Val: Difference between revisions

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This works by instead of doing log(''p'')/log(2) (where ''p'' is prime) we use log(''p'')/log(2.01...) or something to that effect, where 2.01/1 is our altered version of 2/1. The ''val'' produced by a slight alteration is usually the same, so there are actually continuous ranges where the val produced is the same.

For example, let's say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there is two possible mappings to use for 5; all primes up to and including 19 are sharp except for 5 which is quite flat, which causes a lot of inconsistencies; therefore a more natural val to use than the patent val is using the second-best mapping for 5, as log(5)/log(2) * 104 = 241.4805 is very close to exactly off anyways, and given the precision of 104edo, using the second-best mapping is very reasonable, as usually the sharpness of prime 5 cancels out with the sharpness of other primes when constructing ratios from them. But if we basically always want to use the patent val except for a slight modification to a second-best mapping for a handful of primes, do we really need to specify the full val every time? The answer is of course no:

We can specify the ''patent val'' of ''N'' edo as just ''N'', then we can specify each prime we want to map to the second best approximation by a cryptic letter shorthand:

* adding a means you make the mapping of 2 worse

* adding b means you make the mapping of 3 worse

* adding c means you make the mapping of 5 worse

* adding d means you make the mapping of 7 worse

* etc.

So we can refer to "104c" (not to be confused with 104{{cent}} (cents)), where we mnemnonically think "a, b, '''c'''; 3rd letter; 3rd prime is 2, 3, '''5'''; there is one c so we make the mapping of prime 5 worse (further from just) ''once'' compared to patent".

Perhaps a much clearer notation is to specify explicitly which primes are altered how many times in which direction from the patent val; the notation used by [https://sintel.pythonanywhere.com sintel's temperament finder] would be: 104[+5] and if we also "''warted''" prime 23 (which is similarly flat to prime 5), it'd be 104[+5, +23], which corresponds in the other notation to 104ci where i is the 9th letter of the alphabet and 23 is the 9th prime. They differ on how to notate more than one wart though; with [+5, +23] the direction is ''always'' sharpwards, and [-5, -23] would be flatwards, while with "ci" it's based on "second-best, third-best, etc."; so repeated warts with letters have alternating error while repeated +'s and -'s always have the same error. Slight variations of the notation that [[User:sintel|sintel]] uses have been reinvented and suggested by multiple people, so the one in use by a temperament finder is prioritized here.

== Applications ==

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 This works by instead of doing log(''p'')/log(2) (where ''p'' is prime) we use log(''p'')/log(2.01...) or something to that effect, where 2.01/1 is our altered version of 2/1. The ''val'' produced by a slight alteration is usually the same, so there are actually continuous ranges where the val produced is the same.
-{{todo | add 17edo example}}
+For example, let's say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there is two possible mappings to use for 5; all primes up to and including 19 are sharp except for 5 which is quite flat, which causes a lot of inconsistencies; therefore a more natural val to use than the patent val is using the second-best mapping for 5, as log(5)/log(2) * 104 = 241.4805 is very close to exactly off anyways, and given the precision of 104edo, using the second-best mapping is very reasonable, as usually the sharpness of prime 5 cancels out with the sharpness of other primes when constructing ratios from them. But if we basically always want to use the patent val except for a slight modification to a second-best mapping for a handful of primes, do we really need to specify the full val every time? The answer is of course no:
+We can specify the ''patent val'' of ''N'' edo as just ''N'', then we can specify each prime we want to map to the second best approximation by a cryptic letter shorthand:
+* adding a means you make the mapping of 2 worse
+* adding b means you make the mapping of 3 worse
+* adding c means you make the mapping of 5 worse
+* adding d means you make the mapping of 7 worse
+* etc.
+So we can refer to "104c" (not to be confused with 104{{cent}} (cents)), where we mnemnonically think "a, b, '''c'''; 3rd letter; 3rd prime is 2, 3, '''5'''; there is one c so we make the mapping of prime 5 worse (further from just) ''once'' compared to patent".
+Perhaps a much clearer notation is to specify explicitly which primes are altered how many times in which direction from the patent val; the notation used by [https://sintel.pythonanywhere.com sintel's temperament finder] would be: 104[+5] and if we also "''warted''" prime 23 (which is similarly flat to prime 5), it'd be 104[+5, +23], which corresponds in the other notation to 104ci where i is the 9th letter of the alphabet and 23 is the 9th prime. They differ on how to notate more than one wart though; with [+5, +23] the direction is ''always'' sharpwards, and [-5, -23] would be flatwards, while with "ci" it's based on "second-best, third-best, etc."; so repeated warts with letters have alternating error while repeated +'s and -'s always have the same error. Slight variations of the notation that [[User:sintel|sintel]] uses have been reinvented and suggested by multiple people, so the one in use by a temperament finder is prioritized here.
 == Applications ==