Odd limit: Difference between revisions

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== Definition ==

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~~.

'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[Ratio|ratios]]. In the newer sense, discussed [[Odd limit#Odd limit of a ratio or chord|below]], the odd limit ''of a ratio'' is a specific number.

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.

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.

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.

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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~~ ==

== Odd limit of a ratio ==

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.

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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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This conjecture has two implications. First, a given JI chord has an ideal voicing. This voicing may be rather far-flung, and a more compact voicing may be almost as consonant. For example, 1:3:5:7 has a large gap between the two lowest voices, and 2:3:5:7 is more practical. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.

The concept of odd limit can be generalized to prime three in a [[Nonoctave|non-octave]] ("no-twos") tritave-equivalent context such as [[Bohlen-Pierce]]. Just as the words even and odd refer to divisibility by two, mathematicians use the words '''threeven''' and '''throdd''' for divisibility by three. The '''throdd limit''' of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The '''double throdd limit''' of 15/7 is (7,5). Its '''all-throdd voicing''' is 7/5. The 1/1 - 9/7 - 9/5 - 3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35.

==Lists of intervals by odd limit==

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 == Definition ==
-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.
+'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[Ratio|ratios]]. In the newer sense, discussed [[Odd limit#Odd limit of a ratio or chord|below]], the odd limit ''of a ratio'' is a specific number.
-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.
+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.
 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.
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 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 ==
+== Odd limit of a ratio ==
 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.
@@ Line 24: / Line 24: @@
 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 ==
@@ Line 51: / Line 49: @@
 This conjecture has two implications. First, a given JI chord has an ideal voicing. This voicing may be rather far-flung, and a more compact voicing may be almost as consonant. For example, 1:3:5:7 has a large gap between the two lowest voices, and 2:3:5:7 is more practical. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.
+The concept of odd limit can be generalized to prime three in a [[Nonoctave|non-octave]] ("no-twos") tritave-equivalent context such as [[Bohlen-Pierce]]. Just as the words even and odd refer to divisibility by two, mathematicians use the words '''threeven''' and '''throdd''' for divisibility by three. The '''throdd limit''' of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The '''double throdd limit''' of 15/7 is (7,5). Its '''all-throdd voicing''' is 7/5. The 1/1 - 9/7 - 9/5 - 3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35.
 ==Lists of intervals by odd limit==