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SERIES / 3 PUBLISHED

The Mathematics of Musical Pitch

A sequence on ratios, tuning systems, geometry, and the mathematics behind musical pitch.

01 / sequence

Articles and planned notes

  1. 01
    PUBLISHED

    What Is a Musical Interval? Ratios, Tuning, and Consonance

    Why musical intervals are frequency ratios, from vibration and ancient harmonic theory to octaves, fifths, tuning systems, and the limits of ratio-based explanations of consonance.

  2. 02
    PUBLISHED

    The Circle of Fifths Does Not Close: A Diophantine Problem in Music

    Why twelve pure fifths do not equal seven octaves, and how prime factorization turns the Pythagorean comma into a theorem.

  3. 03
    PUBLISHED

    Why Is Pitch Logarithmic? From Frequencies to the Geometry of Pitch

    How logarithms turn frequency ratios into additive pitch distances, explaining octaves, cents, equal temperament, and the pitch-class circle.

  4. 05

    PLANNED

    Harmonics, Beating, and Consonance

    The harmonic series and shared partials explain why simple ratios can matter acoustically, while beating, timbre, and cross-cultural evidence show why ratios alone do not determine consonance.

  5. 06

    PLANNED

    Fourier Analysis of Musical Sound

    Fourier series, transforms, and spectrograms connect periodic harmonic structure to changing real-world sounds and the time-frequency tradeoff.

  6. 07

    PLANNED

    Pitch-Class Algebra and Musical Spaces

    Groups, symmetries, scales, chords, and voice-leading complete the series' mathematical map, showing how different pitch models are chosen and related.