Physics-first music theory
Physics-first music theory
Consonance and dissonance are not binary categories but a continuous spectrum of perceptual qualities arising from multiple interacting mechanisms. At one end: tones that fuse into a single, smooth percept. At the other: tones that clash, beat, and create roughness. The mechanisms are biological; the aesthetic preferences built on top of them are cultural.
🎯 Simple version: Two notes close together fight — their vibrations clash and create a buzzy “roughness.” Notes far apart or in simple ratios cooperate — their vibrations reinforce each other. That’s consonance vs. dissonance. Everyone hears the difference; whether you like the clash is cultural.
Consonance and dissonance arise from at least four interacting mechanisms:
| When two frequencies fall within the same critical bandwidth (see ear-cochlea.md), the basilar membrane cannot fully separate them. The overlapping excitation patterns create beating — periodic amplitude fluctuation at the rate | f₂ - f₁ | . |
Maximum roughness occurs when the frequency separation is approximately 25-30% of the critical bandwidth — roughly 30-40 Hz for two tones around 500 Hz.
This means roughness is not about the ratio of two frequencies but about their absolute Hz separation relative to critical bandwidth. A 1-step-interval (ratio ~1.06) creates strong roughness in the low-to-mid register where critical bandwidth is narrow relative to the step, but less roughness in the extreme high register.
Roughness also applies to harmonic interactions: when two complex tones sound together, every pair of nearby harmonics can contribute roughness. The 1-step-interval between two complex tones produces roughness between nearly every pair of harmonics. The 7-step-interval (3:2 ratio) produces very little, because the harmonics align or fall far apart.
The auditory system continuously searches for harmonic series patterns in the incoming frequency data (see harmonic-series.md). When two or more tones together produce a frequency pattern that fits a single harmonic series template, the brain groups them as fused — heard as “one rich sound” rather than “two separate sounds.”
This mechanism explains why simple frequency ratios produce consonance: they are the ratios that naturally occur between harmonics of a single fundamental. The brain treats them as “one source.”
The nonlinear response of the cochlear amplifier (outer hair cells — see ear-cochlea.md) generates distortion products when two frequencies are present simultaneously. The most prominent:
Difference tone: f₂ - f₁
Cubic difference: 2f₁ - f₂
For consonant intervals, the combination tones fall on notes that reinforce the harmonic series implied by the two tones:
For dissonant intervals, combination tones can add unwanted frequencies that increase roughness and muddy the harmonic picture.
Reinier Plomp and Willem Levelt measured dissonance perception for pairs of pure sine tones (no harmonics) and found a universal curve:
Dissonance
^
| /\
| / \
| / \
| / \
| / \___________________
|/
+--------------------------------→ Frequency separation
0 ~25%CB CB 2×CB ...
For complex tones (with harmonics), the total dissonance is the sum of contributions from every pair of harmonic components. This predicts the consonance ranking of intervals quite accurately:
The 12-step-interval (2:1) > the 7-step-interval (3:2) > the 5-step-interval (4:3) > the 4-step-interval (5:4) > the 3-step-interval (6:5) > …
This ranking emerges from physics alone, without any cultural input. What cultures differ on is whether they prefer the consonant end, the dissonant end, or various mixtures.
When you calculate the total Plomp-Levelt dissonance for all harmonic pairs as a function of interval:
Total
Dissonance
^
| ‖
| ‖ ‖
|‖ ‖ ‖ ‖
| ‖ ‖ ‖ ‖ ‖ ‖
| ‖ ‖ ‖ ‖
+--+--+--+--+--+--+--+--+--+--+--+--+→ Step-interval
1 2 3 4 5 6 7 8 9 10 11 12
Peaks: near step 1, 2 (maximum roughness)
Valleys: at step 5, 7, 12 (minimum roughness — consonance)
Dissonance peaks near step-intervals 1 and 2 (where many harmonic pairs collide within critical bandwidth) and has valleys at step-intervals 5, 7, and 12 (where harmonic pairs either align or are well-separated).
This curve varies with:
Click any button below to hear the two tones separately and together. Listen for roughness at step 1 (maximum clash), fusion at steps 5 and 7, and the perceptual reset at step 12 (octave).
| Step-interval | Interval | Character | Listen |
|---|---|---|---|
| 0 | Unison | Identity — same frequency | |
| 1 | Minor 2nd | Maximum roughness | |
| 5 | Perfect 4th | Open, stable | |
| 7 | Perfect 5th | Maximum fusion | |
| 12 | Octave | Perceptual reset |
A critical distinction established by Josh McDermott and colleagues’ research with the Tsimané people of Bolivia — an indigenous population with minimal exposure to Western music:
| Dimension | Universal (biology) | Cultural (learned) |
|---|---|---|
| Perceiving fusion/roughness difference | ✓ Tsimané can distinguish | ✓ Everyone can |
| Preferring consonance over dissonance | ✗ Tsimané show no preference | ✓ Western listeners strongly prefer consonance |
The mechanisms described above (roughness, template matching, combination tones, Plomp-Levelt) are biological — they operate in every human cochlea and auditory system. But whether the resulting percepts are heard as “pleasant” or “unpleasant” is learned.
This means:
A final complication: the same interval can sound consonant or dissonant depending on musical context.
The brain processes intervals not just as isolated acoustic events but as part of a temporal sequence with built-in expectations. This temporal dimension is the domain of chord progressions.
| PhizMusic | Western | Notes |
|---|---|---|
| Roughness | Dissonance (partial concept) | Western “dissonance” conflates roughness with harmonic function |
| Perceptual fusion | Consonance (partial concept) | Western “consonance” conflates fusion with aesthetic preference |
| Critical bandwidth interaction | — | No standard Western theory term |
| Combination tone | Difference tone, Tartini tone | Same phenomenon |
| Harmonic template matching | — | Psychoacoustics term |
| Plomp-Levelt dissonance | Sensory dissonance | Acoustics term, not in standard music theory |