Odd limit: Difference between revisions

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The '''odd limit''' is a metric ~~that places an upper bound on (i.e.~~ limits) the [[complexity]] of the [[ratio]]s used in a piece of music. Odd limit can refer to the set of all ratios that are within this limit, or it can refer to the metric itself, applied to individual ratios. Integer limit and [[prime limit]] are related concepts.

The '''odd limit''' is a metric hat limits the [[complexity]] of the [[ratio]]s used in a piece of music. Odd limit can refer to the set of all ratios that are within this limit, or it can refer to the metric itself, applied to individual ratios. [[Integer limit]] and [[prime limit]] are related concepts, albeit prime-limit is unbound.

~~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~~.

== Mathematical definitions ==

=== 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.

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).

To find the set of all ratios of ''N''-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to ''N'' in every combination. Discard any ratios that can be simplified (e.g. 5/5 or 15/3). Transposing any of these ratios by an octave doesn't change the odd limit. Hence an odd limit set is theoretically infinite. For convenience, the odd limit set is usually written out in [[octave-reduced]] form.

For example, to find the 5-odd-limit set of ratios, pair off 1, 3 and 5: {1/1, 3/1, 5/1, 1/3, (3/3), 5/3, 1/5, 3/5, (5/5)}. Discard the two redundant ratios in parentheses. Octave-reduced and in ascending order, the 5-odd-limit set is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}.

The 3-odd-limit set of ratios is {1/1, 4/3, 3/2}. The 3-odd-limit set is contained in the 5-odd-limit set, both sets are contained in the 7-odd-limit set, and so on.

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Note that the 5-odd-limit set contains no 2nds or 7ths. It's rare for an entire piece of music to ''strictly'' fall within the 5-odd-limit, because melodies generally do use the 2nd and/or the 7th of the scale. But a piece of music being in a certain odd limit usually means merely that at any given moment, no interval in the piece exceeds the odd limit. In other words, all vertical intervals within a chord, or between a melody note and a chord note, are within the odd limit. Even this looser definition excludes most songs. It perhaps includes {{w|Kumbaya|"Kum Ba Yah"}}, depending on the exact chords used. But even the simplest pentatonic songs usually have a melody note that is a major 2nd from some chord note. The major 2nd ratio is usually 10/9 or 9/8, making the piece 9-odd-limit. An even looser definition ignores the melody notes and requires only 5-limit ''chords''. This definition includes any song that uses only major and minor triads.

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.~~

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

== ~~Mathematical definitions~~ ==

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

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.

~~The~~ '''''q''-odd-limit~~''', where ''q''~~ is ~~an odd positive integer, consists~~ of ~~every number of the form <math>2^i \cdot u/v</math> where ''i'' is an integer~~ and ~~where~~ ''u'' ~~and~~ ''v'' ~~are odd positive integers less than or equal to~~ ''q''~~. It may be identified with the [[Tonality diamond|~~''q''-odd~~-limit tonality diamond]]~~. ~~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 (11~~ is a ~~prime greater than 9) nor 15/7 (since 15 is 3 × 5, both less then 9, but with product greater than 9)~~.

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

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

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.

== Integer limit ==

[[File:WilsonHeightIntegerLimit.png|200px|thumb|right|Diagram by Lériendil showing the integer limit (horizontal axis) versus [[Wilson height]] (vertical axis) of simple intervals.]]The '''integer limit''' of a ratio is simply the larger of the ratio's numerator and denominator. For example, the integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio, and is the same as the [[Weil height]]. The set of all ratios with an integer limit up to ''n'' is the same as the {{w|Farey sequence}} of order ''n''.

The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd limit can be thought of as the best-case integer limit, when assuming [[octave equivalence]].

The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd-limit can be thought of as the best-case integer limit, when assuming [[octave equivalence]].

== Generalizations ==

@@ Line 7: / Line 7: @@
 {{Wikipedia|Limit (music)}}
-The '''odd limit''' is a metric that places an upper bound on (i.e. limits) the [[complexity]] of the [[ratio]]s used in a piece of music. Odd limit can refer to the set of all ratios that are within this limit, or it can refer to the metric itself, applied to individual ratios. Integer limit and [[prime limit]] are related concepts.
+The '''odd limit''' is a metric hat limits the [[complexity]] of the [[ratio]]s used in a piece of music. Odd limit can refer to the set of all ratios that are within this limit, or it can refer to the metric itself, applied to individual ratios. [[Integer limit]] and [[prime limit]] are related concepts, albeit prime-limit is unbound.
-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.
+== Mathematical definitions ==
+=== 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.
+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).
 To find the set of all ratios of ''N''-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to ''N'' in every combination. Discard any ratios that can be simplified (e.g. 5/5 or 15/3). Transposing any of these ratios by an octave doesn't change the odd limit. Hence an odd limit set is theoretically infinite. For convenience, the odd limit set is usually written out in [[octave-reduced]] form.
 For example, to find the 5-odd-limit set of ratios, pair off 1, 3 and 5: {1/1, 3/1, 5/1, 1/3, (3/3), 5/3, 1/5, 3/5, (5/5)}. Discard the two redundant ratios in parentheses. Octave-reduced and in ascending order, the 5-odd-limit set is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}.
 The 3-odd-limit set of ratios is {1/1, 4/3, 3/2}. The 3-odd-limit set is contained in the 5-odd-limit set, both sets are contained in the 7-odd-limit set, and so on.
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 Note that the 5-odd-limit set contains no 2nds or 7ths. It's rare for an entire piece of music to ''strictly'' fall within the 5-odd-limit, because melodies generally do use the 2nd and/or the 7th of the scale. But a piece of music being in a certain odd limit usually means merely that at any given moment, no interval in the piece exceeds the odd limit. In other words, all vertical intervals within a chord, or between a melody note and a chord note, are within the odd limit. Even this looser definition excludes most songs. It perhaps includes {{w|Kumbaya|"Kum Ba Yah"}}, depending on the exact chords used. But even the simplest pentatonic songs usually have a melody note that is a major 2nd from some chord note. The major 2nd ratio is usually 10/9 or 9/8, making the piece 9-odd-limit. An even looser definition ignores the melody notes and requires only 5-limit ''chords''. This definition includes any song that uses only major and minor triads.
-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.
+=== Odd limit as a property of a ratio ===
-== Mathematical definitions ==
-=== Odd limit as a set of ratios ===
+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.
-The '''''q''-odd-limit''', where ''q'' is an odd positive integer, consists of every number of the form <math>2^i \cdot u/v</math> where ''i'' is an integer and where ''u'' and ''v'' are odd positive integers less than or equal to ''q''. It may be identified with the [[Tonality diamond|''q''-odd-limit tonality diamond]]. 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 (11 is a prime greater than 9) nor 15/7 (since 15 is 3 × 5, both less then 9, but with product greater than 9).
-=== Odd limit as a property of a ratio ===
-Given a ratio of positive integers ''p''/''q'', its odd limit is found by first removing factors of two and all common factors from ''p''/''q'', producing a ratio ''a''/''b'' of relatively prime odd numbers. 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.
+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.
 == Integer limit ==
 [[File:WilsonHeightIntegerLimit.png|200px|thumb|right|Diagram by Lériendil showing the integer limit (horizontal axis) versus [[Wilson height]] (vertical axis) of simple intervals.]]The '''integer limit''' of a ratio is simply the larger of the ratio's numerator and denominator. For example, the integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio, and is the same as the [[Weil height]]. The set of all ratios with an integer limit up to ''n'' is the same as the {{w|Farey sequence}} of order ''n''.
-The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd limit can be thought of as the best-case integer limit, when assuming [[octave equivalence]].
+The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd-limit can be thought of as the best-case integer limit, when assuming [[octave equivalence]].
 == Generalizations ==