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GUEST,282RA Origins: Music of the Sphere (13) RE: Origins: Music of the Sphere 21 Apr 07

In Greek mythology, the Fates produced the 5 notes or vowels of the lyre but Apollo added 2 more to make a diatonic scale. Pythagoras (whose name means "Apollo speaking") was said to have added an 8th, which completes the octave. The Greeks did not use our 12-tone equal-tempered (or 12-TET) scale of today. A Chinese musicologist of the 16th century, Zhu Zaiyu, first calculated the value of an equal-tempered half-step to be the 12th root of 2 or about 1.0595. Shockingly enough, the name of Apollo in Greek, Apollon, adds up in Greek isosephia (where every letter in the alphabet has a corresponding number) to 1061. One of his titles, Pythias, adds up to 1059. And Apollo, let us remember, was the god of music.

Some might determine that the Greeks must have known the about 12-TET since that seems far too close to be coincidence, yet in the 19th century, a Western musicologist developed a system that enabled values between two intervals to be converted into a linear scale for comparison. This was useful because, in the Pythagorean scale (the actual ratios with no tempering), the interval ratios are not intuitive. The difference between the interval called a tritone (6 half-steps) and a perfect 5th is only a single half-step but it is hard to know that simply by comparing their ratios—729/512 (about 1.42383) to 3/2. So a linear scale was developed to illustrate this. It utilizes the following formula:

1200 log2 a/b, where a/b represents the two ratios being compared. Hence:

1200 log2 1.5/1.42383 or 1200 x log2 1.053496555 = 90 and the units used here are called "cents." So a half-step in the linear system is 90 cents (but is often rounded off to 100 cents). We can also use a single ratio instead comparing two. A whole step, for example, equals 204 cents (1200 x log2 9/8 but is rounded off to 200 cents). The octave is exactly 1200 cents.

Oddly, a full Pythagorean octave is 531441/262144 which converted into cents is 23.46 or thereabouts. So we see how far the octave drifts from say C to C' without tempering but what is astonishing is that 23.46 is extremely close to the actual tilt of the earth on its axis! The tilt is responsible for our zodiac. So the octave is a cycle that is off by 23.46 cents and the earth is a sphere off its rotational axis by about 23.5 degrees.

Coincidence? Hell, I don't know!

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