# Odd limit

## Definition

The odd limit is a metric that places an upper bound on (i.e. limits) the complexity of the harmonies used in a piece of music, and hence of the music itself. The term also refers to the metric itself, applied to individual ratios. Every ratio has an odd limit, and the q odd limit is the set of all ratios of odd limit of q or less. Integer limit and prime limit are related concepts.

To find the odd limit of a ratio: If either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 > 3, thus the odd limit is 7.

The q odd limit, where q is an odd positive integer, consists of everything of the form 2^i*u/v, or $2^\mathbb Z\frac u v$, where u and v are odd positive integers less than or equal to q. It may be identified with the q-limit diamond. Examples: some ratios in the 9-limit are: 3/2, 5/4, 7/6, 10/7, 12/7, 9/8 and 14/9. But not 11/9 (11 is a prime greater than 9) nor 15/7 (since 15 is 3*5, both less then 9, but with product greater than 9).

## 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. 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 two 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. Example: 10:12:15 has component intervals 6/5, 5/4 and 3/2. The intervals' odd limits are 5, 5 and 3, thus the chord's intervallic limit is 5.

The otonal limit of a chord looks at each number in the extended ratio a:b:c..., and the odd limit of that number. The odd limit of a number is defined as the number itself if odd, and if even, the number divided by two until it is odd. The chord's otonal limit is the largest of these odd limits. Example: 10:12:15 has numbers 10, 12 and 15, the odd limits of which are 5, 3 and 15, thus the chord's otonal limit is 15.

The intervallic limit and the otonal limit of a ratio are both equal to the ratio's odd limit, so both are valid generalizations of odd limit. In either sense, 4:5:6 is 5-limit. Since 10:12:15 is considered more complex than 4:5:6, the otonal limit could be considered the more musically useful of the two.

## Proposed Extensions

Kite Giedraitis has proposed several extensions to the concepts of odd limit and integer limit.

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 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.

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.