Odd limit: Difference between revisions

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=== Odd limit as a set of ratios ===

The n-odd limit is the set of irreducible ratios between 1 and 2 whose numerator and denominator, once all factors of two are removed, are less than or equal to n.

The n-odd limit is the set of irreducible ratios between 1 and 2 whose numerator and denominator, once all factors of two are removed, are both less than or equal to n.

Odd limits are more or less equivalent to what [[Harry Partch]] calls ''[[Tonality diamond|Tonality Diamonds]]''. More precisely, a Tonality Diamond can be viewed as a particular geometric representation of a certain odd-limit, and the two terms are often used together (e.g., the 11-odd-limit Tonality Diamond). The sequence of increasing odd limits can be visualized as as a smaller tonality diamond being embedded in a set of progressively larger ones. Examples: some ratios in the 9-limit are: 3/2, 5/4, 7/6, 10/7, 12/7, 9/8 and 14/9; but not 11/9 nor 13/8 nor 16/15 (these have are odd terms greater than 9, thus not in the set).

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=== Odd limit as a property of a ratio ===

Given a ratio of positive integers ''p''/''q'', its odd limit is found by removing all factors of two and all common factors from ''p''/''q,'' producing a ratio ''a''/''b'' of relatively prime odd numbers. Thus the odd-limit of p/q is the maximum of a and b.

Given a ratio of positive integers ''p''/''q'', its odd limit is found by removing all factors of two and all other common factors from ''p''/''q,'' producing a ratio ''a''/''b'' of relatively prime odd numbers. Thus the odd-limit of p/q is the maximum of a and b.

The odd limit equals max(''a'', ''b''). It's also called the [[Kees expressibility]] of the interval, named after [[Kees van Prooijen]] who showed what this metric looks like geometrically on the lattice. 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 odd numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 is greater than 3, thus the odd limit is 7.

@@ Line 12: / Line 12: @@
 === Odd limit as a set of ratios ===
-The n-odd limit is the set of irreducible ratios between 1 and 2 whose numerator and denominator, once all factors of two are removed, are less than or equal to n.
+The n-odd limit is the set of irreducible ratios between 1 and 2 whose numerator and denominator, once all factors of two are removed, are both less than or equal to n.
 Odd limits are more or less equivalent to what [[Harry Partch]] calls ''[[Tonality diamond|Tonality Diamonds]]''. More precisely, a Tonality Diamond can be viewed as a particular geometric representation of a certain odd-limit, and the two terms are often used together (e.g., the 11-odd-limit Tonality Diamond). The sequence of increasing odd limits can be visualized as as a smaller tonality diamond being embedded in a set of progressively larger ones. Examples: some ratios in the 9-limit are: 3/2, 5/4, 7/6, 10/7, 12/7, 9/8 and 14/9; but not 11/9 nor 13/8 nor 16/15 (these have are odd terms greater than 9, thus not in the set).
@@ Line 26: / Line 26: @@
 === Odd limit as a property of a ratio ===
-Given a ratio of positive integers ''p''/''q'', its odd limit is found by removing all factors of two and all common factors from ''p''/''q,'' producing a ratio ''a''/''b'' of relatively prime odd numbers. Thus the odd-limit of p/q is the maximum of a and b.
+Given a ratio of positive integers ''p''/''q'', its odd limit is found by removing all factors of two and all other common factors from ''p''/''q,'' producing a ratio ''a''/''b'' of relatively prime odd numbers. Thus the odd-limit of p/q is the maximum of a and b.
 The odd limit equals max(''a'', ''b''). It's also called the [[Kees expressibility]] of the interval, named after [[Kees van Prooijen]] who showed what this metric looks like geometrically on the lattice. 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 odd numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 is greater than 3, thus the odd limit is 7.