Odd limit: Difference between revisions
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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. | 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 | 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 === | === Proposed extensions === | ||
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The '''double odd limit''' or '''DOL''' of a ratio is simply the odd limit of each number in the ratio, with the higher one listed first. DOL (12/7) = (7, 3). The DOL is useful as a tiebreaker when comparing the complexity of two ratios with the same odd limit. For example, 50/49 and 49/48 are both odd limit 49. But DOL (50/49) = (49, 25) and DOL (49/48) = (49, 3). Since 3 < 25, 49/48 has a lower DOL. | The '''double odd limit''' or '''DOL''' of a ratio is simply the odd limit of each number in the ratio, with the higher one listed first. DOL (12/7) = (7, 3). The DOL is useful as a tiebreaker when comparing the complexity of two ratios with the same odd limit. For example, 50/49 and 49/48 are both odd limit 49. But DOL (50/49) = (49, 25) and DOL (49/48) = (49, 3). Since 3 < 25, 49/48 has a lower DOL. | ||
The '''double integer limit''' or '''DIL''' of a ratio a/b is (b, a). For any interval, the voicing which has the smallest DIL is the '''all-odd voicing''' or '''AOV''', in which both the numerator and the denominator are odd. The AOV of a ratio is found by taking the odd limit of each number in the ratio, and combining them into a new ratio. For 12/7, the AOV is 7/3. For 3/2, the AOV is 3/1. | The '''double integer limit''' or '''DIL''' of a ratio ''a''/''b'' is (''b'', ''a''). For any interval, the voicing which has the smallest DIL is the '''all-odd voicing''' or '''AOV''', in which both the numerator and the denominator are odd. The AOV of a ratio is found by taking the odd limit of each number in the ratio, and combining them into a new ratio. For 12/7, the AOV is 7/3. For 3/2, the AOV is 3/1. | ||
The concept of integer limit can be generalized to apply to a chord either intervallicly or otonally. Either way, the integer limit is the highest (final) number of the extended ratio. | The concept of integer limit can be generalized to apply to a chord either intervallicly or otonally. Either way, the integer limit is the highest (final) number of the extended ratio. | ||
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==== Nonoctave equaves ==== | ==== Nonoctave equaves ==== | ||
The concept of odd limit can be generalized to prime three in a [[ | 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 1–9/7–9/5–3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35. | ||
==== Equave limit ==== | ==== Equave limit ==== | ||
[[Equave limit]] | See [[Equave limit]]. | ||
== See also == | == See also == | ||
* [[Harmonic limit|''p''-limit]] | * [[Harmonic limit|''p''-limit]] – or prime [[harmonic limit]] | ||
* [[List of 47-odd-limit intervals]] | * [[List of 47-odd-limit intervals]] | ||
* [[Shadow]] (based on integer limit) | * [[Shadow]] (a psychoacoustic effect based on integer limit) | ||
== External links == | == External links == | ||
* [http://tonalsoft.com/enc/l/limit.aspx Limit | * [http://tonalsoft.com/enc/l/limit.aspx Limit – Tonalsoft Encyclopedia of Microtonal Music Theory] | ||
[[Category:Odd limit| ]] <!-- main article --> | [[Category:Odd limit| ]] <!-- main article --> | ||
[[Category:Limit]] | [[Category:Limit]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||