# Odd limit

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The odd limit is a metric that places an upper bound on (i.e. limits) the complexity of the ratios used in a piece of music, and hence of the music itself. Odd limit can refer to the set of all ratios that are within this limit, or it can refer to the metric itself, applied to individual ratios. 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 odd numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 is greater than 3, thus the odd limit is 7.

To find the set of all ratios of N-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to N in every combination. Discard any ratios that can be simplified (e.g. 5/5 or 15/3). Transposing any of these ratios by an octave doesn't change the odd limit. Hence an odd limit set is theoretically infinite. For convenience, the odd limit set is usually written out in octave-reduced form.

For example, to find the 5-odd-limit set of ratios, pair off 1, 3 and 5: {1/1, 3/1, 5/1, 1/3, (3/3), 5/3, 1/5, 3/5, (5/5)}. Discard the two redundant ratios in parentheses. Octave-reduced and in ascending order, the 5-odd-limit set is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}.

The 3-odd-limit set of ratios is {1/1, 4/3, 3/2}. The 3-odd-limit set is contained in the 5-odd-limit set, both sets are contained in the 7-odd-limit set, and so on.

Note that the 5-odd-limit set contains no 2nds or 7ths. It's rare for an entire piece of music to strictly fall within the 5-odd-limit, because melodies generally do use the 2nd and/or the 7th of the scale. But a piece of music being in a certain odd limit usually means merely that at any given moment, no interval in the piece exceeds the odd limit. In other words, all vertical intervals within a chord, or between a melody note and a chord note, are within the odd limit. Even this looser definition excludes most songs. It perhaps includes "Kum Ba Yah", depending on the exact chords used. But even the simplest pentatonic songs usually have a melody note that is a major 2nd from some chord note. The major 2nd ratio is usually 10/9 or 9/8, making the piece 9-odd-limit. An even looser definition ignores the melody notes and requires only 5-limit chords. This definition includes any song that uses only major and minor triads.

Odd limits are more or less equivalent to what Harry Partch calls Tonality Diamonds. 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.

## Mathematical definitions

### Odd limit as a set of ratios

The q-odd-limit, where q is an odd positive integer, consists of every number of the form $2^i \cdot u/v$ where i is an integer and where u and v are odd positive integers less than or equal to q. It may be identified with the q-odd-limit tonality 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 as a property of a ratio

Given a ratio of positive integers p/q, its odd limit is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd numbers. The odd limit equals max(a, b). It's 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. The integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio; it is equal to the exponentiation base two of the Weil height. 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.

## Generalizations

### Odd limit of a chord

Odd limit can be generalized to apply to chords in a number of 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. For example, both 4:5:6 and 10:12:15 have component intervals 3/2, 5/4, and 6/5. The intervals' odd limits are 3, 5, and 5. Thus both chords' intervallic limits are 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. The utonal limit is defined analogously. Combining the otonal and utonal limits, we can define the ambitonal limit, which is the smaller value of the otonal and utonal limits of a chord.

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 1-6/5-10/7-8/5 (left to readers as an exercise).

### 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.

#### Nonoctave equaves

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.