intervals into and how many of each basis element the interval has in the factorization. For example, we can write [[81/80]] = 9<sup>2</sup>/(2<sup>4</sup> 5<sup>1</sup>) in the 2.9.5 subgroup as {{monzo|2^-4, 9^2, 5^-1}}. (We reserve the notation {{monzo|a b c ...}} and {{val|a b c ...}} for the ''p''-limit.)
To notate a subgroup val, we typically precede the "bra" notation with an indicator regarding the subgroup (and choice of basis). For instance, the patent val for 12 equal on the 2.3.7 subgroup is often notated "2.3.7 {{val|12 19 34}}". If the subgroup indicator isn't present, the subgroup can be inferred from context. It is very typical for a val with no explicit subgroup indicator to be interpreted as representing some prime limit, e.g. {{val|a b c}} would represent a 5-limit val.


Vals can be defined the same way in other subgroups as well; they represent how a subgroup is (viewed as being) tuned in terms of that edo's steps, but the basis element and the entry are separated by ~ instead of ^. For example, [[13edo]]'s "2.9.5 [[patent val]]" can be written as {{val|2~13, 9~41, 5~30}} (think "2 is approximately 13 steps, ..."), since [[13edo]]'s best approximation to the 9th harmonic is 41\13 (reduces to 2\13) and its best approximation to the 5th harmonic is 30\13 (reduces to 4\13). To see that this val "tempers out [[81/80]]", we do the same operation (of matching up and multiplying the components and summing the products) as described in the previous section:
Note that we could, for instance, use a different basis for the same subgroup - for instance, we could instead write "2.3.21 {{val|12 19 43}}", which is the 12 equal patent val in the "2.3.21" subgroup. Since the "2.3.21" subgroup is the same as the "2.3.7" subgroup, just written with a different basis, these two apparently "different" svals represent the same map from this subgroup to a rank-1 generator chain. (It is a matter of semantics if these are thought of as "different" svals or "the same sval" written using a different basis.)


&#x27E8;2~13, 9~41, 5~30&#93;&#91;2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
Svals can also be written using subgroups that don't involve primes, e.g. 2.3.7.13/5 {{val|46 73 129 63}}.


== Vals in regular temperaments ==
Note that the notion of a "patent val" for a subgroup val may not agree with the patent val on a prime limit. For instance, [[13edo]]'s "2.9.5 [[patent val]]" can be written as "2.9.5 {{val|13 41 30}}, because the best approximation to 2 is 13 steps, the best approximation to 9 is 41 steps, and the best approximation to 5 is 30 steps. Note that, however, the patent val on the 2.3.5 subgroup instead maps 3/1 to 21 steps, so that the "induced 9" from the 2.3.5 patent val is not the same as the "direct 9" from the 2.9.5 patent val.
Proposed notation: We write a tempered interval (an interval in a [[regular temperament]]) as a (generalized) monzo by taking a set of [[generator]]s (for rank-2 temperaments, this will be the period and the generator), then writing what JI ratio each generator approximates (distinguished from pure-JI subgroups by putting it in quotes), followed by the number of that specified generator that the interval has. For example, the major third in [[meantone]] temperament can be written as {{monzo|"2"^-2, "3/2"^4}}, meaning "4 perfect fifths minus 2 octaves".


Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, [[31edo]]'s tuning of meantone temperament can be written as {{val|"2"~31, "3/2"~18}}.
This notation is also used for subgroup monzos; e.g. [[81/80]] on the 2.9.5 subgroup is "2.9.5 {{val|-4 2 -1}}", and it is thus easy to see that the 2.9.5 13p val above tempers out 81/80:
 
&#x27E8;13, 41, 30|2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
 
== Vals in Regular Temperaments ==
There is also a notion of a "tempered val" on a group of "tempered monzos," representing intervals in some [[regular temperament]]. This name is sometimes abbreviated as [[Tmonzos_and_Tvals "tmonzos" and "tvals"]]. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone temperament]], we can use names like "P8" and "P5," so that the tval "P8.P5 {{val|12 7}}" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a [[Transversal]] can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 {{val|12 7}}, or  "(meantone) 2.3/2 {{val|31 18}}".


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/Val"