Life and career
According to the 3rd century historian Sima Biao, Jing Fang received an appointment as an official in the Bureau of Music under Emperor Yuan of Han . The historian Ban Gu wrote that Jing Fang was an expert at divination and making predictions from the hexagrams of the ancient ''Yijing''. and extended this method fivefold to a scale composed of 60 fifths, finding that after 53 new values became incredibly close to tones already calculated.
He accomplished this calculation by beginning with a suitable large starting value that could be divided by three easily, and proceeded to calculate the relative values of successive tones by the following method:
# Divide the value by three.
# Add this value to the original.
# The new value is now equal to of the original, or a perfect fourth, which is equivalent to a perfect fifth inverted at the octave.
# Proceed now from this new value to generate the next tone; repeat until all tones have been generated.
To produce an exact calculation, some 26 digits of accuracy would have been required. Instead, by rounding to about 6 digits, his calculations are within 0.0145 of exactness, which is a difference much finer than is usually perceptible. The final value he gave for the ratio between this 53rd fifth and the original was .
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