Consonance

Two tones close together do not sound like two tones

Play two sine waves a couple of hertz apart and you do not hear two pitches. You hear one pitch that pulses, getting louder and softer a few times a second. That is beating: the two waves drift in and out of phase, reinforcing and cancelling.

Push them further apart and the pulsing speeds up until you stop hearing it as a rhythm and start hearing it as a texture, a rough, buzzing quality. Push further still and the roughness fades, and you finally hear two separate, clean pitches.

That progression is not a quirk of taste. It is a consequence of how the inner ear resolves frequency. The basilar membrane maps frequency onto position, and two frequencies close enough to excite overlapping regions cannot be told apart. The width of that overlap is the critical band, and it is proportionally wider at low frequencies, which is why a semitone in the bass sounds muddy and the same semitone two octaves up sounds clean.

Now do that for every pair of partials at once

Two complex tones bring a dozen or more partials between them. Every pair contributes some roughness, depending on how far apart they sit relative to the critical band at that frequency. Add it all up and you get a single number for how much any given interval grates.

Sweep one tone against the other and plot that number. Nobody has told this graph where the good intervals are.

What just happened

The valleys are the consonances. They land at 2:1, 3:2, 4:3 and 5:4, within about a cent, and they are there because simple frequency ratios make partials coincide rather than sit close enough to fight. When the upper tone is exactly 3:2 above the lower one, its second partial lands exactly on the lower tone's third. Coinciding partials contribute no roughness at all. Near misses contribute the most.

Those are the same ratios you found inside a single note in lesson 2. That is not a coincidence, it is the same fact viewed twice: the harmonic series puts partials at integer multiples, and intervals built on small integer ratios are exactly the ones whose partials line up.

Two things are worth trying before moving on, because they are the reason this lesson exists rather than a list of consonant intervals.

Switch to the pure sine. The valleys do not get shallower, they stop existing. What is left is one smooth hump: roughness rising as the two partials separate, then falling away as they resolve, with nothing to distinguish a fifth from anything either side of it. One partial each leaves a single pair to interact, and a single pair has no ratios to care about. Consonance is not a property of an interval. It is a property of an interval and a spectrum.

Then drag the stretching slider. The partials stop being integer multiples, and every valley walks away from the ratio it was sitting on. The intervals that sound good are the intervals whose partials happen to line up, and if you change what the partials are doing, you change which intervals those are. Gamelan tuning is not a mistake; it is a scale matched to the spectra of the instruments that play it.

A caveat worth stating plainly, because Battuto claims physics elsewhere and this part is not quite physics. The roughness model here is fitted to listener data rather than derived from mechanics. The robust finding is the shape: roughness peaks around a quarter of a critical band and vanishes at both ends. The exact constants are one published fit among several, and researchers still argue about whether consonance is roughness at all or partly a matter of the brain matching harmonic templates. What is not in dispute is that coinciding partials do not beat.