PhronesisThe kernel — grant it
Sound two tones close together and you hear the difference between them: a slow throb, beats at the difference frequency, the two waves drifting in and out of step. Widen the gap and something else happens. When the partials of the two tones fall inside one critical band — the ear's resolving width at that pitch — the beating speeds up past the point of hearing individual pulses and turns to roughness, the grating edge we call dissonance.
Helmholtz named the mechanism; Plomp and Levelt measured it. For small frequency differences the beats are heard singly and the sound stays smooth; push the difference toward roughly a quarter of the critical bandwidth and roughness peaks; widen past the whole band and the sound settles back to consonant and agreeable. Roughness is not worst at zero distance — a few beats a second is smooth. It is worst in the middle, and it clears at both ends.
A true interval is the far-cleared end reached by arithmetic. For tones with harmonics — the tones real instruments make — a simple ratio lines the overtones up. At the octave, 2:1, every partial of the upper tone lands exactly on a partial of the lower: nothing beats. At the fifth, 3:2, half the partials coincide and the rest sit well clear. At the fourth, 4:3, the same coincidence logic holds a rung looser. Simple ratio, coinciding partials, few beats, low roughness.
So the kernel of the seed is true. A matched, simple-ratio interval physically bestows smoothness — beat-free, low-roughness sound. This is measurable, and it holds for anyone with ears.
What it does not bestow
"Bestows" over-reaches the moment it is read to mean bestows feeling, meaning, healing, or beauty. The smoothness is one thing; the value laid on the smoothness is another, and the two come from different places.
Sensory consonance — the smoothness itself — is largely shared: the critical band is a fact of the cochlea, roughly the same ear to ear. But the preference for smooth over rough, whether consonance reads as pleasant or beautiful, is substantially learned. The Tsimané of the Bolivian Amazon, with little exposure to Western music, rated consonant and dissonant chords as equally pleasant. They were not deaf to roughness — they discriminated it and responded to rough sounds much as Westerners do. What was absent was the specific preference for the consonant chord over the dissonant one. City-dwellers in the same country showed the preference; US listeners showed it strongest.
The interval bestows a physical property. The value laid on it is learned.
So the smoothness travels; the loveliness does not, or not on its own. A true interval hands you the same clean sound everywhere. Whether that clean sound is heard as sweet is a thing your culture taught you, not a thing the ratio gave you.
No woo
This is acoustics and perception. It ends there.
A true interval bestows no health effect, no tuning of the body, no repair. There is no credible scientific evidence that 432 Hz tuning or the "solfeggio" frequencies confer any benefit or possess any acoustic superiority; the solfeggio set is a twentieth-century numerological invention, and the single small pilot comparing 432 to 440 Hz found only a borderline, unreplicated difference in heart rate. Smoothness is a property of the sound, not a medicine. Said plainly so it is not mistaken: the interval is not therapy, and the site's tone instruments — binaural, lung — are careful to claim nothing more than what they are.
The proposal
In the site's language, offered as one line, not a law: a true interval is a low-cost acoustic configuration — least beating, least roughness — a kind of ground state for a pair of tones. Of all the ways two tones can sit together, the simple ratio is the one that costs the ear the least work to hold. The unison is the floor; the octave, fifth, and fourth are the near shelves above it. That is all the claim is — a resting place in the space of intervals, named by arithmetic.
The limit
Two caveats keep the proposal honest, and it is stronger for them.
First, consonance is not a property of the ratio alone — it depends on the timbre. Sethares showed that the smooth intervals move with the spectrum: for the harmonic tones of ordinary instruments the roughness-minima sit at the simple ratios, recovering the familiar consonances, but for inharmonic spectra — bells, struck metal, gamelan bars — the minima shift off the small integers entirely. Almost any interval can be made smooth or rough by sculpting the timbre. The ratio account is the harmonic-tone special case of a wider rule, not a spectrum-free law.
Second, the pure ratios are rarely played exactly. Equal temperament — the tuning most Western instruments use — detunes every interval but the octave a little, trading pure ratios for the freedom to change key. The fifth you hear on a piano is close to 3:2, not exactly it. The true interval is the ideal the tuning approximates, not the note that sounds.
The live instrument is at /field/interval — sound the ratios and hear the smoothness clear.
Kin to the interval, binaural, lung, and The Logic of the Circle — the wave, sounded.
Rests on: Helmholtz's beating/roughness account (Die Lehre von den Tonempfindungen, 1863; trans. Ellis, On the Sensations of Tone, 1875) and Plomp & Levelt, "Tonal Consonance and Critical Bandwidth," J. Acoust. Soc. Am. 38 (1965): 548–560. Cultural variation in preference: McDermott, Schultz, Undurraga & Godoy, "Indifference to dissonance in native Amazonians reveals cultural variation in music perception," Nature 535 (2016): 547–550. Timbre dependence: Sethares, Tuning, Timbre, Spectrum, Scale (2nd ed., 2005). The physics is standard and established; reading a true interval as a ground state for two tones is the proposal, offered to be argued with.