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
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