intervaller

Intervaller, or intervals, is a term used in mathematics and music to describe a relation between two endpoints or pitches. The concept denotes a portion of a continuum bounded by two points and can be analyzed in terms of boundaries, size, and the way the endpoints are included or excluded.

In mathematics, an interval is a connected subset of real numbers. Finite intervals include open intervals (a, b), closed intervals [a, b], and half-open or half-closed intervals [a, b) or (a, b]. Infinite intervals extend to infinity, such as (a, ∞) or [a, ∞). Intervals are ordered by their endpoints, and their endpoints determine properties like openness or closeness. Operations on intervals include union and intersection, which produce new intervals or sets that are themselves intervals in many cases. Interval arithmetic deals with estimating the range of values resulting from expressions where inputs are known only to lie within certain intervals.

In music, an interval is the perceived distance between two pitches. It is named by a diatonic interval number (second, third, fourth, etc.) and a quality (diminished, minor, major, perfect, augmented). Simple intervals span within an octave; compound intervals exceed an octave. Distances are often measured in semitones or diatonic steps. Examples include the major third (four semitones) and the perfect fifth (seven semitones). Inversion pairs intervals to complementary sizes within an octave, such as a major third becoming a minor sixth. Tuning systems, such as equal temperament and just intonation, affect the exact size of intervals in practice.

Intervaller thus encompasses both mathematical ranges and musical distances, each with its own conventions and applications in theory and analysis.