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The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example (,) [,] = (,]. If R {\displaystyle \mathbb {R} } is viewed as a metric space , its open balls are the open bounded intervals ( c + r , c − r ) , and its closed balls ...
4 members of a sequence of nested intervals. In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers =,,, … as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
The unordered pitch class interval i(a, b) may be defined as (,) = , , ,where i a, b is an ordered pitch-class interval (Rahn 1980, 28).. While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including Robert Morris, [1] prefer to use braces, as in i{a, b}.
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of ; these are the widths of the Riemann rectangles (hereafter "boxes"). Because the right Riemann sum is to be used, the sequence of x coordinates for the boxes will be x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} .
The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval (a, b). For example, for any summable sequence of positive numbers and any enumeration () of the rational numbers, the monotonically increasing function () = is continuous exactly at every irrational number (cf. picture).
In Western music and music theory, augmentation (from Late Latin augmentare, to increase) is the lengthening of a note or the widening of an interval. Augmentation is a compositional device where a melody, theme or motif is presented in longer note-values than were previously used. Augmentation is also the term for the proportional lengthening ...