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; 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 '''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.

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.

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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; 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 '''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.
 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.