Val: Difference between revisions

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Also note that in practice vals are ''very far'' from just any list of positive integers; rather, they are generally equal to or one off from the lists of integers that correspond to a ''patent val''.

==~~= Generalized~~ patent ~~vals =~~==

== Patent val and generalized patent val ==

{{Main| ~~Generalized patent val }}~~

~~{{See also| patent~~ val }}

~~The algorithm/~~process for producing a val does not actually require us to use a purely-tuned ~~2/1 (~~octave); instead we can stretch or compress the octave, resulting in potentially different mappings for primes, which is more common the more off the prime is and the more we alter the octave. This can give us a sense in which certain vals which are not patent vals are patent in a more broad sense, hence ''generalized''.

As discussed, a patent val is a val derived from rounding prime harmonics to the nearest edosteps. This process for producing a val does not actually require us to use a purely-tuned octave; instead we can stretch or compress the octave, resulting in potentially different mappings for primes, which is more common the more off the prime is and the more we alter the octave. This can give us a sense in which certain vals which are not patent vals are patent in a more broad sense, hence ''generalized''.

This works by instead of ~~doing log~~2(''p'') (where ''p'' is prime) we use ~~log2.01…(~~''p'') 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.

This works exactly like ordinary vals, but instead of plugging integer ''N'' into ''N''⋅log2(''p'') where ''p'' is a prime, we use a non-integer ''N'' or something to that effect. 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 us 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~~2(5) ~~× 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.

For example, let us say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there are 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 104⋅log2(5) = 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.

== Shorthand notations ==

@@ Line 88: / Line 88: @@
 Also note that in practice vals are ''very far'' from just any list of positive integers; rather, they are generally equal to or one off from the lists of integers that correspond to a ''patent val''.
-=== Generalized patent vals ===
+== Patent val and generalized patent val ==
-{{Main| Generalized patent val }}
+{{Main| Patent val }}
-{{See also| patent val }}
-The algorithm/process for producing a val does not actually require us to use a purely-tuned 2/1 (octave); instead we can stretch or compress the octave, resulting in potentially different mappings for primes, which is more common the more off the prime is and the more we alter the octave. This can give us a sense in which certain vals which are not patent vals are patent in a more broad sense, hence ''generalized''.
+As discussed, a patent val is a val derived from rounding prime harmonics to the nearest edosteps. This process for producing a val does not actually require us to use a purely-tuned octave; instead we can stretch or compress the octave, resulting in potentially different mappings for primes, which is more common the more off the prime is and the more we alter the octave. This can give us a sense in which certain vals which are not patent vals are patent in a more broad sense, hence ''generalized''.
-This works by instead of doing log<sub>2</sub>(''p'') (where ''p'' is prime) we use log<sub>2.01…</sub>(''p'') 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.
+This works exactly like ordinary vals, but instead of plugging integer ''N'' into ''N''⋅log<sub>2</sub>(''p'') where ''p'' is a prime, we use a non-integer ''N'' or something to that effect. 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 us 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<sub>2</sub>(5) × 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.
+For example, let us say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there are 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 104⋅log<sub>2</sub>(5) = 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.
 == Shorthand notations ==