Generation of the Pentatonic Scale
"Up-and-Down Principle" recorded in the "Guan-Zi"

A passage in the "Guan-Zi" explains the mathematical procedure to generate a set of five notes, namely :

Starting with a number A, we add 1/3 to it to get the second note. As seen in the animation, we subtract 1/3 from the second note to get the third, and so on till we get the fifth note.

The Guan-Zi has arbitarily defined this number A to be 81. It was suggested that 81 was chosen so that integers can be obtained to represent the other notes. This is because in generating the rest of the four notes in the scale, the procedure requires the number A to be divided by 3 four times, and the number 81 is the smallest integer that can be evenly divided by 3 four times.

To help you to visualize this, imagine a string orginally 81 centimetres long. Plucking this string will produce a sound with an associated frequency. We will call this note (gong).

Note that the length of a string is inversely proportional to the frequency of the sound produced when it is plucked. And the number used to describe the notes in the Guan-Zi are representative of their associated frequencies. Thus we decrease the length of this string by 1/3, ie. shorten the string to (81*2/3) = 54 to get the second note, zhi. We repeat this process of lengthening and shortening the string till we get the set of all five notes. (Note that the sounds you hear in the animation above are reproduced by us this way)

Representing the procedure mathematically, we have :

Arbitarily define as 81 (for the note gong)
81 X (1 + 1/3) = 108 (for the note zhi)
108 X (1 - 1/3) = 72 (for the note shang)
72 X (1 + 1/3) = 96 (for the note yu)
96 X (1 - 1/3) = 64 (for the note jue)

The five notes are rearranged with the lowest note being the zhi note and the highest note being jue.

ie : zhi, yu, gong, shang, jue (Click here to listen to the rearranged sequence)