Val: Difference between revisions

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If you know what intervals (frequency ratios) that you want to combine (multiply) are, then ''combining their approximations'' (in the edo) '''''does not''''' give you the same result as ''multiplying their ratios first'' and ''then'' using the nearest approximation of ''that'' in the edo. When this happens, we say that the arithmetic is ''inconsistent''. (Therefore when this doesn't happen, we say that the result is [[consistent]].) (We will work through an example in a moment in [[#What is a val exactly and how do we use it]] to help understanding.)

Unfortunately, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals in question better so that the multiplication (or division) of their approximations is consistent, but that isn't actually necessary, and plus, even if you did that, then more complex ratios (or just ''different'' ratios) will be inconsistent still, and so on (because an approximation can't be perfect), so you can't truly eliminate the inconsistency completely. Rather than giving up and saying that we can't guarantee that 6/5 * 5/4 = 3/2 or 9/1 * 5/1 = 45/1 or 135/128 * 24/25 = 81/80, etc. in our chosen edo, we ''can'' guarantee this, if we are willing to allow one or more of these ratios to '''not''' use the closest approximation. (This may seem strange in this example, as one likely wants at least 6/5 * 5/4 = 3/2, but in principle we probably don't mind if something more complex is inconsistent, like 11 * 11 * 75 = 9075, if we can guarantee that the arithmetic never fails us.) How? By using a val! So, [[#What is a val exactly and how do we use it|what is a val exactly and how do we use it]]?

Unfortunately, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals in question better so that the multiplication (or division) of their approximations is consistent, but that isn't actually necessary, and plus, even if you did that, then more complex ratios (or just ''different'' ratios) will be inconsistent still, and so on (because an approximation can't be perfect), so you can't truly eliminate the inconsistency completely. Rather than giving up and saying that we can't guarantee that 6/5 * 5/4 = 3/2 or 9/1 * 5/1 = 45/1 or 135/128 * 24/25 = 81/80, etc. in our chosen edo, it turns out we ''can'' actually guarantee this if we are willing to allow one or more of these ratios to '''not''' use the closest approximation. (This may seem strange in this example, as one likely wants at least 6/5 * 5/4 = 3/2, but in principle we probably don't mind if something more complex is inconsistent, like 11 * 11 * 75 = 9075, if we can guarantee that the arithmetic never fails us.) How? By using a val! So, [[#What is a val exactly and how do we use it|what is a val exactly and how do we use it]]?

== What is a val exactly and how do we use it ==

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 If you know what intervals (frequency ratios) that you want to combine (multiply) are, then ''combining their approximations'' (in the edo) '''''does not''''' give you the same result as ''multiplying their ratios first'' and ''then'' using the nearest approximation of ''that'' in the edo. When this happens, we say that the arithmetic is ''inconsistent''. (Therefore when this doesn't happen, we say that the result is [[consistent]].) (We will work through an example in a moment in [[#What is a val exactly and how do we use it]] to help understanding.)
-Unfortunately, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals in question better so that the multiplication (or division) of their approximations is consistent, but that isn't actually necessary, and plus, even if you did that, then more complex ratios (or just ''different'' ratios) will be inconsistent still, and so on (because an approximation can't be perfect), so you can't truly eliminate the inconsistency completely. Rather than giving up and saying that we can't guarantee that 6/5 * 5/4 = 3/2 or 9/1 * 5/1 = 45/1 or 135/128 * 24/25 = 81/80, etc. in our chosen edo, we ''can'' guarantee this, if we are willing to allow one or more of these ratios  to '''not''' use the closest approximation. (This may seem strange in this example, as one likely wants at least 6/5 * 5/4 = 3/2, but in principle we probably don't mind if something more complex is inconsistent, like 11 * 11 * 75 = 9075, if we can guarantee that the arithmetic never fails us.) How? By using a val! So, [[#What is a val exactly and how do we use it|what is a val exactly and how do we use it]]?
+Unfortunately, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals in question better so that the multiplication (or division) of their approximations is consistent, but that isn't actually necessary, and plus, even if you did that, then more complex ratios (or just ''different'' ratios) will be inconsistent still, and so on (because an approximation can't be perfect), so you can't truly eliminate the inconsistency completely. Rather than giving up and saying that we can't guarantee that 6/5 * 5/4 = 3/2 or 9/1 * 5/1 = 45/1 or 135/128 * 24/25 = 81/80, etc. in our chosen edo, it turns out we ''can'' actually guarantee this if we are willing to allow one or more of these ratios  to '''not''' use the closest approximation. (This may seem strange in this example, as one likely wants at least 6/5 * 5/4 = 3/2, but in principle we probably don't mind if something more complex is inconsistent, like 11 * 11 * 75 = 9075, if we can guarantee that the arithmetic never fails us.) How? By using a val! So, [[#What is a val exactly and how do we use it|what is a val exactly and how do we use it]]?
 == What is a val exactly and how do we use it ==