Odd limit: Difference between revisions

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== Relationship to other limits ==

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.

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; it is equal to the exponentiation base two of the [[Weil height]]. 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.

Odd limit can be generalized to apply to chords in a number of 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. For example, both 4:5:6 and 10:12:15 have component intervals 3/2, 5/4, and 6/5. The intervals' odd limits are 3, 5, and 5. Thus both chords' intervallic limits are 5.

The '''otonal limit''' of a chord looks at each number in the extended ratio a:b:c~~...~~, and the odd limit of that number. The odd limit of a number is defined as the number itself if odd, and if even, the number divided by two until it is odd. The chord's otonal limit is the largest of these odd limits. ~~Example: 10:12:15 has numbers 10, 12~~ and ~~15, the odd~~ limits ~~of which are 5~~, ~~3 and 15, thus~~ the ~~chord~~'~~s otonal~~ limit is 15.

The '''otonal limit''' of a chord looks at each number in the extended ratio ''a'':''b'':''c'':…, and the odd limit of that number. The odd limit of a number is defined as the number itself if odd, and if even, the number divided by two until it is odd. The chord's otonal limit is the largest of these odd limits. The '''utonal limit''' is defined analogously. Combining the otonal and utonal limits, we can define the '''ambitonal limit''', which is the smaller value of the otonal and utonal limits of a chord.

~~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~~.

For example, 10:12:15 has numbers 10, 12 and 15, the odd limits of which are 5, 3 and 15, and thus the chord's otonal limit is 15. By contrast, 4:5:6's otonal limit is 5. 10:12:15 is sometimes considered more complex than 4:5:6, and the otonal limit is the measure that reflects that. However, 10:12:15 can be written as 1/(6:5:4), so the chord's utonal limit is 5, same as 4:5:6's otonal limit. Thus the ambitonal limits of both chords are 5, bringing them back to the same complexity level by recognizing each chord's more prominent otonal or utonal identity.

Note that the ambitonal limit is often equal to the intervallic limit, but not always, e.g. the chord 1-6/5-10/7-8/5 (left to readers as an exercise).

== Proposed Extensions ==

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 == Relationship to other limits ==
-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.
+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; it is equal to the exponentiation base two of the [[Weil height]]. 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.
+Odd limit can be generalized to apply to chords in a number of 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. For example, both 4:5:6 and 10:12:15 have component intervals 3/2, 5/4, and 6/5. The intervals' odd limits are 3, 5, and 5. Thus both chords' intervallic limits are 5.
-The '''otonal limit''' of a chord looks at each number in the extended ratio a:b:c..., and the odd limit of that number. The odd limit of a number is defined as the number itself if odd, and if even, the number divided by two until it is odd. The chord's otonal limit is the largest of these odd limits. Example: 10:12:15 has numbers 10, 12 and 15, the odd limits of which are 5, 3 and 15, thus the chord's otonal limit is 15.
+The '''otonal limit''' of a chord looks at each number in the extended ratio ''a'':''b'':''c'':…, and the odd limit of that number. The odd limit of a number is defined as the number itself if odd, and if even, the number divided by two until it is odd. The chord's otonal limit is the largest of these odd limits. The '''utonal limit''' is defined analogously. Combining the otonal and utonal limits, we can define the '''ambitonal limit''', which is the smaller value of the otonal and utonal limits of a chord.
-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.
+For example, 10:12:15 has numbers 10, 12 and 15, the odd limits of which are 5, 3 and 15, and thus the chord's otonal limit is 15. By contrast, 4:5:6's otonal limit is 5. 10:12:15 is sometimes considered more complex than 4:5:6, and the otonal limit is the measure that reflects that. However, 10:12:15 can be written as 1/(6:5:4), so the chord's utonal limit is 5, same as 4:5:6's otonal limit. Thus the ambitonal limits of both chords are 5, bringing them back to the same complexity level by recognizing each chord's more prominent otonal or utonal identity.
+Note that the ambitonal limit is often equal to the intervallic limit, but not always, e.g. the chord 1-6/5-10/7-8/5 (left to readers as an exercise).
 == Proposed Extensions ==