User:Dummy index/アドリミット
(under construction)
定義
アドリミット (odd-limit) は2つの意味を持つ。1つは純正音程（周波数比）の集合のシリーズである。もう1つ（後述）は任意のある純正音程（周波数比）に対しその特徴を取り出す関数である。
アドリミットは、周波数比の分母と分子から2の因数を除去（奇数になるまで2で割る）した上でそれらが（つまり分母と分子どちらも）ある値以下になる周波数比を集めた集合である。必然的にこのある値は正の奇数を挙げれば分別に十分となる。含まれる周波数比が少ない方から1アドリミット、3アドリミット、5アドリミット、…とそれぞれの集合が定義される。この列において左にある集合は右にある集合の部分集合となる。
アドリミットはまあおおよそハリー・パーチによるTonality Diamondsと同じものである。正確に言うなら、特定のアドリミットに対する特定の幾何学的表現としてTonality Diamondを見ることができる。この2つの用語はしばしば一緒に使われる（e.g., the 11-odd-limit Tonality Diamond）。アドリミットの集合の階層性を、小さいTonality Diamondが大きいTonality Diamondに埋め込まれていることをもって可視化することができる。
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]\displaystyle{ 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.
See also
- p-limit - or prime harmonic limit