Musical consonance

Why do some combinations of notes feel settled while others create tension?

Play two notes together and one pair sits still while another seems to want to go somewhere — a peculiar thing for air pressure to do. Before reaching for taste or training, ask whether anything in the sound itself distinguishes the two cases. If something does, we should find it before we start explaining the feeling.

First correct the picture of a note as a single frequency. A plucked string or a sung vowel does not vibrate only at its lowest rate; it vibrates at whole-number multiples of it too, so a string at 200 vibrations per second also puts out components at 400, 600, 800 and beyond. Sounding "one note" is already sounding a stack. If each note is a stack, what happens when two sound together?

Take 200 and 300 — a ratio of 3 to 2, the interval we call a fifth. The lower stack runs 200, 400, 600, 800, 1000, 1200; the upper runs 300, 600, 900, 1200. Notice 600 and 1200: not merely close but identical. Compare the octave, 200 with 400: every component of the upper tone is already present in the lower. Now take a ratio built from larger numbers and do the same arithmetic — exact coincidences become rare, and you get instead pairs of components sitting a few hertz, or a few tens of hertz, apart.

What does the ear do with two components close but not equal? They drift in and out of step, reinforcing and cancelling, so the combined loudness pulses at their difference. Two hertz apart is a slow wobble; twenty or thirty apart is too fast to count and is heard as a grating roughness. That roughness is not metaphorical — it depends on how close the pair falls relative to the ear's resolving power at that pitch, and it fades once they are far enough apart to be handled separately.

So a physical distinction really is available: simple ratios give many exact coincidences and few close-but-unequal pairs, complex ratios the reverse. Two accounts of smoothness follow, and they differ. One makes it the absence of roughness; the other says the ear favours sounds fitting the pattern of a single voice, which two tones in a simple ratio do, since together they resemble one deeper tone's stack.

Which is the honest hinge. Nothing so far tells us that smooth means pleasant, or that rough means tension wanting resolution. That needs a further ingredient: which intervals a style treats as stable, which as needing to move, where a piece may end. Those are conventions, and they change — in early Western polyphony the third was not among the stable consonances it later became. When researchers tested listeners in an Amazonian community with little exposure to Western music, they found little preference for consonant chords over dissonant ones, though those listeners heard other acoustic properties much as anyone else does; the result is read as evidence that the preference is shaped heavily by exposure. That reading is contested, and where the ear's work ends and a culture's begins remains open.

One more piece of evidence draws the line. The story above depends on partials being whole-number multiples of the lowest. On instruments where they are not — certain struck metal bars and bells — different intervals come out smooth, and tuning systems built around such instruments differ to match. The acoustics travel; the particular chords do not.

• Why equal temperament deliberately mistunes almost every interval, and what it buys in exchange.
• How the same beating that makes roughness is used deliberately to tune a piano.
• Why a chord's dissonance depends on the instrument playing it as much as on the notes.

Every term the collection defines is gathered in the glossary.