Odd limit: Difference between revisions

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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.

The '''odd limit''' is a metric that 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.

== ~~Mathematical definitions~~ ==

== As a set of ratios ==

The ''q''-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 ''q''.

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

Odd limits are more or less equivalent to what [[Harry Partch]] calls ''[[Tonality diamond]]s''. 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-odd-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).

~~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).

To find the set of all ratios of ''q''-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to ''q'' 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 does not 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.

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}.

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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.

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.

Note that the 5-odd-limit set contains no seconds or sevenths. It is rare for an entire piece of music to ''strictly'' fall within the 5-odd-limit, because melodies generally do use the second and/or the seventh 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 many 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 second from some chord note. The major second 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 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 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.

== As a property of a ratio ==

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

The odd limit is also called the [[Kees semi-height]] 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 ==

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 {{Wikipedia|Limit (music)}}
-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.
+The '''odd limit''' is a metric that 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.
-== Mathematical definitions ==
+== As a set of ratios ==
+The ''q''-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 ''q''.
-=== Odd limit as a set of ratios ===
+Odd limits are more or less equivalent to what [[Harry Partch]] calls ''[[Tonality diamond]]s''. 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-odd-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).
-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).
+To find the set of all ratios of ''q''-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to ''q'' 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 does not 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.
-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}.
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 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.
-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.
+Note that the 5-odd-limit set contains no seconds or sevenths. It is rare for an entire piece of music to ''strictly'' fall within the 5-odd-limit, because melodies generally do use the second and/or the seventh 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 many 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 second from some chord note. The major second 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 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 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.
+== As a property of a ratio ==
+Given a ratio of positive integers ''n''/''d'', its odd limit is found by removing all factors of two and all other common factors from ''n''/''d'', producing a ratio ''a''/''b'' of relatively prime odd numbers. Thus the odd-limit of ''n''/''d'' is the maximum of ''a'' and ''b'', max(''a'', ''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.
+The odd limit is also called the [[Kees semi-height]] 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 ==