3reduced-limit

From TD Xenharmonic Wiki
Jump to navigation Jump to search

3reduced-limits:

Definition[edit]

3reduced-limit has two meanings. In the original sense of the term, discussed first, a 3reduced-limit is an array of ratios. In the newer sense, discussed below, the 3reduced-limit of a ratio is a specific number.

A 3reduced-limit is the set of all ratios for which neither the numerator nor denominator exceed some maximum value, once all powers of 3 are removed. Typically, the maximum value is some small 3reduced-number, such as 2, 4, 5, 7, 8, 10, 11, etc. Each 3reduced-number gives rise to a different 3reduced-limit, so that there is a 2-3reduced-limit, a 4-3reduced-limit, and so on. This forms an increasing sequence of 3reduced-limits, so that each numbered 3reduced-limit in this sequence is a subset of the next, so that the 3-3reduced-limit is a subset of the 5-3reduced-limit, which is in turn a subset of the 7-3reduced-limit, and so on.

3reduced-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 3reduced-limit, and the two terms are often used together (e.g., the 11-3reduced-limit Tonality Diamond). The sequence of increasing 3reduced limits can be visualized as as a smaller tonality diamond being embedded in a set of progressively larger ones.

The purpose of a 3reduced-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 3 makes it so that for any interval that is viewed as "simple. enough," the array of all its tritave 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 array. 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 5-3reduced-limit is the array of intervals[11] {1/1, 2/1, 1/2, 4/1, 1/4, 5/1, 1/5, 5/2, 2/5, 5/4, 4/5}, as well as every tritave transpositions of the above (e.g. 3/1, 9/1, 3/2, 6/1, 6/5 and so on).

As a result, the 3reduced-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, odd limit and prime limit are related concepts.

Mathematical Definition[edit]

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

3reduced-limit of a ratio or chord[edit]

From the definition above, we can see that an interval like 3/2 is not only part of the 2-3reduced-limit, but also the 4-3reduced-limit, the 5-3reduced-limit, and so on. However, it is also useful to refer to the *smallest* such 3reduced-limit that some interval fits into. This is often simply just called the "3reduced-limit" of the ratio.

To find the 3reduced-limit of a ratio: If either the numerator or the denominator mod 3 is false, divide it by three until it is true. The larger of the two numbers is the 3reduced-limit. Example: 12/7 becomes 4/7, and 7 > 4, thus the 3reduced-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[edit]

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 3reduced-limit is far more common, because the integer limit depends on the voicing of the interval, and the 3reduced-limit does not. For example, 12/7 voiced a tritave wider is 36/7, integer limit 36. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the 3reduced-limit. For 12/7, voicings 7/4 and 4/7 both have integer limit 7. Thus the 3reduced-limit can be thought of as the best-case-scenario integer limit. The 3reduced-limit reflects the complexity of the ratio in a context in which tritave equivalence is assumed.

3reduced-limit can be generalized to apply to chords in two ways. The intervallic limit looks at each interval of the chord, and the 3reduced-limit of that interval. The chord's 3reduced-limit is the largest of these 3reduced-limits. Example: 10:12:15 has component intervals 6/5, 5/4 and 3/2. The intervals' 3reduced-limits are 5, 5 and 2, 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 3reduced-limit of that number. The 3reduced-limit of a number is defined as the number mod 3 if true, and if false, the number divided by three until it is true. The chord's otonal limit is the largest of these 3reduced-limits. Example: 10:12:15 has numbers 10, 12 and 15, the 3reduced-limits of which are 10, 4 and 5, thus the chord's otonal limit is 10.

The intervallic limit and the otonal limit of a ratio are both equal to the ratio's 3reduced-limit, so both are valid generalizations of 3reduced-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[edit]

Kite Giedraitis has proposed several extensions to the concepts of 3reduced-limit and integer limit.

The double 3reduced-limit or D3L of a ratio is simply the 3reduced-limit of each number in the ratio, with the higher one listed first. D3L (12/7) = (7, 4). The D3L is useful as a tiebreaker when comparing the complexity of two ratios with the same 3reduced-limit. For example, 51/50 and 50/49 are both 3reduced-limit 50. But D3L (51/50) = (50, 17) and D3L (50/49) = (50, 49). Since 17 < 49, 51/50 has a lower D3L.

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-3reduced-voicing or A3V, in which both the numerator and the denominator are 3reduced. The A3V of a ratio is found by taking the 3reduced-limit of each number in the ratio, and combining them into a new ratio. For 12/7, the A3V is 7/4. For 3/2, the A3V is 2/1.

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 A3V, in which every number of the extended ratio is 3reduced. The A3V of a chord is found by taking the 3reduced-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 A3V is 2:4:5. For 10:12:15, the A3V is 4:5:10.

Kite has conjectured that the all-3reduced-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 3reduced-limit. For example, narrow all-3reduced-ratios like 67/65 ≈ 52¢ are better voiced widened by a tritave. Also, the best voicing of 301/225 is not 301/25 but 301/75, because 301/225 is very close to a ratio with a much smaller 3reduced-limit, 4/3. 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 a tritave, either 14/3 or 19/4 may be preferred to 24/5.

See also[edit]

Retrieved from "https://tdxen.replit.app/index.php?title=3reduced-limit&oldid=16959"