Jing Fang

Jing Fang , born Li Fang , courtesy name Junming , was a , mathematician and astrologer born in present-day during the Han Dynasty . He was the first to notice how closely a succession of 53 just fifths approximates 31 octaves. This observation would much later lead to the discovery of 53 equal temperament in the seventeenth century. He was also an advocate of the theory that the light emanating from the spherical Moon was of sunlight.

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. $177147/3\; =\; 59049$
# Add this value to the original. $177147\; +\; 59049\; =\; 226196$
# The new value is now equal to $4/3$ 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 $177147\; /\; 176776$.