Intervals

Story · Guided tour

How did one finding
open the next?

Start with one stretched string. Seven pages each add one finding and one sound experiment.

Discovery 5 of 7
1523 CE

The circle refuses to close

Twelve fifths overshoot seven octaves.

Twelve pure fifths should land on the same pitch as seven octaves. They miss.

The miss is 23.46 cents, called the Pythagorean comma. Powers of 3 cannot equal powers of 2. A keyboard with fixed pipes or strings must place that gap somewhere. Eleven pure fifths leave a sour twelfth interval at 678 cents, called a wolf. Pietro Aron described a meantone method in 1523 CE that tuned common major thirds close to 5:4. Temperament carries the problem through seven playable steps, including a meantone wolf.

Play the pure fifth, the wolf, then the comma alone. Listen for the 3:2 fifth to settle, the 678-cent wolf to sound wide, and the close comma tones to beat. Twelve pure fifths overshoot seven octaves, so a fixed-key instrument must store or spread the 23.46-cent gap. The rough wolf makes that tuning bargain audible.

Hear the comma and the wolf in Temperament ↗

Phonology · The vibrating string

This is a monochord.

One string is stretched over a sounding box between two fixed ends. The rule beneath it measures the string, and the movable bridge stops it wherever you set it. Musicians and early students of nature both worked with this instrument, because it turns a length you can measure into a pitch you can hear. Set the equal parts, watch the bridge slide, then pluck. The shorter the sounding length, the higher the pitch.

One string over a sounding box, a rule that measures it, and a bridge that stops it. Musicians and early students of nature both used this instrument to turn a measured length into a pitch. Set the equal parts, watch the bridge slide, then pluck. The shorter the sounding length, the higher the pitch.

Move the bridgewhole string · 110 Hz
Sounding length1