Why do some notes sound good together while others clash?

Two notes sound good together when their vibrations fit tidily, and they clash when the vibrations rub against each other. The old rule of thumb is that consonance goes with simple ratios: the lower the whole numbers in the frequency ratio of two notes, the more consonant they sound. The octave is the extreme case, because the partials of the higher note are also partials of the lower one, so the two sounds blend almost into one.

The clash has a physical signature you can hear. Play two notes whose frequencies are just slightly different and you get a beating "wah-wah-wah". Speed the beating up past about twenty fluctuations a second and it stops sounding like a pulse and turns into roughness, a buzzy, fluttering quality. Roughness is strongest when two tones fall inside what the ear treats as one frequency band, a critical band, which is about as wide as a minor third in the low register and as narrow as a minor second high up.

But the story is not only physics. A major second would have been a clash in a Bach prelude, yet the same interval may sound consonant in Debussy. Some musical cultures treat the roughness as a treasure: Bosnian ganga singers, the Indian tambura drone and Indonesian gamelan all prize that slight buzzing. And in Western music dissonance is not the villain: a dissonance is tension that asks for an onward motion to a stable chord, which is exactly what makes the arrival feel good.