Val: Difference between revisions

Line 12:

''The immediate question is:'' why would you need an algorithm instead of just looking at the nearest 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 ''nearest approximation'' of each of these frequency ratios (6:5, 5:4, 3:2, etc.) in the edo. Why? Because of something called ''inconsistency'', which means (read carefully):

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 ''nearest approximation'' of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo. Why? Because of something called ''inconsistency'', which means (read carefully):

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.)

@@ Line 12: / Line 12: @@
 ''The immediate question is:'' why would you need an algorithm instead of just looking at the nearest 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 ''nearest approximation'' of each of these frequency ratios (6:5, 5:4, 3:2, etc.) in the edo. Why? Because of something called ''inconsistency'', which means (read carefully):
+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 ''nearest approximation'' of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo. Why? Because of something called ''inconsistency'', which means (read carefully):
 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.)