Music and mathematics

Music and mathematics explores numerical relationships and formal structures in musical sound, including rhythm, pitch, tuning, acoustics, and composition.

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Musical interval: The pitch distance between two notes, often expressed as a frequency ratio or number of semitones. Intervals turn relationships between pitches into quantities that can be compared and calculated.
Fourier analysis: A mathematical method that represents a signal as a sum of sinusoidal components. It decomposes complex musical sounds into component frequencies and amplitudes.
Rhythmic notation: A system of symbols that represents note durations, rests, and rhythmic grouping. Notation translates durations and subdivisions into readable proportional values.
Pythagorean tuning: A tuning system that derives musical intervals from ratios of small whole numbers. It exemplifies an early mathematical account of musical consonance.
Just intonation: A tuning approach that uses intervals defined by simple ratios of whole-number frequencies. Its pure ratios contrast with equal temperament’s evenly spaced, compromise intervals.
Frequency: The number of repeating cycles of a wave that occur per unit of time. A musical tone’s frequency provides the measurable basis for pitch and tuning.
Equal temperament: A tuning system that divides the octave into equal logarithmic pitch steps. Its geometric frequency spacing makes transposition between keys practical.
Twelve-tone technique: A compositional method that organizes all twelve chromatic pitch classes through an ordered series. Its row operations use systematic permutations to shape pitch relationships.
Pythagoras: An ancient Greek philosopher associated with a school that studied numerical relations in nature and music. Pythagorean traditions linked musical intervals to numerical ratios.
Well temperament: A family of tuning systems that permits playing in multiple keys while giving them distinct interval character. It offers a historical alternative to equal temperament’s identical semitone steps.