SERIES / 1 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
- 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.
- 02
PLANNED
The Circle of Fifths Does Not Close: A Diophantine Problem in Music
Repeated pure fifths can approach but never exactly reach a stack of octaves, turning the familiar Pythagorean comma into a concrete theorem about prime factors.
- 03
PLANNED
Why Pitch Is Logarithmic: From Frequencies to the Geometry of Pitch
Logarithms convert multiplicative frequency relationships into additive distances, giving a natural mathematical explanation for octaves, cents, and pitch-class geometry.
- 04
PLANNED
Why Twelve Notes? Equal Temperament as Diophantine Approximation
The impossibility of closing pure fifths becomes a design problem: rational approximation explains why twelve-tone equal temperament is useful and how other equal divisions offer different compromises.
- 05
PLANNED
Prime Numbers Make Chords: Just Intonation and the Lattice of Musical Intervals
Factoring rational intervals into powers of 2, 3, and 5 turns just intonation into a lattice, making the structure of thirds, triads, and harmonic limits visible as integer geometry.
- 06
PLANNED
Commas, Temperaments, and Quotient Spaces: When Music Deliberately Identifies Different Ratios
Temperament can be understood as identifying selected differences in an interval lattice, connecting musical commas to kernels, integer matrices, and quotient-group intuition.
- 07
PLANNED
A Musical Note Is Not One Frequency: The Harmonic Series
Standing-wave structure explains why a musical note contains a fundamental and many partials, and why familiar small ratios appear naturally among the harmonics.
- 08
PLANNED
Why the Fifth Sounds Like a Fifth: Shared Harmonics, Beating, and Consonance
Shared harmonics, beating, roughness, and timbre show why simple ratios can matter acoustically without becoming a complete or universal theory of consonance.
- 09
PLANNED
Fourier’s Idea: Every Periodic Musical Sound Is Made of Simpler Oscillations
Fourier series turn the harmonic intuition into analysis by decomposing periodic waveforms into sinusoidal components and recovering the integer multiples behind musical spectra.
- 10
PLANNED
From Fourier Transform to Spectrogram: Seeing Music in Time and Frequency
The Fourier transform and short-time Fourier transform provide a mathematical way to study changing spectra, transients, instrument timbre, and the tradeoff between time and frequency resolution.
- 11
PLANNED
Notes on a Circle: Groups, Symmetry, and the Mathematics of Chords
Under octave equivalence and twelve-tone equal temperament, pitch classes become a finite algebraic space where transposition, inversion, interval classes, and chord symmetries can be studied together.
- 12
PLANNED
From Pythagoras to Harmonic Analysis: What Is the Mathematical Space of Music?
The final article gathers the series' models into one map, explaining why multiplication, logarithms, quotients, groups, periodicity, and Fourier harmonics recur across musical pitch and sound.