#phytagora

[geometrymatters]

The chords of the Universe

by Eli Maor for Aeon

It’s no surprise that mathematics has influenced music. But did you know that the influence goes both ways?

Music and mathematics have always been intimately intertwined. Anyone who has ever played a musical instrument is aware of the presence of mathematics on every page of the score – from the time signature that sets a piece’s rhythm, to the metronome number that determines the speed at which the piece should be played; and, of course, the very act of playing music requires us to count 1, 2, 3… and arrange these numbers into groups, called bars or measures. So it comes as no surprise that mathematics has had a significant influence on music. Much less known is that the influence extends both ways.

The Greek philosopher Pythagoras, active during the 6th century BCE, might have been the first to uncover a quantitative relation between music and mathematics. Experimenting with taut strings, he found that shortening the effective length of a string to one-half its original length raises the pitch of its sound by an agreeable interval, an octave. Other ratios of string lengths produce smaller intervals: 2:3 corresponds to the musical interval of a fifth (so called because it is the fifth note up the scale from the base note); 3:4 corresponded to a fourth; and so on. Pythagoras also discovered that multiplying two ratios is equivalent to adding their intervals: (2:3) x (3:4) = 1:2, so a fifth plus a fourth equals an octave. In doing so, he unknowingly came up with the first logarithmic law in history.

The octave, the fifth and the fourth produced pleasing combinations of tones, or consonants, whereas more complicated ratios, such as 9:8 or 15:16, led to dissonances. This revelation made a deep impression on the Pythagoreans, prompting their belief that everything in the Universe – from the laws of musical harmony to the motion of the Sun, Moon and the five planets – was governed by simple ratios of whole numbers. Number rules the Universe was the Pythagorean motto. It would dominate scientific thought for the next two millennia.

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[geometrymatters]

The Pythagorean Theorem Makes Construction and GPS Possible 

"You see, to make a rectangular foundation for, say, a temple, you need to make right angles. But how can you do that? By eyeballing it? This wouldn't work for a large structure. But, when you have the length and width, you can use the Pythagorean theorem to make a precise right angle to any precision." - Donald Allen

Beyond that, "This theorem and those related to it have given us our entire system of measurement," Allen says. "It allows pilots to navigate in windy skies, and ships to set their course. All GPS measurements are possible because of this theorem."

In navigation, the Pythagorean theorem provides a ship's navigator with a way of calculating the distance to a point in the ocean that's, say, 300 miles north and 400 miles west (480 kilometers north and 640 kilometers west). It's also useful to cartographers, who use it to calculate the steepness of hills and mountains.

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