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Art and culture · Music and sound

Why does harmony sound pleasant? Simple ratios and the monochord

Some notes sound sweet together, others tense. From the Pythagoreans' string measurements to today's piano, discover the secret of simple ratios such as the perfect fifth (3:2) and perfect fourth (4:3) in strings and waves.

  • 18 min
  • 2 simulations
  • Mini quiz

Updated:

Preview

The question

Press C and G on a piano at the same time and you get a full, calm sound. Press C together with the C sharp right next to it and the sound becomes tense, almost "buzzing". In both cases you're only playing two notes. Why does your ear find one peaceful and the other unsettling?

Try it yourself

No account needed: change the values and check your answer. Some simulations ask for a prediction before they start; it doesn't matter whether it's right or wrong.

Sound lab

Build a perfect fifth

Get roughly close with the slider, then fine-tune in 1 Hz steps with the plus and minus buttons. Watch the sum wave at the bottom: the simpler the ratio, the shorter the interval at which the pattern repeats. If you like, press Play to listen, then press Check.

Prediction

Predict first

What do you think the frequencies of two notes should be like for them to sound most consonant together?

It doesn’t matter whether you’re right; the simulation opens once you choose.

Sound lab

Three quarters on the monochord

Adjust the frequency of the second tone; the ratio display and the waves will guide you. If you like, listen to the sounds together, then press Check.

Prediction

Predict first

If you vibrate only 3/4 of a string, what happens to its frequency?

It doesn’t matter whether you’re right; the simulation opens once you choose.

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Frequently asked

5 questions

Don't sounds with close frequencies sound more harmonious?

Quite the opposite. Two very close sounds, such as 220 and 225 Hz, produce fast beats and sound tense. What makes a pair harmonious is not closeness but frequencies linked by a simple ratio such as 2:1, 3:2 or 4:3.

Did Pythagoras find the ratios from a blacksmith's hammers?

That is a legend written centuries after Pythagoras. The sound of a hammer does not change with its weight in that way, so the story cannot be physically true. The ratios are measured with strings, for example on a monochord; these measurements are the legacy of the Pythagoreans.

Why do simple ratios sound pleasant?

According to Helmholtz's explanation, every sound has overtones at multiples of its fundamental frequency. Sounds of 220 and 330 Hz share many overtones exactly, such as 660 and 1320 Hz. As the ratio gets more complex, overtones that are close but not equal produce rough beats.

Do harmonious sounds please everyone equally?

Not exactly. In a study with the Tsimane people, who had heard little Western music, participants did not find dissonant chords more unpleasant than consonant ones. The physics is real, but habit and culture also play a part in why we like harmony.

Why isn't a perfect fifth on the piano exactly 3:2?

Twelve pure 3:2 fifths in a row do not land exactly on seven octaves: 3/2 to the power 12 is about 129.7, while 2 to the power 7 is 128. To play in every key, the octave is split into 12 equal semitones; in this system the fifth above a 220 Hz A is 329.6 Hz, not 330 Hz.