A description for the generation of chromatic scales is given in the Lu-Shi Chun-Qiu. It is highly obvious that the procedure given is based on the Up-and-Down procedure described by the Guan-Zi. Explicit definitions for the up and down generations are given in the passage. (Click here to view the passage)
The animation above illustrates how the chromatic scale is generated. The notes "C", "D", ... are annotations used in modern day music and are the rough equivalents of the Chinese notes so generated.
In the generating of the sounds, we have arbitarily fixed the base note (Huang-Zhong) to be equivalent to modern day "C". We then follow the up and down generations by 1/3 to produce the rest of the sounds.
For comparison, the number 81 from the Guan-Zi procedure is again used as the starting number. We then get the following sequence of values, after applying the corresponding up and down generations.
As we have mentioned previously, the semitone is the interval between two adjacent notes. We can obtain the mathematical value of this semitone by dividing one note with its preceeding note. Performing 12 divisions, you will realise that you will only get 2 different ratios (ie. (major) 2048/2187 and (minor) 243/256). This implies that in this chormatic scale, there are 2 different kinds of semitones. (The modern music scales have just one kind of semitone)
At this point, there is a need to explain why there are 12 divisions because it may not be immediately obvious. For the scale to repeat in an octave, the note immediately superceeding (Ying-Zhong) (B) should be (Huang-Zhong) (C) again. This (Huang-Zhong) differs from the base (Huang-Zhong) by an octave, ie its representative value should be 81/2. Taking 81/2 divided by 128/3 (value for Ying-Zhong), we get the last minor semitone of 243/256.
From these twelve divisions, we get 5 major tones and 7 semitones as shown below.
Following such a system, as the diagram shows, we get a scale with two different kinds of semitones, ending in a round of octave.
A last point to mention about this scale is the deviation from the up and down sequence in the (Da-Lu) (C#) note. Instead of a down generation akin the sequence, we do a up generation to get Da-Lu instead. The sequence resumes thereafter with a down generation of (Yi-Ze) (G#). It was explicitly stated in the Lu-Shi Chun-Qiu for Da-Lu to be up generated. This deviation from the norm can be explained by a need to keep the Da-Lu note within the same octave as the rest of the notes. If the deviation was not made, but instead we were to continue the generation for the (Da-Lu) note with a down generation. We will get a note which is an octave higher than the rest. We would thus not be able to form a scale spanning one octave. We thus have to up-generate the note instead. Refer to the animation below to explain why up or down generating from a note will give you two octaves of a note.
Refer to the animation above. It shows that starting from a base number ß say, we get a note ß' say from up-generation and another note ß'' say from down-generation. It can be shown that ß' and ß'' are exactly an octave apart. (ie ß'' / ß' = 1/2)
ß' = 4/3(ß) and ß'' = 2/3(ß)
then (ß'' / ß') = 2/3(ß) / (4/3)(ß) = 1/2