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Why Two Notes Sound Good Together

About 11 minutes

Press two piano keys next to each other and hold them. The sound wobbles and scrapes and you want it to stop.

Now let two keys go and press two that are further apart. Sometimes you get the same scraping. Sometimes you get something that sounds like one rich sound rather than two notes at all.

The keys did nothing different. You pressed two of them both times. So what changed?

A note is a count

Every note is something wobbling very fast, over and over. A guitar string, a stretched drum skin, the column of air inside a recorder, the skin of a speaker.

What makes one note different from another is how many wobbles it fits into a second. Low notes wobble slowly, high notes wobble quickly. That is the whole of it.

So when you play two notes at once, you are running two counts side by side. And two counts either fit together or they do not.

Two counts, one pattern

Say one note wobbles twice in the time the other wobbles three times.

After that short stretch, both are back where they started, at the same instant, ready to do exactly the same thing again. The combined shape they make repeats, and it repeats quickly.

Two wobbles that lock quickly, and two that take far longer Two framed boxes side by side, each holding a pair of wavy lines with a clear band of space between them so the lines never touch. Both boxes have an upright bar at each end, and every wave sets off flat from the first bar and arrives flat on the second. In the left box the upper line makes two long arches and the lower line makes three, and the numbers two and three are written beside them. In the right box the arches are much smaller and there are many more of them, seven on the upper line and eight on the lower, with the numbers seven and eight written beside them. Both pairs do eventually line up, but the pair made of small numbers gets there in a couple of arches while the pair made of big numbers needs seven. 2 3 7 8
Both boxes hold two wobbles that eventually line up. On the left it takes two arches; on the right it takes seven. Your ear notices the difference between a pattern that repeats almost at once and one that keeps you waiting.

Now say one wobbles seven times while the other wobbles eight. They will line up eventually — but not for seven whole trips. Until then, every repeat is slightly different from the last one.

Your ear is very good at spotting a pattern that repeats. Give it one that settles in two steps and it relaxes. Give it one that takes seven and it never quite gets hold of it, and that unsettled feeling is what you hear as scraping.

Small whole numbers settle. Big ones grind.

The pairs musicians use most

The pairs with the smallest numbers turn up in every kind of music on Earth, which is a strong hint that this is about ears rather than about taste.

Six pairs of notes. Using only the numbers, decide whether each pair locks quickly or keeps the ear waiting. You do not need to have heard them.

  • 2 against 1
  • 3 against 2
  • 5 against 4
  • 9 against 8
  • 16 against 15
  • 45 against 32

Hear it on one rubber band

You can find the 2-against-1 pair yourself, with a rubber band, in about a minute.

  1. Stretch the rubber band round the box so it crosses the opening. Pluck it. Listen to the note.
  2. Press the pencil down onto the band, right in the middle, so the band is held in two halves. Pluck one half.
  3. That new note is exactly an octave above the first one. Half the length, twice the wobbles, 2 against 1.
  4. Now slide the pencil so it splits the band into two thirds and one third. Pluck the longer part. That is 3 against 2 — a fifth.
  5. Move the pencil slowly along and listen for the places where the note feels like it belongs with the open band, and the places where it does not.

The good places are not spread evenly along the band. They sit at the halves, the thirds and the quarters, which is exactly where the small whole numbers are.

You play two notes and one wobbles 200 times a second while the other wobbles 300 times. What should you expect to hear?

Why this does not settle everything

Small numbers explain a great deal, and they do not explain all of it. A note from a violin and the same note from a trumpet are the same count and sound nothing alike, and some music is built deliberately out of pairs that grind, because a chord that refuses to settle is exactly what makes you want the next one.

But if you ever wondered why the notes musicians use are the ones they use, rather than any of the endless others in between, the answer is sitting in that rubber band. Halves and thirds and quarters. Nothing more complicated than that.