Odd limit: Difference between revisions

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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 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 (i.e. factoring out all the twos), 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 '''multiple integer limit''' or '''MIL''' of a chord is simply the numbers of the extended ratio, listed highest to lowest. For any chord, the voicing which has the smallest MIL is the AOV, in which every number of the extended ratio is odd. The AOV of a chord is found by taking the odd limit of each number in the extended ratio, sorting them by size, and assembling them into a new extended ratio. For 4:5:6, the AOV is 1:3:5. For 10:12:15, the AOV is 3:5:15.

A chord in AOV is often impractically wide. The '''condensed all-odd voicing''' or '''CAOV''' octave-reduces every interval between adjacent notes. For example, 1:3:5 has a large gap between the two lowest voices, and 2:3:5 is more practical. To find the CAOV, begin with the AOV. Starting at the top, when you come to an interval wider than an octave, double all the numbers below it. Keep going until you reach the bottom. For example, the AOV of 10:12:15 is 3:5:15, and the CAOV is 6:10:15.

Kite has conjectured that the all-odd voicing of a just intonation ratio or chord is in general the most consonant voicing, with several caveats. Timbre matters. Register matters. Musical context matters. This conjecture may fail for ratios and chords with a high odd limit. For example, narrow all-odd ratios like 65/63 = 54¢ are better voiced widened by an octave. Also, the best voicing of 301/200 is not 301/25 but 301/100, because 301/200 is very close to a ratio with a much smaller odd limit, 3/2. Finally, it's difficult to judge the consonance of extremely wide intervals such as 11/1.

This conjecture has two implications. First, a given JI chord has an ideal voicing~~. This voicing may be rather far-flung, and a more compact voicing may be almost as consonant. For example, 1:3:5:7 has a large gap between the two lowest voices, and 2:3:5:7 is more practical~~. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.

This conjecture has two implications. First, a given JI chord has an ideal voicing. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.

==== Non-octave settings ====

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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 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 (i.e. factoring out all the twos), 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 '''multiple integer limit''' or '''MIL''' of a chord is simply the numbers of the extended ratio, listed highest to lowest. For any chord, the voicing which has the smallest MIL is the AOV, in which every number of the extended ratio is odd. The AOV of a chord is found by taking the odd limit of each number in the extended ratio, sorting them by size, and assembling them into a new extended ratio. For 4:5:6, the AOV is 1:3:5. For 10:12:15, the AOV is 3:5:15.
+A chord in AOV is often impractically wide. The '''condensed all-odd voicing''' or '''CAOV''' octave-reduces every interval between adjacent notes. For example, 1:3:5 has a large gap between the two lowest voices, and 2:3:5 is more practical. To find the CAOV, begin with the AOV. Starting at the top, when you come to an interval wider than an octave, double all the numbers below it. Keep going until you reach the bottom. For example, the AOV of 10:12:15 is 3:5:15, and the CAOV is 6:10:15.
 Kite has conjectured that the all-odd voicing of a just intonation ratio or chord is in general the most consonant voicing, with several caveats. Timbre matters. Register matters. Musical context matters. This conjecture may fail for ratios and chords with a high odd limit. For example, narrow all-odd ratios like 65/63 = 54¢ are better voiced widened by an octave. Also, the best voicing of 301/200 is not 301/25 but 301/100, because 301/200 is very close to a ratio with a much smaller odd limit, 3/2. Finally, it's difficult to judge the consonance of extremely wide intervals such as 11/1.
-This conjecture has two implications. First, a given JI chord has an ideal voicing. This voicing may be rather far-flung, and a more compact voicing may be almost as consonant. For example, 1:3:5:7 has a large gap between the two lowest voices, and 2:3:5:7 is more practical. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.
+This conjecture has two implications. First, a given JI chord has an ideal voicing. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.
 ==== Non-octave settings ====