Toward a Cross-cultural History of Mathematics. Between the Chinese, and the Arabic Mathematical Theories of Music: the Puzzle of the Indian Case

A comparison is proposed between the Chinese, the Indian, and the Arabic theories of music. Chinese theories mainly concerned the pipes of 5 or 12 {\it l\"ul\"u}. In the {\it Qian Hanshu [Former Han books]}, {\it l\"ul\"u} were tuned according to some ratios. Here, even the diameters of the pipes were changed following the same ratios; because the {\it l\"ul\"u} were considered solid in this respect. Arabic theories mainly derived from the well known Greek ones, and therefore they were a matter of strings. These scholars read the relevant Euclid's, Ptolemy's, and Aristoxenus' books. However, because of the algebra (and all that), they sometimes used numbers which were different from the Greek ones. In some case, Arabic scholars made themselves free from the Pythagorean prohibition about irrationals. Now the question is: were there anybody who forgot the Pythagorean musical ratios as well, to invent new ones? The Indian case really is a puzzle. Singing the {\it mantra} of {\it Sama Veda [Veda of melodies]} followed precise rules which should be learned, and transmitted without errors, as other religious rituals did. Yet Sanskrit texts lack any ratio among musical tones. The {\it gita} [singing], and the tuning of instruments for the {\it gandharva} [music] was a matter of ear, and not of any number measuring lengths. The ear guided Indian tuning faraway from the Chinese, or the Arabic ones. Taking account of music, we weaken the hypothesis that deep scientific exchanges were active among these cultures. Each retained its own character.