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MusicAsEasyAsPi - Music, Ancient Greek Ratios, Geometry and Speculations.

Once upon a time some two and a half thousand years ago the Ancient Greek philosophers thought about space, time and musical sounds. Pythagoras became best know in our era for his concept that the sum of the squares of the adjacent sides of a right angle triangle are equal to the square of the hypotenuse. (a^2 + b^2 = c^2) Stated like this, some people find it difficult to visualize, but a simple drawing makes it abundantly clear. Since the original drawings were probably marked with a stick in sand my untidy proportions are comparable to the first attempts.

A picture as with music, a sound is worth a million words. What does all this have to do with music? You can justifiably ask.
Geometry is a mapping system of space. Music is a mapping system of sound and other vibrating patterns.
The Ancient Greeks can not claim the discovery of Pi, but they were certainly aware of it, for it has existed in all known cultures. The Chinese of the 12th century B.C. assumed Pi to be 3, and since then the mathematicians of many cultures have produced more accurate values.

The traditional geometry which we were all taught at school is based on the ideas of the ancient Greeks. We assume that a point has location but no dimensions; a straight line has length but no width; a square has area but no depth. It has served civilisation reasonably well for more than two thousand years. The most familiar model is the right angle triangle where the square of the length of one side adjacent to the right angle plus the square of the other adjacent side equals the square of the hypotenuse. Usually know as a squared plus b squared equals c squared.

Sacred geometry uses this idea and others to construct the Platonic solids: octohedron, cube, dodecahedron, iscosahedron, which may all be arranged within concentric spheres. These spheres have radii, which are in strict proportions, linked to the Golden Mean, which is the square root of five, plus one, all divided by 2. i.e. (5^(1/2)+1)/2 = 1.618. [The reciprocal of which is 1.618 - 1 = 0.618.]

The ancient Greeks had used gears for calculating and hence could have a particular interest in representing all ratios as integers. A mechanical device, which some believe was a simple analogue computer from the time of Christ, was found under the Mediterranean near one of the Greek islands. It seems to have been used for celestial calculations and employed cogs, which if rotating completely would always need to have an integer number of teeth. Hence the tendency for the ancient Greeks to express ratios as simple integers?

Comparison of proportional and rotational logic

There are many ways to calculate Pi as Leonhard Euler (1707-1783) and other great mathematicians have shown. Pi is an irrationa number; that is it cannot 355/113: and 1146408/364913, but none of these fractions can accurately describe it. Pi is also a transcendental number in that it cannot be the root of any algebraic equation. You could spend the rest of your own and all your descendents lives waiting for your computer to accurately calculate Pi, for the number patterns never repeat. For practical purposes it can be assumed to be slightly less than 3.1415926536.


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Related Pages
MusicAsEasyAsPi - Chapter One
Pi has a fascinating history and many approximations. For entertaining and informative book on the subject read A History of Pi by Peter Beckmann published in 1971 by St. Martin's Press, New York or more recently...... More info. on Pi (The Joy of Pi - website)