Odd limit: Difference between revisions

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== Definition ==

~~The~~ '''odd limit''' is ~~a metric that places an upper bound on (i.e. limits)~~ the ~~complexity~~ of the ~~harmonies used in a piece of music, and hence of the music itself. The term also refers to the metric itself, applied to individual ratios. Every ratio has an odd limit~~, ~~and the q odd limit is the set of~~ all ~~ratios of odd limit~~ of ~~q or less. Integer limit and [[Prime limit|'''prime limit''']]~~ are ~~related concepts~~.

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.

~~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 > 3~~, ~~thus~~ the odd limit is 7.

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.

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.

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).

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).

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

== Mathematical Definition ==

The '''q''' '''odd limit''', where ''q'' is an odd positive integer, consists of everything of the form <code>2^i*u/v</code>, or <math>2^\mathbb Z\frac u v</math>, where ''u'' and ''v'' are odd positive integers less than or equal to q. It may be identified with the [[Diamonds|q-limit 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 of a ratio or chord ==

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.

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 > 3, thus the odd limit is 7.

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.

Each chord also has an associated odd limit, which is the largest odd limit of all the dyads in the chord. Note that the smallest odd limit of 10:12:15 is not the 15-odd-limit, but rather the 5-odd-limit, since each dyad is in the 5-odd-limit (as this is just the utonal inverse of 4:5:6).

== Relationship to other limits ==

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 == Definition ==
-The '''odd limit''' is a metric that places an upper bound on (i.e. limits) the complexity of the harmonies used in a piece of music, and hence of the music itself. The term also refers to the metric itself, applied to individual ratios. Every ratio has an odd limit, and the q odd limit is the set of all ratios of odd limit of q or less. Integer limit and [[Prime limit|'''prime limit''']] are related concepts.
+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.
-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 > 3, thus the odd limit is 7.
+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.
+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.
+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).
+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).
+As a result, the '''odd limit''' is a metric that places an upper bound on (i.e. limits) the complexity of the harmonies used in a piece of music, and hence of the music itself. Integer limit and [[Prime limit|'''prime limit''']] are related concepts.
+== Mathematical Definition ==
 The '''q''' '''odd limit''', where ''q'' is an odd positive integer, consists of everything of the form <code>2^i*u/v</code>, or <math>2^\mathbb Z\frac u v</math>, where ''u'' and ''v'' are odd positive integers less than or equal to q. It may be identified with the [[Diamonds|q-limit 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 of a ratio or chord ==
+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.
+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 > 3, thus the odd limit is 7.
+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.
+Each chord also has an associated odd limit, which is the largest odd limit of all the dyads in the chord. Note that the smallest odd limit of 10:12:15 is not the 15-odd-limit, but rather the 5-odd-limit, since each dyad is in the 5-odd-limit (as this is just the utonal inverse of 4:5:6).
 == Relationship to other limits ==