intervals

Intervals are sets of numbers that lie between two endpoints, commonly used in mathematics and music. In mathematics, an interval is a connected subset of the real numbers. Finite intervals have endpoints a and b with a ≤ b, and are described by open, closed, or half-open notations: open (a,b) excludes endpoints; closed [a,b] includes them; half-open [a,b) or (a,b] include one endpoint. Unbounded intervals extend to infinity, such as (a, ∞), (-∞, b], or (-∞, ∞). The length or measure of a finite interval is b−a, time interval as a measure, and intervals are closed under certain operations: the intersection of intervals is always an interval (possibly empty), while the union of two intervals may be another interval or a union of disjoint intervals. Interval notation uses brackets and parentheses to indicate inclusivity of endpoints. Interval arithmetic uses the endpoints to bound results when combining intervals.

In music, an interval is the pitch distance between two notes. It is described by the number of letter names spanned and the number of semitone steps. Intervals are classified by quality and size, such as perfect, major, minor, augmented, and diminished. Within an octave, common simple intervals include the unison, second, third, fourth, fifth, sixth, and seventh; an octave increases the count to eight notes and twelve semitones. Intervals can be inverted, a process that moves the lower note above the upper one and changes quality in specific ways. Intervals underpin harmony and melody, contributing to consonance, dissonance, and tonal function.