Generation
of the Pentatonic Scale

"Up-and-Down Principle" recorded in the "*Guan-Zi*"

"Up-and-Down Principle" recorded in the "

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)**