Interval

If T is a totally ordered set, and S is a subset of T, then S is called an interval if it has the property that whenever x and y are in S and x < z < y then z is in S.

A particularly important case is when T = R, the set of real numbers. The rest of this article will deal with this special case. The intervals of R are precisely the convex subsets of R, and are of the following eleven different types (where a and b are real numbers, with a < b):

(i) (a,b) = { x | a<x<b }

(ii) [a,b] = { x | a≤x≤b }

(iii) [a,b) = { x | a≤x<b }

(iv) (a,b] = { x | a<x≤b }

(v) (a,∞) = { x | x>a }

(vi) [a,∞) = { x | x≥a }

(vii) (-∞,b) = { x | x<b }

(viii) (-∞,b] = { x | x≤b }

(ix) R itself, the set of all real numbers

(x) {a}

(xi) the empty set

Note that a square bracket [ or ] indicates that the number is included in the interval, while a round bracket ( or ) indicates that it is not.

Intervals of type (i), (v), (vii), (ix) and (xi) are called open intervals; the intervals (ii), (vi), (viii), (ix), (x) and (xi) are called closed intervals; the intervals (i), (ii), (iii), (iv), (x) and (xi) are called bounded intervals; the intervals (v), (vi), (vii), (viii), (ix) are unbounded intervals.

The length of the bounded intervals (i),(ii),(iii),(iv) is b-a in each case. The total length of a sequence of intervals is the sum of the lengths of the intervals. No allowance is made for the intersection of the intervals. For instance, the total length of the sequence {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5.

Intervals are precisely the connected subsets of R. Since a continuous image of a connected set is connected, it follows that if f: R→R is a continuous function and I is an interval, then its image f(I) is also an interval. This is one formulation of the intermediate value theorem.

Intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define (see above). One then has to extend the concept of measure to more complicated sets.

Many theorems in analysis require that certain functions be continuous or differentiable on an open interval.

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An interval in musical theory is the difference in pitch? between two notes.

Intervals are inclusive of the two notes being considered; for example the interval between a C and a G is a fifth. As well as the number of tones between notes, also described can be the nature of the interval.

In a major scale intervals are named as follows:

Fig 1. Intervals in the C major scale

In the minor scales the minor interval is introduced.

Fig 2. Intervals in the C minor melodic scale

Fig 3. Intervals in the C minor harmonic scale

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