Val: Difference between revisions

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== Motivation ==

If you want to find an approximation to a just interval, the immediate question is: why would you need an algorithm instead of just looking at the [[direct approximation]] possible in the edo? The answer is to avoid contradictions. For example, it might not be true that [[~]][[6/5]] × [[~]][[5/4]] = [[~]][[3/2]] or that [[~]]9/1 × [[~]]5/1 = [[~]]45/1 if you are just always using the direct approximation of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo, because of something called ''inconsistency'', which means ~~(read carefully):~~

If you want to find an approximation to a just interval, the immediate question is: why would you need an algorithm instead of just looking at the [[direct approximation]] possible in the edo? The answer is to avoid contradictions. For example, it might not be true that [[~]][[6/5]] × [[~]][[5/4]] = [[~]][[3/2]] or that [[~]]9/1 × [[~]]5/1 = [[~]]45/1 if you are just always using the direct approximation of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo, because of something called ''inconsistency'', which means 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]].

~~: 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]]. ~~(If this is difficult to follow, we'll soon work through an example 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 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. So how do we do that? By using a val. Which brings us to...

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 == Motivation ==
-If you want to find an approximation to a just interval, the immediate question is: why would you need an algorithm instead of just looking at the [[direct approximation]] possible in the edo? The answer is to avoid contradictions. For example, it might not be true that [[~]][[6/5]] × [[~]][[5/4]] = [[~]][[3/2]] or that [[~]]9/1 × [[~]]5/1 = [[~]]45/1 if you are just always using the direct approximation of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo, because of something called ''inconsistency'', which means (read carefully):
+If you want to find an approximation to a just interval, the immediate question is: why would you need an algorithm instead of just looking at the [[direct approximation]] possible in the edo? The answer is to avoid contradictions. For example, it might not be true that [[~]][[6/5]] × [[~]][[5/4]] = [[~]][[3/2]] or that [[~]]9/1 × [[~]]5/1 = [[~]]45/1 if you are just always using the direct approximation of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo, because of something called ''inconsistency'', which means 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]].
-: 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]]. (If this is difficult to follow, we'll soon work through an example 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 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. So how do we do that? By using a val. Which brings us to...