Val: Difference between revisions

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For a more mathematically intensive introduction to vals, see [[Vals and tuning space]]. For the characterization of higher-rank temperaments, see [[Mapping]].

== Relationship with (equal) temperaments ==

== Relationship with equal temperaments ==

~~Despite having~~ no contradictions, ~~stacking~~ the ~~tempered intervals of~~ the ~~val will inevitably cause error to accumulate, when compared to~~ the ~~JI counterpart that is supposed to be represented~~. This ~~is because temperaments temper~~ out an infinite set of ~~commas~~, ~~which can be derived from~~ a ~~select set~~ of simple/musically relevant commas that are ~~all nullified~~ in the val.

The guarantee that there are no contradictions comes with an interesting feature: somehow, you have managed to approximate JI in an ''internally-consistent'' way despite the fact that the approximations get worse the more you combine the errors. This corresponds to [[tempering out]] an infinite set of [[comma]]s, though there is a finite number of simple/musically relevant commas; that set is simply the set of all intervals that are mapped to 0 steps ([[1/1]]) by the val. This explains where the additional "structure" went – if there are two or more primes, then you need to specify two or more integers in the exponents of the prime factorization (i.e. in the monzo). So we have lost information and structure by simplifying everything to a single integer coordinate; exactly the information that corresponds to ''equating'' any two intervals whose difference is one of the commas tempered, so we have found a precise sense in which we can equate two nearby intervals that are not actually equal – by mapping according to a val that maps the difference to zero. In fact, you do not have to use an [[edo]] tuning as you could use multiple vals ''simultaneously'' to map a single interval if you want to preserve more of the information in JI rather than just increasing the size of the edo; this corresponds to [[regular temperaments]] generally rather than just the 1-dimensional ("rank-1") case that vals correspond to. Therefore, a val specifies a rank-1 temperament a.k.a. an equal temperament.

~~All temperaments compromise JI by reducing the number of primes used, so for instance, 5-limit requires 2,3,5 to represent any pitch. If a 5-limit comma is tempered out~~, ~~the structure~~ is ~~collapsed, and error is introduced to compensate for something that was not~~ a ~~unison now being one. In mathematical terms, this is equivalent to making one of the basis vectors~~ of ~~JI linearly dependent.~~

Furthermore, there is actually a lot of applications of vals and monzos that are not necessarily about approximating things in edos or even regular temperaments for that matter, discussed in [[#Applications]], though all of them do still use the idea of the ''mapping'' provided by the val, so really, a val is a ''mapping'' from JI to the numbers with certain properties.

When tempering out enough commas, JI is collapsed onto a quantized line; an equal temperament or rank-1 tuning. This is where vals come into play. Each of the primes is determined by a certain number of quanta, corresponding to octave divisions (edosteps) in [[EDO|edos]], tritave divisions in [[EDT|edts]], et cetera. There are many applications of vals and monzos ~~disjoint from RTT~~, discussed in [[#Applications]], though all of them still ~~treat vals as providing~~ ''~~mappings~~'' from JI to the numbers, with ~~constraints~~.

== Patent val and generalized patent val ==

@@ Line 82: / Line 82: @@
 For a more mathematically intensive introduction to vals, see [[Vals and tuning space]]. For the characterization of higher-rank temperaments, see [[Mapping]].
-== Relationship with (equal) temperaments ==
+== Relationship with equal temperaments ==
 {{Todo|inline=1| improve readability }}
-Despite having no contradictions, stacking the tempered intervals of the val will inevitably cause error to accumulate, when compared to the JI counterpart that is supposed to be represented. This is because temperaments temper out an infinite set of commas, which can be derived from a select set of simple/musically relevant commas that are all nullified in the val.
+The guarantee that there are no contradictions comes with an interesting feature: somehow, you have managed to approximate JI in an ''internally-consistent'' way despite the fact that the approximations get worse the more you combine the errors. This corresponds to [[tempering out]] an infinite set of [[comma]]s, though there is a finite number of simple/musically relevant commas; that set is simply the set of all intervals that are mapped to 0 steps ([[1/1]]) by the val. This explains where the additional "structure" went – if there are two or more primes, then you need to specify two or more integers in the exponents of the prime factorization (i.e. in the monzo). So we have lost information and structure by simplifying everything to a single integer coordinate; exactly the information that corresponds to ''equating'' any two intervals whose difference is one of the commas tempered, so we have found a precise sense in which we can equate two nearby intervals that are not actually equal – by mapping according to a val that maps the difference to zero. In fact, you do not have to use an [[edo]] tuning as you could use multiple vals ''simultaneously'' to map a single interval if you want to preserve more of the information in JI rather than just increasing the size of the edo; this corresponds to [[regular temperaments]] generally rather than just the 1-dimensional ("rank-1") case that vals correspond to. Therefore, a val specifies a rank-1 temperament a.k.a. an equal temperament.
-All temperaments compromise JI by reducing the number of primes used, so for instance, 5-limit requires 2,3,5 to represent any pitch. If a 5-limit comma is tempered out, the structure is collapsed, and error is introduced to compensate for something that was not a unison now being one. In mathematical terms, this is equivalent to making one of the basis vectors of JI linearly dependent.
+Furthermore, there is actually a lot of applications of vals and monzos that are not necessarily about approximating things in edos or even regular temperaments for that matter, discussed in [[#Applications]], though all of them do still use the idea of the ''mapping'' provided by the val, so really, a val is a ''mapping'' from JI to the numbers with certain properties.
-When tempering out enough commas, JI is collapsed onto a quantized line; an equal temperament or rank-1 tuning. This is where vals come into play. Each of the primes is determined by a certain number of quanta, corresponding to octave divisions (edosteps) in [[EDO|edos]], tritave divisions in [[EDT|edts]], et cetera. There are many applications of vals and monzos disjoint from RTT, discussed in [[#Applications]], though all of them still treat vals as providing ''mappings'' from JI to the numbers, with constraints.
 == Patent val and generalized patent val ==