Collected Illustrations of the Three Realms, Volume 73

The origin of all numbers is rooted in the natural Qi The vital energy or "breath" of the universe of Heaven and Earth; it is not something that can be manufactured by human effort. This is known as the mathematical theory of pitch.

Nine times nine makes eighty-one, which is established as the Gong The first and most fundamental note of the pentatonic scale, often called the "Palace" note. Subtracting one-third yields fifty-four, which becomes Zhi The fifth note of the scale. Adding one-third to that yields seventy-two, which becomes Shang The second note of the scale. Subtracting one-third yields forty-eight, which becomes Yu The sixth note of the scale. Adding one-third yields sixty-four, which becomes Jue The third note of the scale.

Furthermore, it is said that the Yellow Bell The foundational pitch pipe from which all others are derived is eight cun and seven-tenths plus one Ancient Chinese units: cun is roughly an inch, and fen is a tenth of a cun in length; the Forest Bell is five cun and seven-tenths plus four; the Great Tai-cu The pitch of the first lunar month is seven cun and seven-tenths plus one; the Southern Pipe is four cun and seven-tenths plus eight; and Ablution is six cun and seven-tenths plus four. In each instance, a sequence of "seven-tenths plus one" is added to the fractional remainders. Readers who fail to observe this pattern often mistakenly believe these figures to be errors.

Only Zhang Xingcheng A Song dynasty scholar known for his mathematical studies of the I Ching correctly noted that the method for crafting these pipes—where three parts out of ten are removed to create the mouthpiece—aligns with the "Great Number" of seventy-eight for the Yellow Bell found in the Tai Xuan The Tai Xuan Jing or "Canon of Supreme Mystery," a Han dynasty philosophical text.

To expand on this, Ablution occupies the position of the Jue tone. It is said that in the mathematical relationship between eighty-one (Gong) and sixty-four (Jue), the five tones cannot be tuned without Gong, and the sound of Gong cannot be fulfilled without Jue. This single statement encompasses the entire principle.