The short answer to your question is that The Art of War is referring to a pentatonic scale that was understood to be the core of "elegant music" (雅樂), which was a lofty form of music used for court rituals. In the context of the passage you've referenced, it's just a metaphor for how a limited number of categories can still produce a wide variety of combinations, and not meant to be a literal claim about there only being five musical notes.

Another comment has already noted that there may be a link to the "five phases" system (五行). Though the five phases system hadn't reached its maturity at the time of The Art of War, the fact that the next two examples used in the aforementioned passage are the "five colours" and the "five flavours" strongly implies a "five phases" influence. As far as music is concerned, though, archaeologists have found Shang dynasty ocarinas designed to produce a five-tone pentatonic scale, and this significantly predates the "five phases" system as we know it. So, perhaps we can think of this system as contributing to the continued emphasis on a pentatonic scale in premodern China, but not as its origin.

If I can ask a follow-up, why was Sun Tzu talking about musical notes at all during something as focused on military strategy as the Art of War?