Val: Difference between revisions

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: If you know what the intervals that you want to combine are, then combining their approximations in the edo does not give you the same result as multiplying their ratios ''first'' and ''then'' using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]]. (We will work through an example in a moment in [[#Using a val]] 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 is not actually necessary. Plus, even if you did that, there would still be other inconsistent ratios because an approximation cannot be perfect, so you cannot truly eliminate the inconsistency completely. Rather than giving up and saying that we cannot 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 by using a val. This may seem strange in this example, as one likely wants at least ~6/5 × ~5/4 = ~3/2, but in principle we probably do not mind if something more complex is inconsistent, like ~11 × ~11 × ~75 = ~9075, if we can guarantee that the arithmetic never fails us.

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 is not actually necessary. Plus, even if you did that, there would still be other inconsistent ratios because an approximation cannot be perfect, so you cannot truly eliminate the inconsistency completely. Rather than giving up and saying that we cannot 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, by using a val. This may seem strange in this example, as one likely wants at least ~6/5 × ~5/4 = ~3/2, but in principle we probably do not mind if something more complex is inconsistent, like ~11 × ~11 × ~75 = ~9075, if we can guarantee that the arithmetic never fails us.

== Definition ==

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 : If you know what the intervals that you want to combine are, then combining their approximations in the edo does not give you the same result as multiplying their ratios ''first'' and ''then'' using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]]. (We will work through an example in a moment in [[#Using a val]] 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 is not actually necessary. Plus, even if you did that, there would still be other inconsistent ratios because an approximation cannot be perfect, so you cannot truly eliminate the inconsistency completely. Rather than giving up and saying that we cannot 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 by using a val. This may seem strange in this example, as one likely wants at least ~6/5 × ~5/4 = ~3/2, but in principle we probably do not mind if something more complex is inconsistent, like ~11 × ~11 × ~75 = ~9075, if we can guarantee that the arithmetic never fails us.
+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 is not actually necessary. Plus, even if you did that, there would still be other inconsistent ratios because an approximation cannot be perfect, so you cannot truly eliminate the inconsistency completely. Rather than giving up and saying that we cannot 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, by using a val. This may seem strange in this example, as one likely wants at least ~6/5 × ~5/4 = ~3/2, but in principle we probably do not mind if something more complex is inconsistent, like ~11 × ~11 × ~75 = ~9075, if we can guarantee that the arithmetic never fails us.
 == Definition ==