Odd limit
Definition
Odd limit has two meanings. In the original sense of the term, discussed first, an odd limit is a set of ratios. In the newer sense, discussed below, the odd limit of a ratio is a specific number.
An odd limit is the set of all ratios for which neither the numerator nor denominator exceed some maximum value, once all powers of 2 are removed. Typically, the maximum value is some small odd number, such as 3, 5, 7, 9, 11, etc. Each odd number gives rise to a different odd limit, so that there is a 3-odd-limit, a 5-odd-limit, and so on. This forms an increasing sequence of odd-limits, so that each numbered odd limit in this sequence is a subset of the next, so that the 3-odd-limit is a subset of the 5-odd-limit, which is in turn a subset of the 7-odd-limit, and so on.
Odd-limits are more or less equivalent to what Harry Partch calls Tonality Diamonds, in his theory. More precisely, a Tonality Diamond can be viewed as a particular geometric representation of a certain odd-limit, and the two terms are often used together (e.g., the 11-odd-limit Tonality Diamond). The sequence of increasing odd limits can be visualized as as a smaller tonality diamond being embedded in a set of progressively larger ones.
The purpose of an odd-limit or tonality diamond is to provide a "simple" subset of JI intervals to play, given one particularly natural definition of "simple." The removal of powers of 2 makes it so that for any interval that is viewed as "simple enough," the set of all its octave transpositions is also included in the set. Increasing the cutoff number increases the set of ratios viewed as being "simple enough" to be in the set. These are musically useful because such intervals will often tend to be play nicely with one another when forming chords (or at least, more so than some random JI intervals).
As an example, the 5-odd-limit is the set of intervals {1/1, 3/1, 1/3, 5/1, 1/5, 5/3, 3/5}, as well as every octave transpositions of the above (e.g. 2/1, 4/1, 3/2, 6/1, 5/4 and so on).
As a result, 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. Integer limit and prime limit are related concepts.
Mathematical Definition
The q odd limit, where q is an odd positive integer, consists of everything of the form 2^i*u/v, or [math]2^\mathbb Z\frac u v[/math], 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).
Odd limit of a ratio
From the definition above, we can see that an interval like 3/2 is not only part of the 3-odd-limit, but also the 5-odd-limit, the 7-odd-limit, and so on. However, it is also useful to refer to the *smallest* such odd limit that some interval fits into. This is often simply just called the "odd limit" of the ratio.
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.
This is also called the Kees expressibility of the interval, named after Kees van Prooijen who showed what this metric looks like geometrically on the lattice.
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.
The concept of odd limit can be generalized to prime three in a 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
- p-limit - or prime harmonic limit