Odd limit: Difference between revisions

Line 1:

== Definition ==

The '''odd limit''' is a metric that places an upper bound on (i.e. limits) the complexity of the ratios 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.

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 = 7. The '''~~integer~~ limit''' of ~~a ratio is simply the larger~~ of the ~~ratio~~'~~s two numbers, which is always~~ the ~~numerator~~. ~~The integer~~ limit of 12/7 is 12.

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.

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).

== Relationship to other limits ==

The integer limit more directly reflects the complexity of the ratio. But the odd limit is far more common, because the integer limit depends on the voicing of the interval, and the odd limit does not. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd limit can be thought of as the best-case-scenario integer limit. The odd limit reflects the complexity of the ratio in a context in which octave equivalence is assumed.<math>2^\mathbb Z\frac u v</math>The '''q''' '''odd limit''', where ''q'' is an odd positive integer, consists of everything of the form <code>2^i*u/v</code>, or , 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).

The '''integer limit''' of a ratio is simply the larger of the ratio's two numbers, which is always the numerator. The integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio. But the odd limit is far more common, because the integer limit depends on the voicing of the interval, and the odd limit does not. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd limit can be thought of as the best-case-scenario integer limit. The odd limit reflects the complexity of the ratio in a context in which octave equivalence is assumed.

Odd limit can be generalized to apply to chords in two ways. The '''intervallic limit''' looks at each interval of the chord, and the odd limit of that interval. The chord's odd limit is the largest of these odd limits. Example: 10:12:15 has component intervals 6/5, 5/4 and 3/2. The intervals' odd limits are 5, 5 and 3, thus the chord's intervallic limit is 5.

Line 11:

Line 17:

The intervallic limit and the otonal limit of a ratio are both equal to the ratio's odd limit, so both are valid generalizations of odd limit. In either sense, 4:5:6 is 5-limit. Since 10:12:15 is considered more complex than 4:5:6, the otonal limit could be considered the more musically useful of the two.

== Extensions ==

== Proposed Extensions ==

[[KiteGiedraitis|Kite Giedraitis]] has proposed several extensions to the concepts of odd limit and integer limit.

@@ Line 1: / Line 1: @@
+== Definition ==
 The '''odd limit''' is a metric that places an upper bound on (i.e. limits) the complexity of the ratios 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.
-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 = 7. The '''integer limit''' of a ratio is simply the larger of the ratio's two numbers, which is always the numerator. The integer limit of 12/7 is 12.
+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.
+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).
+== Relationship to other limits ==
-The integer limit more directly reflects the complexity of the ratio. But the odd limit is far more common, because the integer limit depends on the voicing of the interval, and the odd limit does not. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd limit can be thought of as the best-case-scenario integer limit. The odd limit reflects the complexity of the ratio in a context in which octave equivalence is assumed.<math>2^\mathbb Z\frac u v</math>The '''q''' '''odd limit''', where ''q'' is an odd positive integer, consists of everything of the form <code>2^i*u/v</code>, or , 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).
+The '''integer limit''' of a ratio is simply the larger of the ratio's two numbers, which is always the numerator. The integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio. But the odd limit is far more common, because the integer limit depends on the voicing of the interval, and the odd limit does not. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd limit can be thought of as the best-case-scenario integer limit. The odd limit reflects the complexity of the ratio in a context in which octave equivalence is assumed.
 Odd limit can be generalized to apply to chords in two ways. The '''intervallic limit''' looks at each interval of the chord, and the odd limit of that interval. The chord's odd limit is the largest of these odd limits. Example: 10:12:15 has component intervals 6/5, 5/4 and 3/2. The intervals' odd limits are 5, 5 and 3, thus the chord's intervallic limit is 5.
@@ Line 11: / Line 17: @@
 The intervallic limit and the otonal limit of a ratio are both equal to the ratio's odd limit, so both are valid generalizations of odd limit. In either sense, 4:5:6 is 5-limit. Since 10:12:15 is considered more complex than 4:5:6, the otonal limit could be considered the more musically useful of the two.
-== Extensions ==
+== Proposed Extensions ==
 [[KiteGiedraitis|Kite Giedraitis]] has proposed several extensions to the concepts of odd limit and integer limit.