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{{Wikipedia|Limit (music)}}
{{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|below]], the odd limit ''of a ratio'' is a specific number.
== Definition and explanation ==
 
== Definition ==
{{odd-limit navigation}}
{{odd-limit navigation}}


An '''odd limit''' is the set of all ratios for which neither the numerator nor denominator exceed some maximum value, once all powers of 2 are removed. Typically, the maximum value is some small odd number, such as 3, 5, 7, 9, 11, etc. Each odd number gives rise to a different odd limit, so that there is a 3-odd-limit, a 5-odd-limit, and so on. This forms an increasing sequence of odd-limits, so that each numbered odd limit in this sequence is a subset of the next, so that the 3-odd-limit is a subset of the 5-odd-limit, which is in turn a subset of the 7-odd-limit, and so on.  
The '''odd limit''' is a metric that places an upper bound on (i.e. limits) the complexity of the [[Ratio|ratios]] 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|'''prime limit''']] are related concepts.  


Odd limits are more or less equivalent to what Harry Partch calls ''[[Tonality diamond|Tonality Diamonds]]'', in his theory. 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.
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 purpose of an odd limit or tonality diamond is to provide a "simple" subset of JI intervals to play, given one particularly natural definition of "simple." The removal of powers of 2 makes it so that for any interval that is viewed as "simple enough," the set of all its octave transpositions is also included in the set. Increasing the cutoff number increases the set of ratios viewed as being "simple enough" to be in the set. These are musically useful because such intervals will often tend to be play nicely with one another when forming chords (or at least, more so than some random JI intervals).
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.


As an example, the 5-odd-limit is the set of intervals {1/1, 3/1, 1/3, 5/1, 1/5, 5/3, 3/5}, as well as every octave transpositions of the above (e.g. 2/1, 4/1, 3/2, 6/1, 5/4 and so on).
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}.  


As a result, the odd limit is a metric that places an upper bound (i.e. limit) on the [[complexity]] of the harmonies used in a piece of music, and hence of the music itself. Integer limit and prime limit are related concepts.
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.


=== Mathematical definition ===
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.


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 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 of a ratio ==
== Mathematical definitions ==


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


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


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.
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.


== Relationship to other limits ==
== Relationship to other limits ==
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