Overview
An interval denotes the portion or span that lies between two specified bounds. Depending on the discipline it can mean a contiguous subset of numbers, a duration of time, the distance in pitch between two musical tones, or a range of plausible values in statistics. In every use it captures the idea of ‘‘in-between’’ rather than a single point.
Mathematical intervals
On the real number line an interval is any set containing all numbers between any two of its members. Common types include:
- Closed interval [a, b]: includes endpoints a and b.
- Open interval (a, b): excludes both endpoints.
- Half-open intervals [a, b) and (a, b]: include one endpoint and exclude the other.
- Unbounded intervals such as (a, ∞) or (−∞, b].
Intervals are order-convex and connected subsets of R; many basic topological and measure properties (length, interior, closure) are easily expressed for intervals. The length of a bounded interval [a,b] is b−a.
Other meanings and applications
In music an interval measures the pitch separation between two notes. Intervals are named (octave, fifth, third, etc.), counted in steps or semitones, and classified as melodic (sequential notes) or harmonic (simultaneous notes). In timekeeping an interval is a duration with a start and end, commonly used in scheduling, logging and historical chronology.
Statistics, computing and graphs
A confidence interval gives a range of parameter values considered consistent with observed data under a chosen statistical procedure; its interpretation depends on the statistical framework. Interval arithmetic treats numbers as ranges and defines arithmetic operations that propagate bounds, supporting rigorous numerical computations. In computer science, intervals model tasks with start and finish times; collections of intervals lead to interval graphs used in algorithms and resource allocation.
History, notation and distinctions
The English word derives from Latin intervallum, ‘‘space between’’. Notation varies by convention—square and round brackets are the most common—and context matters: an interval on the real line differs from a geometric segment only in terminology and emphasis on inclusion of endpoints. Understanding which endpoints are included is essential to correct interpretation.