Notwos-3reduced-limit

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Notwos-3reduced-limits:

Definition[edit]

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

A notwos-3reduced-limit is all the notwos ratios (numerator and denominator odd) 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 odd 3reduced-number, such as 5, 7, 11, 13, 17, 19, 23, etc. Each odd 3reduced-number gives rise to a different notwos-3reduced-limit, so that there is a 5-3reduced-limit, a 7-3reduced-limit, and so on. This forms an increasing sequence of notwos-3reduced-limits, so that each numbered notwos-3reduced-limit in this sequence is a subset of the next, so that the 7-3reduced-limit is a subset of the 11-3reduced-limit, which is in turn a subset of the 13-3reduced-limit, and so on.

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

The purpose of a notwos-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 7-3reduced-limit is the array of intervals[7] {1/1, 5/1, 1/5, 7/1, 1/7, 7/5, 5/7}, as well as every tritave transpositions of the above (e.g. 3/1, 9/1, 5/3, 7/3, 9/5 and so on).

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

Mathematical Definition[edit]

The q notwos-3reduced-limit, where q is an odd 3reduced-positive-integer, consists of everything of the form 3^i*u/v, or , where u and v are odd 3reduced-positive-integers less than or equal to q. It may be identified with the q-limit diamond. Examples: some ratios in the 7-limit are: 3/1, 5/1, 7/3, 5/7, 3/7, 9/1 and 15/7. But not 11/9 (11 is a prime greater than 7) nor 25/9 (since 25 is 5*5, both less then 7, but with product greater than 7) nor 5/4 (because factors of 2 are not supported).

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

From the definition above, we can see that an interval like 5/3 is not only part of the 5-notwos-3reduced-limit, but also the 7-notwos-3reduced-limit, the 11-notwos-3reduced-limit, and so on. However, it is also useful to refer to the *smallest* such notwos-3reduced-limit that some interval fits into. This is often simply just called the "notwos-3reduced-limit" of the ratio.

To find the notwos-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: 15/7 becomes 5/7, and 7 > 5, thus the notwos-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 15/7 is 15. The integer limit more directly reflects the complexity of the ratio. But the notwos-3reduced-limit is far more common, because the integer limit depends on the voicing of the interval, and the notwos-3reduced-limit does not. For example, 15/7 voiced a tritave wider is 45/7, integer limit 45. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the notwos-3reduced-limit. For 15/7, voicings 7/5 and 5/7 both have integer limit 7. Thus the notwos-3reduced-limit can be thought of as the best-case-scenario integer limit. The notwos-3reduced-limit reflects the complexity of the ratio in a context in which tritave equivalence is assumed.

Notwos-3reduced-limit can be generalized to apply to chords in two ways. The intervallic limit looks at each interval of the chord, and the notwos-3reduced-limit of that interval. The chord's 3reduced-limit is the largest of these 3reduced-limits. Example: 15:21:35 has component intervals 7/5, 5/3 and 7/3. The intervals' notwos-3reduced-limits are 7, 5 and 7, thus the chord's intervallic limit is 7.

The otonal limit of a chord looks at each number in the extended ratio a:b:c..., and the notwos-3reduced-limit of that number. The notwos-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 notwos-3reduced-limits. Example: 15:21:35 has numbers 15, 21 and 35, the notwos-3reduced-limits of which are 5, 7 and 35, thus the chord's otonal limit is 35.

The intervallic limit and the otonal limit of a ratio are both equal to the ratio's notwos-3reduced-limit, so both are valid generalizations of notwos-3reduced-limit. In either sense, 3:5:7 is 7-limit. Since 15:21:35 is considered more complex than 3:5:7, 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 notwos-3reduced-limit and integer limit.

The double notwos-3reduced-limit or DN3L of a ratio is simply the notwos-3reduced-limit of each number in the ratio, with the higher one listed first. DN3L (15/7) = (7, 5). The DN3L is useful as a tiebreaker when comparing the complexity of two ratios with the same notwos-3reduced-limit. For example, 51/49 and 49/47 are both 3reduced-limit 49. But DN3L (51/49) = (49, 17) and DN3L (49/47) = (49, 47). Since 17 < 47, 51/49 has a lower DN3L.

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-notwos-3reduced-voicing or AN3V, in which both the numerator and the denominator are 3reduced. The AN3V of a ratio is found by taking the 3reduced-limit of each number in the ratio, and combining them into a new ratio. For 15/7, the AN3V is 7/5. For 5/3, the AN3V is 5/1.

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 AN3V, in which every number of the extended ratio is 3reduced. The AN3V of a chord is found by taking the notwos-3reduced-limit of each number in the extended ratio, sorting them by size, and assembling them into a new extended ratio. For 3:5:7, the AN3V is 1:5:7. For 15:21:35, the AN3V is 5:7:35.

Kite has conjectured that the all-notwos-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 notwos-3reduced-limit. For example, narrow all-notwos-3reduced-ratios like 67/65 ≈ 52¢ are better voiced widened by a tritave. Also, the best voicing of 377/225 is not 377/25 but 377/75, because 377/225 is very close to a ratio with a much smaller 3reduced-limit, 5/3. 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, 1:5:7:11 has a large gap between the two lowest voices, and 3:5:7:11 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.

The concept of notwos-3reduced-limit can be generalized to prime two in an octave octave-equivalent context such as 12edo. 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.

Lists of intervals by notwos-3reduced-limit[edit]

See also[edit]

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