The chakra system itself is old, described across centuries of Sanskrit tantric and yogic literature using its own vocabulary of seed syllables, colours, petals and presiding deities specific to each centre. Assigning the seven chakras to the seven natural notes of the Western diatonic scale is a much younger overlay, emerging from early twentieth century Theosophical writing on the chakras, most influentially C.W. Leadbeater's work, and elaborated considerably through the later twentieth century New Age chakra literature that followed Theosophy's lead. No classical Sanskrit source pairs Muladhara, the Root chakra, with the pitch C or with any other Western note name, because the concept of fixed Western pitch played no part in the tradition that first described the chakras.
The correspondence as commonly given today, C for Root, rising through the scale to B for Crown, reflects a specific modern choice to map an ascending musical scale onto an ascending sequence of bodily centres, a tidy correspondence that made the system easy to teach and remember but that should be read as a twentieth century synthesis rather than as inherited doctrine. Other correspondence schemes exist in the wider literature and do not always agree with this one, which is itself evidence that no single version carries the weight of settled tradition.
Sources The Serpent PowerSir John Woodroffe (Arthur Avalon) with Rainbow Body: A History of the Western Chakra System from Blavatsky to BrennanKurt Leland