Physics-first music theory
Physics-first music theory
Most acoustic instruments fall into three source archetypes with different vibration equations and different spectral outcomes. The crucial consequence is not just “different sound” but different harmonic structure, which changes interval fusion behavior and even preferred tuning strategies.
🎯 Simple version: A guitar string vibrates in whole-number patterns, so it makes clean harmonics. A pipe can make all harmonics or only odd ones. A drum skin vibrates in more complicated patterns that are not whole-number multiples, so its overtones are inharmonic. Different physics -> different musical roles.
For an ideal string of length L, tension T, and linear mass density mu:
f_n = n * (1 / 2L) * sqrt(T / mu), n = 1,2,3,...
Properties:
n = 1,2,3...) -> harmonic spectrumExcitation controls harmonic amplitudes (timbre), not mode frequencies:
So strings are a classic harmonic source family used for melody and harmony.
Air columns follow boundary-condition physics.
f_n = n * v / (2L), n = 1,2,3,...
All integer harmonics are available.
f_n = (2n - 1) * v / (4L), n = 1,2,3,...
Only odd harmonics appear ideally (1,3,5,...).
Perceptual effect:
This odd-harmonic constraint is the core physical reason closed-pipe families (for example clarinet-like bores in their low register approximation) differ from open-pipe families.
Circular membranes (drum heads) are 2D systems governed by Bessel-function mode families. Modal frequencies are proportional to Bessel zeros rather than integer multiples.
A normalized low-order sequence is approximately:
1.00, 1.59, 2.14, 2.30, 2.65, ...
These are inharmonic partials. Consequences:
Membrane physics explains why many drum sounds are perceived more by attack profile and spectral centroid than by stable scale degree.
This is the bridge to timbre.md: interval systems are not independent of source spectra.
This is Sethares’ coupling principle in physical form: “best” tuning depends on what kind of spectrum your instrument family actually produces.
Harmonic-template matching in consonance-dissonance.md works best when source spectra themselves are harmonic. With inharmonic spectra, fusion cues shift, and cultures often optimize for different interval compromises and beating textures.
So instrument construction is not downstream of theory; it co-defines theory.
Use the Timbre Designer below to explore how different source archetypes produce different spectra. Select a preset (String, Clarinet, Membrane, etc.) or edit individual partial ratios and amplitudes, then play sounds to hear the difference.
| PhizMusic | Western | Notes |
|---|---|---|
| String harmonic series source | String family acoustics | Same physical model |
| Open-open pipe all-harmonic model | Open pipe resonance | Flute-like idealization |
| Closed-open odd-harmonic model | Closed pipe resonance | Clarinet-like low-register idealization |
| Inharmonic membrane mode set | Drumhead modes | Bessel-zero governed, non-integer partials |