Odd limit: Difference between revisions

Line 31:

== Integer limit ==

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''.

[[File:WilsonHeightIntegerLimit.png|200px|thumb|right|Diagram by Lériendil showing the Wilson height (vertical axis) versus integer limit (horizontal axis) of simple intervals.]]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 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]].

@@ Line 31: / Line 31: @@
 == Integer limit ==
-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''.
+[[File:WilsonHeightIntegerLimit.png|200px|thumb|right|Diagram by Lériendil showing the Wilson height (vertical axis) versus integer limit (horizontal axis) of simple intervals.]]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 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]].