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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 | axb }
(iii) [a,b) = { x | ax<b }
(iv) (a,b] = { x | a<xb }
(v) (a,∞) = { x | x>a }
(vi) [a,∞) = { x | xa }
(vii) (-∞,b) = { x | x<b }
(viii) (-∞,b] = { x | xb }
(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: RR 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.

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 can be perfect or major. A Unison is the interval between a note and itself (meaning normally just one note heard.) Fig 1. Intervals in the C major scale

In 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

Compound intervals

When an interval exceeds an octave, it is called a compound interval. For example, if a note was a 10th above middle C in a major scale it would be known as a 'compound major third.'

Concodant and discordant intervals

Concordant intervals usually 'sound' right. Discordant intervals jar, and can sound as if one of the notes wants to move up (this is the basis of suspensions.) Concordant intervals include:

  • Perfect intervals - 4ths, 5ths, 8ves
  • Imperfect intervals - minors/majors 3rds and 6ths

Discordant intervals include:

  • All other intervals - 2nds, 7ths, augmented or dimished notes

Modifying intervals

It is possible to modify intervals. Naming follows these rules:

  • If the bottom note of a perfect or major interval is lowered a semitone (or the top is raised,) the interval has been augmented.
  • If the bottom note of a perfect or minor interval is raised a semitone (or the top etc...,) the interval becomes diminished.
  • If the bottom note of a minor interval is lowered a semitone, the interval becomes major.
  • If the bottom note of a major interval is raised a semitone, the interval becomes minor.

Modified intervals often have more than one name. For example, a minor 7th can also be written as an augmented 6th. These are called enharmonic intervals. Typically names with 'minor' or 'major' in them are preferred, so the more correct way to write the interval given in the example is 'minor 7th.'