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 numerator and denominator. For example, the integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio, and is the same as the [[Weil height]]. The set of all ratios with an integer limit up to ''n'' is the same as the {{w|Farey sequence}} of order ''n''.

The '''integer limit''' of a ratio is simply the larger of the ratio's ~~two numbers~~. ~~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~~.

The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. 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 integer limit, when assuming [[octave equivalence]].

== Generalizations ==

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 == Relationship to other limits ==
+The '''integer limit''' of a ratio is simply the larger of the ratio's numerator and denominator. For example, the integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio, and is the same as the [[Weil height]]. The set of all ratios with an integer limit up to ''n'' is the same as the {{w|Farey sequence}} of order ''n''.
-The '''integer limit''' of a ratio is simply the larger of the ratio's two numbers. 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.
+The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. 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 integer limit, when assuming [[octave equivalence]].
 == Generalizations ==