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The Harmonious Blacksmith

The ancient Greek philosopher Pythagoras (ca. 580 - 500 B.C.) is generally credited with having discovered that musical intervals which are recognized as concordant are related by small integer ratios. Prior to Pythagoras, musicians appreciated that particular notes when sounded together created a pleasant effect, and tuned their lyres so that plucking two strings would generate such a harmony. However, the early musicians had no understanding of why particular notes were harmonious and had no objective system for tuning their instruments. Instead they tuned their lyres purely by ear until a state of harmony was established--a process that Plato called torturing the tuning pegs.

Iamblichus, the fourth-century scholar who wrote nine books about the Pythagorean sect, describes how Pythagoras came to discover the underlying principles of musical harmony:

  "Once he was engrossed in the thought of whether he could devise a mechanical aid for the sense of hearing which would prove both certain and ingenious. Such an aid would be similar to the compasses, rules and optical instruments designed for the sense of sight. Likewise the sense of touch had scales and the concepts of weights and measures. By some divine stroke of luck he happened to walk past the forge of a blacksmith and listened to the hammers pounding iron and producing a variegated harmony of reverberations between them, except for one combination of sounds."

According to Iamblichus, Pythagoras immediately ran into the forge to investigate the harmony of the hammers. He noticed that most of the hammers could be struck simultaneously to generate a harmonious sound, whereas any combination containing one particular hammer always generated an unpleasant noise. He analyzed the hammers and realized that those that were harmonious with each other had a simple mathematical relationship--their masses were simple ratios or fractions of each other. That is to say that hammers half, two- thirds, or three-quarters the weight of a particular hammer would all generate harmonious sounds. On the other hand, the hammer that was generating disharmony when struck along with any of the other hammers had a weight that bore no simple relationship to the other weights.

Here is Händels theme for the Harmonious Blacksmith, from his fifth suite for harpsichord.

Frontispiece to Theorica Musice, Franchino Gafurio, 1492

The upper left illustration depicts Jubal, the biblical father of music, and six blacksmiths with differing size hammers striking an anvil. This relates to the story that the young Pythagoras was first moved to investigate musical intervals on hearing the notes produced by different size hammers at a blacksmith's shop.

The upper right illustration depicts Pythagoras testing the interval of an octave between bells of size 16 and 8 and between glasses filled in the proportion 16 and 8.

The lower left illustration shows Pythagoras testing intervals on a stringed instrument and the lower right illustration shows Pythagoras and his pupil Philolaus testing intervals by means of flutes.

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In collaboration with Eivind Grovens Institutt for renstemming