Val: Difference between revisions

Line 12:

== Motivation ==

One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it ~~creates~~ contradictions. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily 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]].

One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2, though practically this is usually true for tunings of interest, so the realistic examples appear for more complex cases like (the direct approximations of) [[~]][[9/8]] and [[~]][[5/4]] combined not being equal to the direct approximation of [[45/32]] (= 9/8 * 5/4), as discussed in the 26edo example below. More generally, combining the approximations in an edo does not necessarily 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]].

Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.

== Definition ==

@@ Line 12: / Line 12: @@
 == Motivation ==
-One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it creates contradictions. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily 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]].
+One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2, though practically this is usually true for tunings of interest, so the realistic examples appear for more complex cases like (the direct approximations of) [[~]][[9/8]] and [[~]][[5/4]] combined not being equal to the direct approximation of [[45/32]] (= 9/8 * 5/4), as discussed in the 26edo example below. More generally, combining the approximations in an edo does not necessarily 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]].
 Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.
 == Definition ==