Why some notes sound right together

Play two notes together and one of two things happens. Either they settle into something that sounds stable and agreeable, or they grind against each other. Musicians call this consonance and dissonance, and it is tempting to treat it as pure convention. It is not. Underneath sits some straightforward physics about vibrating objects, and a genuine compromise that Western music made a few centuries ago and has lived with ever since.

Pitch is frequency

A sound is a vibration travelling through air, and pitch is how fast that vibration repeats. Faster vibration, higher note. The unit is the hertz, meaning cycles per second, and human hearing spans roughly 20 Hz at the bottom to somewhere near 20,000 at the top — the upper limit falling steadily with age, which is why some people can hear a whining television and others in the same room genuinely cannot.

The interesting part is not the individual frequencies but the relationships between them, because the ear is far more attentive to ratios than to absolute values. A tune played high or low is recognisably the same tune, even though not a single frequency in it is shared.

The octave is a doubling

Start at any note and double its frequency and you arrive at a note that sounds, in a strange way, like the same note again. Higher, obviously, but somehow equivalent — so much so that most musical cultures give the two the same name.

That is the octave, and it is the most consonant interval there is because the ratio is the simplest possible: 2 to 1. Every second peak of the lower wave lines up with a peak of the higher one. Nothing fights.

The next simplest ratio, 3 to 2, gives the interval Western music calls a perfect fifth — the gap between C and G. Then 4 to 3 gives a fourth, 5 to 4 a major third. As the numbers involved get larger and less tidy, the intervals sound progressively less settled. There is a rough and genuine correlation between mathematical simplicity and perceived consonance.

Why a violin and a flute differ

If pitch were the whole story, every instrument playing the same note would sound identical. They plainly do not, and the reason is that almost nothing in nature vibrates at a single frequency.

A plucked string vibrates along its whole length, producing its fundamental pitch. But it also vibrates in halves, in thirds, in quarters, all simultaneously, each producing a fainter higher tone. These are overtones, and their frequencies are whole-number multiples of the fundamental. What distinguishes a violin from a flute from a human voice is largely which overtones are present and how loud each one is. That mixture is timbre.

This also supplies a deeper explanation for consonance. When two notes an octave apart sound together, the overtone series of the higher note is a subset of the lower one's — the two share most of their content. Notes in less simple ratios have overtones that fall close to one another without matching, and closely spaced frequencies produce an audible roughness called beating. Dissonance is, to a significant degree, the sound of overtones colliding.

The problem nobody can solve

Here is where it gets awkward, and where a genuine compromise enters.

Stack twelve perfect fifths, each a clean 3-to-2 ratio, and you should arrive back at your starting note seven octaves higher. You do not. You land slightly sharp — by a small but clearly audible amount known as the Pythagorean comma. The discrepancy is not a measurement error or a limitation of instruments. It is arithmetic: powers of three never exactly equal powers of two, so a system built from pure fifths cannot close.

Musicians have known this for well over two thousand years, and every tuning system is a different answer to it. Tune the fifths pure and some keys sound beautiful while others are unusable. Adjust some intervals and you can spread the error around.

The compromise inside your piano

The solution that Western music largely settled on is equal temperament: divide the octave into twelve equal steps and accept that almost every interval is slightly out of tune.

Only the octave remains pure. The fifths are very slightly narrow, the major thirds noticeably wide. The gain is that every key sounds identical in character, so a piece can move freely between them and an instrument needs no retuning. That freedom is what made a great deal of later Western music possible.

The cost is real, though most listeners never consciously notice it. A choir or a string quartet, having no frets or fixed keys, will often drift toward pure ratios instinctively, tuning chords more sweetly than a piano can. If unaccompanied singing has ever sounded richer to you than the same harmony on a keyboard, that is why.

How much of this is universal

It would be neat to conclude that consonance is simply physics and therefore identical everywhere, but the honest position is more mixed.

The octave appears near-universally across musical cultures, and the fifth very widely, which is about what the overtone argument predicts. Beyond that, practice diverges considerably. Many traditions divide the octave differently, use intervals that Western ears hear as out of tune, or treat intervals as unstable that Western theory calls consonant. Gamelan tunings do not map onto twelve equal steps at all.

So the physics constrains without dictating. There are real acoustic reasons why certain combinations sound stable, and a great deal of latitude in what a culture builds on top of them. Which is roughly the most interesting possible answer: not arbitrary, not determined, but a long conversation between the behaviour of vibrating objects and what people decided to do about it.