Perfect fifth

The simple ratio

Two tones sound the more fused together, the simpler the ratio of their frequencies. The octave has 2:1, the perfect fifth 3:2, the perfect fourth 4:3, the major third 5:4. The larger the numbers become, the less often the vibrations coincide and the rougher the combination sounds. At 3:2 the common pattern repeats after every two and three vibrations respectively, a very short period that the ear perceives as one sound.

That two strings whose lengths stand in the ratio 3:2 produce, after the octave, the most pleasant-sounding interval has been ascribed to Pythagoras (around 500 BC) since antiquity; the attribution is considered legendary. If you stack pure fifths on top of each other, twelve different notes arise before the circle returns approximately to the starting note: that is the foundation of the circle of fifths.

Pure and tempered

Twelve pure fifths on top of each other give (3/2) to the power of twelve, about 129.746. Seven octaves give 2 to the power of seven, that is 128. The two numbers are not equal: a power of 3 is never a power of 2. The difference is called the Pythagorean comma, 23.46 cents. Equal temperament spreads it evenly, so every fifth is then 2 cents too low. Meantone tuning, on the other hand, shrinks every fifth a little to make the major thirds pure, and pays for it with one out-of-tune fifth, the wolf fifth.

A fifth in the equal-tempered system beats slowly, depending on register about once every one to three seconds (slower in the low register, faster in the high). Piano tuners set the temperament by beat rates, not by pure fifths.

Common mistakes

• Mixing up fifth and fourth: Fifth seven, fourth five semitones.
• Thinking all fifths are equally pure: On the piano no fifth is exactly pure.
• Hearing the fifth as a major chord: Without a third it has no colour.