Val: Difference between revisions

=== Example: 26edo ===

* prime 2 is ''mapped'' to 26 steps (by definition, as we are equally dividing [[2/1]] into 26 steps, each representing an interval with frequency ratio 2^1/26/1)

* prime 3 is ''mapped '' to log₂(3) × 26 = 41.209… steps, which we round to 41 steps (because otherwise we are using frequency ratios not present in 26edo), meaning 2^41/26 = 2.983…/1 is the frequency ratio of 26edo that we use to approximate the [[3/1]] frequency ratio, so the correct statement is prime 3 is ''mapped to 41 steps'' (not a fractional amount)

* by the same procedure, prime 5 is ''mapped'' to 60 steps, as a result of rounding log₂(5) × 26 = 60.370…, meaning 2^60/26 = 4.950…/1 is the frequency ratio of 26edo that we use to approximate the [[5/1]] frequency ratio.  

Note that when we take the closest approximation of each prime (corresponding to rounding rather than e.g. using the second-best approximation possible) we call it a ''[[patent val]]'', therefore, the "5-limit patent val" of 26edo is {{val| 26 41 60 }}, where the limit is 5 because that is the highest prime we are considering.

== Using a val ==

Using the 26edo val {{val| 26 41 60 }} as our example, say we want to figure out how [[9/8]], [[5/4]] and 9/8 × 5/4 = [[45/32]] are mapped.

First we have to find the {{w|Integer factorization|prime factorization}} of our intervals (then we deduce the mapping from the prime factorization and the val):

That is a successful use of a val. The arithmetic works out nicely: the approximation of 5/4 times the approximation of 9/8 is the approximation of 45/32. Using [[backslash notation]] to denote the number of steps in an edo, these are 4\26, 8\26, and 12\26, respectively, so that the underlying "logic" of the approximations being followed is 4\26 + 8\26 = 12\26.

So you may have noticed that there was quite a bit of repeated addition we can simplify, so let us note how we can simplify it exactly. Specifically, using exponent notation and not omitting implicit "to the power of 0"'s, the prime factorizations we worked out above are equivalent to:

Did you notice that {{monzo| -3 2 0 }} + {{monzo| -2 0 1 }} = {{monzo| -5 2 1 }}? Specifically, try adding corresponding numbers in the lists; -3 with -2 gives -5, 2 with 0 gives 2, 0 with 1 gives 1. That is not a coincidence, but the same thing as multiplication except we are doing it with an additive notation.

Now equipped with our funny notation for a val and a monzo, we can do the exact same thing we did before — calculating the mappings — but using this very dense but efficient notation, where we express "mapping the interval described by the monzo by a val" as simple juxtaposition with angle brackets on the outskirts, where we ''multiply'' corresponding numbers in the list and then add them together, like so:

This is all very tedious, but in practice using a val is much simpler, because you do not need to do this, all you need to know is [[5/4]] is mapped to 8\26 and [[3/2]] is mapped to 15\26, therefore [[9/4]] is mapped to 30\26, therefore [[9/8]] is mapped to (30 - 26)\26 = 4\26, so that since we know 9/8 × 5/4 = 45/32, the mapped version of 45/32 will just be 4 + 8 = 12. This method guarantees that you never contradict yourself, even if you are technically using suspicious approximations.

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 so can get arbitrarily inconsistent. 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 (a.k.a. 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.