Physics-first music theory
Physics-first music theory
A dissonance curve plots total sensory dissonance as a function of frequency ratio for a given spectrum. The critical insight, formalized by William Sethares in 1993: the intervals that minimize dissonance depend on the timbre of the tones being played. There is no universal set of “consonant intervals” — only intervals that are consonant for a particular spectral content.
🎯 Simple version: Different instruments have different overtone recipes. When you calculate which note-combinations sound smoothest for each recipe, different instruments prefer different intervals. A xylophone’s “sweet spots” aren’t the same as a violin’s.
In Tuning, Timbre, Spectrum, Scale (1993), William Sethares demonstrated a revolutionary connection: the intervals a listener perceives as “consonant” are not fixed by nature but emerge from the interaction between a tone’s spectral content and the Plomp-Levelt dissonance model (see consonance-dissonance.md for the model itself).
The logic chain:
For harmonic timbres (integer-multiple partials: 1f, 2f, 3f, 4f…), the dissonance minima fall at ratios like 2:1, 3:2, 4:3, 5:4 — exactly the familiar step-intervals. For inharmonic timbres (non-integer partials), the minima shift to entirely different ratios. This means tuning and timbre are coupled parameters, not independent choices.
🎯 Simple version: Sethares proved that the reason the 7-step-interval (the “perfect fifth”) sounds good on a violin is because of violin overtones — not because 3:2 is inherently magical. Change the overtones, and the 7-step-interval might sound terrible while some completely different ratio sounds great.
The algorithm sweeps a frequency ratio from unison (1:1) to just past the octave (2:1) and computes total sensory dissonance at each point.
Setup:
{f₁, f₂, ..., fₙ} and amplitudes {a₁, a₂, ..., aₙ}α: its frequencies become {α·f₁, α·f₂, ..., α·fₙ} with the same amplitude profileFor each ratio α:
D_total(α) = Σᵢ Σⱼ d_PL(fᵢ, α·fⱼ, lᵢ, lⱼ)
where d_PL is the Plomp-Levelt dissonance function for a single pair of pure tones, and lᵢ and lⱼ are loudness values derived from the amplitudes.
Key details:
min(l₁, l₂) — so a pair contributes significant dissonance only when both components are reasonably loudThe sweep produces a curve with peaks (maximum roughness — many partial pairs within critical bandwidth) and valleys (minimum roughness — partial pairs either aligned or well-separated). The valleys indicate the least-dissonant intervals for that spectrum.
For a tone with harmonic partials (1f, 2f, 3f, 4f, 5f, 6f, 7f, 8f) and natural amplitude roll-off (1/n), the dissonance curve shows clear valleys at:
Ratio Step-interval (Western name)
1.000 0.00 (unison)
1.200 ~3 (minor third)
1.250 ~4 (major third)
1.333 5.00 (perfect fourth)
1.500 7.02 (perfect fifth)
1.667 ~9 (major sixth)
2.000 12.00 (octave)
These minima land near the step-intervals that 12-TET approximates: 0, 3, 4, 5, 7, 9, 12. This is not coincidence — it is the reason 12-TET works. The 12-tone equal-temperament system was designed (over centuries of empirical refinement) to serve instruments with harmonic spectra: strings, voices, brass, woodwinds. The dissonance curve for harmonic timbres has its valleys at almost exactly the positions 12-TET places its steps.
🎯 Simple version: If you plot “clashiness vs. interval” for violin-like sounds, the smooth spots land right where piano keys are. That’s not luck — pianos were designed for violin-like sounds.
When partials are not integer multiples of the fundamental, the dissonance minima shift — sometimes dramatically.
Many percussion instruments have partials at frequencies like 1f, 2.1f, 3.15f, 4.2f… — slightly “stretched” from the harmonic series due to the stiffness of bars and plates (see instrument-physics.md). For these spectra:
This directly explains gamelan tuning. The Indonesian gamelan ensemble uses metallophones and gongs with distinctly inharmonic spectra. The traditional slendro (5-note) and pelog (7-note) tuning systems are not “out-of-tune 12-TET” — they are tuning systems optimized for their instruments’ spectral content. Sethares computed dissonance curves for measured gamelan spectra and found that the traditional tunings closely match the predicted minima.
A pure sine tone has only one partial (the fundamental). The dissonance curve reduces to the raw Plomp-Levelt curve for a single pair: one broad hump of dissonance near unison, then smooth consonance everywhere beyond about one critical bandwidth. There are no special minima at step-intervals 5, 7, or 12 — those emerge only when multiple harmonics interact.
A square wave contains only odd harmonics (1f, 3f, 5f, 7f…). Its dissonance curve has fewer valleys than a full harmonic spectrum because fewer partial pairs are present. The 7-step-interval minimum weakens (since the 2nd harmonic, crucial for the 3:2 interaction, is absent), while other minima shift subtly.
Use the explorer below to see how different spectra produce different dissonance curves. Select a preset or adjust individual harmonic amplitudes, then use the ratio slider to hear any interval with that timbre.
Attribution: The interactive dissonance curve visualization below is heavily inspired by and based on Aatish Bhatia’s excellent Dissonance Curve Explorer. The underlying algorithm follows William Sethares’ model from Tuning, Timbre, Spectrum, Scale. We gratefully acknowledge Bhatia’s work in making this concept accessible and interactive — much of the visualization approach, including the spectral editing concept and curve rendering, was adapted from that project.
Sethares’ work reveals that tuning system and instrument timbre are not independent design choices — they are co-dependent parameters. Selecting a tuning system implicitly assumes something about the spectral content of the instruments that will play in it. Selecting an instrument implicitly constrains which tuning systems will sound smooth.
This coupling has always existed in practice. Western music theory evolved around instruments with (mostly) harmonic spectra — bowed strings, blown pipes, the human voice. The resulting tuning systems (Pythagorean, meantone, and eventually 12-TET) all target the dissonance minima of harmonic timbres. But this historical path was not recognized as a choice until Sethares made the timbre dependence explicit.
Twelve-tone equal temperament is not “the natural tuning” or “the best tuning.” It is the tuning system that minimizes dissonance for instruments with harmonic spectra while also providing equal transposability across all step-numbers. For a gamelan metallophone, 12-TET is a poor fit — the dissonance minima of its stretched partials fall between 12-TET steps, not on them.
The Sethares model validates what ethnomusicologists have long argued: non-Western tuning systems are not “primitive approximations” of 12-TET. They are independent engineering solutions optimized for their own instruments’ spectra. Javanese gamelan tunings, Thai 7-TET, and other systems are each locally optimal for their respective timbres — just as 12-TET is locally optimal for its own.
This is precisely why the PhizMusic framework insists on physics-first vocabulary. When we say “the 7-step-interval is consonant,” we must qualify: consonant for harmonic spectra. The dissonance curve makes this qualification unavoidable and explicit, grounding music theory in measurable spectral properties rather than cultural assumption.
| PhizMusic | Western | Notes |
|---|---|---|
| Dissonance curve | Sensory dissonance curve | Same concept |
| Spectral content | Timbre, tone color | PhizMusic is explicit about the spectrum |
| Step-interval of minimum | Consonant interval | Western term conflates sensory and cultural consonance |
| Timbre-tuning coupling | — | No standard Western theory term |
| Inharmonic spectrum | Inharmonic partials | Same concept |
| Stretched partials | Inharmonicity | Audio engineering term |