Odd limit: Difference between revisions
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{{Wikipedia|Limit (music)}} | {{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, and hence of the music itself. 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 that places an upper bound on (i.e. limits) the complexity of the [[ | |||
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. | 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. | ||
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. | 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}. | 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. | 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 "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 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 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. | ||
== Mathematical definitions == | == Mathematical definitions == | ||
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== Proposed extensions == | == Proposed extensions == | ||
[[ | [[Kite Giedraitis]] has proposed several extensions to the concepts of odd limit and integer limit. | ||
The '''double odd limit''' or '''DOL''' of a ratio is simply the odd limit of each number in the ratio, with the higher one listed first. DOL (12/7) = (7, 3). The DOL is useful as a tiebreaker when comparing the complexity of two ratios with the same odd limit. For example, 50/49 and 49/48 are both odd limit 49. But DOL (50/49) = (49, 25) and DOL (49/48) = (49, 3). Since 3 < 25, 49/48 has a lower DOL. | The '''double odd limit''' or '''DOL''' of a ratio is simply the odd limit of each number in the ratio, with the higher one listed first. DOL (12/7) = (7, 3). The DOL is useful as a tiebreaker when comparing the complexity of two ratios with the same odd limit. For example, 50/49 and 49/48 are both odd limit 49. But DOL (50/49) = (49, 25) and DOL (49/48) = (49, 3). Since 3 < 25, 49/48 has a lower DOL. | ||
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== See also == | == See also == | ||
* [[p-limit]] - or prime [[harmonic limit]] | * [[Harmonic limit|''p''-limit]] - or prime [[harmonic limit]] | ||
== External links == | == External links == | ||
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[[Category:Limit]] | [[Category:Limit]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||