PYTHAGOREAN HARMONICS : FROM PYTHAGORAS TO NEWTON

Author: * Sileinos Socrates - 2 Posts on this thread out of 12 Posts sitewide.
Date: Apr 25, 2005 - 20:49

Musical chords and ratios, as demonstrated by Pythagoras. A woodcut from the THEORICA MUSICE, Naples, 1492, by Franchino Gafurio, music theorist and choirmaster of Milan cathedral.

In this late medieval or early Renaissance woodcut, Pythagoras is shown discovering the physical and mathematical ratios that underlie the basic chords and the most harmonious musical intervals. Tradition says that after noticing the different sounds of blacksmiths’ hammers, Pythagoras embarked on the experiments shown here: striking bells, water glasses and strings; and blowing flute pipes. The hammers, bells, glasses, strings and pipes are labelled 4, 6, 8, 9, 12 and 16, to indicate their weights and sizes; and each experiment supposedly demonstrates the same interval or chord: the octave, whose ratio is 1/2 or 1:2, shown here as the ratio 8:16.

In the panel at upper left Pythagoras watches the blacksmiths take turns hitting an anvil. One strikes the base note, soon to be followed by its octave, supposedly sounded by the 8-pound hammer at the upper right. In a charming attempt at reconciling Christian piety with pagan wisdom, the artist has relabelled the Pythagoras figure as IV BAL — Latin lettering for JUBAL, whom the Bible declares to be the father of music.

In the panel at upper right Pythagoras is shown striking two bells at once: a size 8, or 8-pound bell, and a bell twice as big (#16), supposedly producing an octave, whose ratio is therefore 8/16 or 1/2. Just below he is shown performing the same experiment with glasses filled with varying amounts of water. Here 8 and l6 ounces also produce the high and low tones of an octave. By adding several more glasses, and rubbing their rims instead of striking them with sticks, Pythagoras could have invented the Glass Harmonica that Benjamin Franklin (re)invented and for which Mozart composed a delightful piece. In ancient Greece, however, even centuries after Pythagoras, even unblown glass was surprisingly rare and expensive. So it is doubtful that Pythagoras ever tinkered with glassware of any sort. Or with bells, for that matter.

In the panel at lower left Pythagoras is shown striking with sticks six stretched strings of ox sinews and sheep guts. Instead of a single string “stopped” at different lengths, or several strings of different lengths, the strings are "tuned" by different tensions — stretched by different weights. Pythagoras is striking the two strings whose weights are labelled 8 and l6, again supposedly producing an octave. The artist has been a bit careless, however, and it looks as if Pythagoras might be hitting instead the strings stretched by the 9 and l2 pound weights. In that case, he would be sounding the chord called a "fourth," whose ratio is 3:4 (= 9:12).

In the lower right panel Pythagoras, assisted by his disciple Philolaus, demonstrates the harmonic chords and ratios with flute pipes. Here also the chord being played is the octave, with Philolaus sounding the base note on a pipe l6 units long, while his master blows a pipe half its length, one labelled 8.

What these five experiments supposedly show is that the harmonic chord or interval called the octave always corresponds to a mathematical and physical difference whose ratio is 1:2.

By using other pairs of the instruments in each series Pythagoras and his followers could demonstrate the mathematical ratios underlying other basic chords or intervals. Thus, bells, strings or pipes sized, weighted, cut or stopped in the ratios 6 to 9 (= 2:3) were said to produce the chord or harmonic interval called a "fifth." And those sized or weighted 9 and 12 (= 3:4) would produce a chord or interval called the "fourth."

Why, then, does the series of numbers in the woodcut run from 4 to 16? For simplicity, why aren't there just four basic weights and lengths here, labelled 1, 2, 3 and 4? The reason is that theorists wanted to include the ratio for the interval between two whole tones on the musical scale, which has the seven whole notes plus the octave: Do re mi fa sol la ti do. The ratio for the physical difference between two successive whole notes is 8:9. In other words, a string has to vibrate along 8/9ths of its length to produce a tone that is one whole note above the base note produced by its entire unstopped length. Therefore the ratio for the interval between these two notes is 8:9. And the series of numbers from 4 to 16 is the smallest series that includes this ratio in addition to the ratios for the octave (1:2 = 8:16) and those for the fifth and fourth.

THE PYTHAGORAS WOODCUT & MODERN PHYSICS

In 1492 the medieval technique of printing from woodcuts was still the only cheap way of reproducing pictures in quantity. To make a woodcut, the artist usually draws the illustration directly on the surface of the woodblock, then cuts into the surface along the drawn lines. The block is then inked with a roller that spreads the ink over the uncut surface while leaving uninked the cut-out lines and areas below the surface. When a sheet of white paper is pressed against the block, the inked areas on it are transferred to the paper, leaving the uninked spaces down in the block to appear white on the uninked areas of the paper. The printed Pythagoras woodcut would therefore normally and most cheaply show the design as white lines on a black background - and most reproductions show it that way. So either this illustration was reproduced with colors reversed, or it came from a later, more expensive edition that was cut from harder wood, with the areas around the lines cut out, instead of vice versa. Its details are certainly clearer, which is why I chose it.

In this woodcut some of the panels may have been created by different craftsmen, explaining why “Pythagoras” is spelled differently in the lower left panel from the two panels on the right. Also, in the upper left panel, the head of the hammer hitting the anvil and the foot of one smith have not been cut away, leaving these areas a distracting black.

But what strikes us in this woodcut is not these errors but the care and clarity of the thinking, drawing, carving and printing. It is laid out so clearly that anyone can repeat the experiments and calculations. The rise of printing from moveable woodblocks may thus have been an important step in the rise of modern science — a science which depends, not on the reciting of ancient authorities, but on the repeatability by anyone of carefully made apparatus and carefully measured experiments.

Ironically this very woodcut may have led to experiments falsifying and correcting its long-held theory, thereby playing a small part in the rise of modern physics. In the 16th century the musician Vicenzo Galilei repeated these experiments, only to discover that, except for the inverse ratio between string length and pitch, most of the discoveries attributed to Pythagoras were wrong. Vicenzo’s son Galileo continued the experiments, as did other scientists, until the French friar Marin Mersenne showed, in his Harmonie universelle of 1636, that though Pythagoras was right about the inverse ratio between pitch and string length, in the experiment with the weights on the strings the tension must be quadrupled, not doubled, to produce a tone an octave higher. That is, the ratio between pitch and string tension varies inversely as the square of the tension on the string.

Isaac Newton saw in this his own inverse-square law between the gravitational attractions (“weights”) of the planets falling towards the sun and their distances (“string lengths”) from it. An ardent Pythagorean, Newton insisted Pythagoras himself knew all this, but had hidden from the vulgar such mystical “square” secrets beneath his simple arithmetical ratios!

In his Notebooks of 1664 and 1666, and later in his Opticks, Newton defended the Pythagorean doctrine that the harmonic ratios found in music operate throughout the entire universe, arguing that the rainbow of colors in the visible spectrum progresses in a series of mathematical ratios like those Pythagoras discovered in the world of sound.