Val: Difference between revisions

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Mathematically, a val is a type of function that inputs a [[Wikipedia: Rational number|rational number]] (ratio) and outputs an integer (whole number) that represents what interval of the edo we use to approximate that frequency ratio. If ''a''/''b'' is our ratio and ''k'' is the output of the function, then the interval* of ''N'' edo is 2<sup>''k''/''N''</sup> (if we [[#Warts and generalized patent vals|assume a pure octave tuning]]) which is often written with the shorthand ''k''\''N'' and pronounced "''k'' steps of ''N'' edo", so that ''k'' is how many [[step]]s the interval is mapped to.

It is not just any such function though; it is a function with a special property called {{Wikipedia: Linearity|~~Linearity}}~~ that allows our arithmetic to be "internally consistent" (having an "internal logic") in the way described above; here "internally consistent" is meant in the English sense, so shouldn't be confused with [[consistency]] in the aforediscussed technical sense. The most obvious use of a val (the one discussed in the example) is to algorithmically determine ''JI interpretations'' of intervals in edo, which is called using the edo as an ''equal temperament'' or ''rank 1 temperament'', where "rank 1" means that it corresponds to a 1 dimensional grid of notes related by the same (usually [[Wikipedia: Irrational number|irrational]]) frequency ratios.

It is not just any such function though; it is a function with a special property called [[Wikipedia: Linearity|linearity]] that allows our arithmetic to be "internally consistent" (having an "internal logic") in the way described above; here "internally consistent" is meant in the English sense, so shouldn't be confused with [[consistency]] in the aforediscussed technical sense. The most obvious use of a val (the one discussed in the example) is to algorithmically determine ''JI interpretations'' of intervals in edo, which is called using the edo as an ''equal temperament'' or ''rank 1 temperament'', where "rank 1" means that it corresponds to a 1 dimensional grid of notes related by the same (usually [[Wikipedia: Irrational number|irrational]]) frequency ratios.

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'', which [[#For example, in 26edo:|as a reminder,]] is "when we take the closest approximation of each prime (corresponding to rounding rather than (EG) using the second-best approximation possible)".

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 Mathematically, a val is a type of function that inputs a [[Wikipedia: Rational number|rational number]] (ratio) and outputs an integer (whole number) that represents what interval of the edo we use to approximate that frequency ratio. If ''a''/''b'' is our ratio and ''k'' is the output of the function, then the interval* of ''N'' edo is 2<sup>''k''/''N''</sup> (if we [[#Warts and generalized patent vals|assume a pure octave tuning]]) which is often written with the shorthand ''k''\''N'' and pronounced "''k'' steps of ''N'' edo", so that ''k'' is how many [[step]]s the interval is mapped to.
-It is not just any such function though; it is a function with a special property called {{Wikipedia: Linearity|Linearity}} that allows our arithmetic to be "internally consistent" (having an "internal logic") in the way described above; here "internally consistent" is meant in the English sense, so shouldn't be confused with [[consistency]] in the aforediscussed technical sense. The most obvious use of a val (the one discussed in the example) is to algorithmically determine ''JI interpretations'' of intervals in edo, which is called using the edo as an ''equal temperament'' or ''rank 1 temperament'', where "rank 1" means that it corresponds to a 1 dimensional grid of notes related by the same (usually [[Wikipedia: Irrational number|irrational]]) frequency ratios.
+It is not just any such function though; it is a function with a special property called [[Wikipedia: Linearity|linearity]] that allows our arithmetic to be "internally consistent" (having an "internal logic") in the way described above; here "internally consistent" is meant in the English sense, so shouldn't be confused with [[consistency]] in the aforediscussed technical sense. The most obvious use of a val (the one discussed in the example) is to algorithmically determine ''JI interpretations'' of intervals in edo, which is called using the edo as an ''equal temperament'' or ''rank 1 temperament'', where "rank 1" means that it corresponds to a 1 dimensional grid of notes related by the same (usually [[Wikipedia: Irrational number|irrational]]) frequency ratios.
 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'', which [[#For example, in 26edo:|as a reminder,]] is "when we take the closest approximation of each prime (corresponding to rounding rather than (EG) using the second-best approximation possible)".