Odd limit: Difference between revisions

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'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[interval]]s. In the newer sense, discussed [[Odd limit#Odd limit of a ratio ~~or chord~~|below]], the odd limit ''of a ratio'' is a specific number.

'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[interval]]s. In the newer sense, discussed [[Odd limit#Odd limit of a ratio|below]], the odd limit ''of a ratio'' is a specific number.

== Definition ==

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From the definition above, we can see that an interval like 3/2 is not only part of the 3-odd-limit, but also the 5-odd-limit, the 7-odd-limit, and so on. However, it is also useful to refer to the ''smallest'' such odd limit that some interval fits into. This is often simply just called the "odd limit" of the ratio.

To find the odd limit of a ratio: If either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 > 3, thus the odd limit is 7.

To find the odd limit of a ratio: If either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 is greater than 3, thus the odd limit is 7.

This is also called the [[Kees expressibility]] of the interval, named after [[Kees van Prooijen]] who showed what this metric looks like geometrically on the lattice.

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 {{Wikipedia|Limit (music)}}
-'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[interval]]s. In the newer sense, discussed [[Odd limit#Odd limit of a ratio or chord|below]], the odd limit ''of a ratio'' is a specific number.
+'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[interval]]s. In the newer sense, discussed [[Odd limit#Odd limit of a ratio|below]], the odd limit ''of a ratio'' is a specific number.
 == Definition ==
@@ Line 28: / Line 28: @@
 From the definition above, we can see that an interval like 3/2 is not only part of the 3-odd-limit, but also the 5-odd-limit, the 7-odd-limit, and so on. However, it is also useful to refer to the ''smallest'' such odd limit that some interval fits into. This is often simply just called the "odd limit" of the ratio.
-To find the odd limit of a ratio: If either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 > 3, thus the odd limit is 7.
+To find the odd limit of a ratio: If either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 is greater than 3, thus the odd limit is 7.
 This is also called the [[Kees expressibility]] of the interval, named after [[Kees van Prooijen]] who showed what this metric looks like geometrically on the lattice.