This method of generating the chromatic scale is found on the Marquis Yi Niu set-bells. In this method, we use the Up-and-Down procedure listed in the Guan-Zi to generate four fundamental notes and from these four notes, we generate the remaining eight notes to make up a chromatic scale. These eight notes are derived from the basic four notes by applying two special intervals (jue and zeng). These two intervals are "applied" to each of the four fundamental notes to get two extra notes.
The animation above shows clearly how the notes could have been generated. Starting from the four basic notes (zhi, you, gong and shang), eight new notes (two from each basic note) are generated with the jue and zeng intervals.
The question now is that it is not known what exactly these two interval ratios are. Calculated guesses however can be made.
The figure above shows the layout of the Yi Niu bells. This arrangement of the bells display a chromatic structure rising in ascending order from the left to right of each row, from the bottom row to the top most row. The diagram has labelled the starting note as the "C" we know nowadays to illustrate the chromatic ascendence.
Measurements have also been made of the frequencies of the bells by three agencies in China. By examining the frequencies of these bells, as well as the arrangement of the various notes in the chromatic scale, Dr Chen Ching-Yih concluded in "Early Chinese Work in Natural Science" that the jue and zeng intervals correspond to the major third and minor sixth. He thus approximated the major third interval to be either 64/81 or 6561/8192 and the minor sixth interval to be either 81/128 or (64/81)^2. He based these deductions on the work of Guan-Zi and Lu-Shi Chun-Qiu. This thus gives rise to four possible combinations for the two intervals, namely :
Using these 4 sets of different ratios combinations, we get 4 different chromatic scales. However, out of the four, 3 of them are actually isomorphic to each other. Looking at the diagram below, observe that Sets 1 to 3 consist of exactly the same semitones, ie. 5 major tones (2048/2187) and 7 minor tones (243/256). They can be obtained from each other by simple rotations about the centre of the octaval circles.
Note that the generated notes can be rearranged to reflect the layout of the Marquis Yiu-Niu bells.
Because of the lack of sufficient information and research in this area, we have been unable to find out exactly which ofthe 4 possibilities is the closest to the early scale produced by this method. However, several issues of interest are to be found by this method.
The first point to note, is as stated above, 3 out of the 4 possible scales actually correspond to the scale generated by the Lu-Shi Chun-Qiu method.
The second point, which requires more work, is that this method can actually produce a scale which matches that of the Just Innotation. This scale is generated by using the exact ratios for the major third (4/5) and minor sixth (5/8), as in the Just Innotation. The resulting octaval circle is reproduced below.
This suggests that the Chinese acutally may have gotten the same scale as Pythagoras did, but many years earlier! The implications of these will be discussed in a later session.