Odd limit: Difference between revisions
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== Definition == | == Definition == | ||
{{odd-limit navigation}} | |||
'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[Ratio|ratios]]. In the newer sense, discussed [[Odd limit#Odd limit of a ratio or chord|below]], the odd limit ''of a ratio'' is a specific number. | '''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[Ratio|ratios]]. In the newer sense, discussed [[Odd limit#Odd limit of a ratio or chord|below]], the odd limit ''of a ratio'' is a specific number. | ||
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The concept of odd limit can be generalized to prime three in a [[Nonoctave|non-octave]] ("no-twos") tritave-equivalent context such as [[Bohlen-Pierce]]. Just as the words even and odd refer to divisibility by two, mathematicians use the words '''threeven''' and '''throdd''' for divisibility by three. The '''throdd limit''' of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The '''double throdd limit''' of 15/7 is (7,5). Its '''all-throdd voicing''' is 7/5. The 1/1 - 9/7 - 9/5 - 3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35. | The concept of odd limit can be generalized to prime three in a [[Nonoctave|non-octave]] ("no-twos") tritave-equivalent context such as [[Bohlen-Pierce]]. Just as the words even and odd refer to divisibility by two, mathematicians use the words '''threeven''' and '''throdd''' for divisibility by three. The '''throdd limit''' of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The '''double throdd limit''' of 15/7 is (7,5). Its '''all-throdd voicing''' is 7/5. The 1/1 - 9/7 - 9/5 - 3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35. | ||
== See also == | == See also == | ||
* [[p-limit]] - or prime [[harmonic limit]] | * [[p-limit]] - or prime [[harmonic limit]] | ||
* [http://tonalsoft.com/enc/l/limit.aspx Limit - Tonalsoft Encyclopedia of Microtonal Music Theory] | * [http://tonalsoft.com/enc/l/limit.aspx Limit - Tonalsoft Encyclopedia of Microtonal Music Theory] | ||