What is the oldest musical composition that exists both as verifiable historical notation (or some other primary source) and as available recording on CD (or better e.g. on YouTube)?
The Book of Rites, in a chapter on Touhu, records an apparently complete composition for drums. Known as the Lu Drums (
魯鼓) or the Xue Drums (
薛鼓), it seems to be intended for a match of the semi-ritualistic game of pitch-pot. This is considered the oldest extant musical score in Chinese history.
As the image shows, it uses very simple notations consisting of only two symbols: a square and a circle. The accompanying text explains that a circle means beat the drum, while a square means beat the smaller war drum.
This composition is difficult to date. The Book of Rites, in its modern form, was compiled during the Western Han dynasty, 206 BC – AD 9. However, that was a reproduction of much more ancient contents, purportedly produced during the Eastern Zhou dynasty, 770 BC - 256 BC. Much of it were destroyed during the Qin dynasty and recreated from memory by the old masters.
However, the chapter on Touhu is known to be part of a large collection supposedly found hidden in the walls of Confucius' old home. These were turned over to the Han dynasty government to assist in the reconstruction of the Book of Rites. Assuming this is correct, the drum score contained within Touhu must have been created before 221 BC or so.
Seikilos epitaph is a Greek song which is the oldest complete musical composition. It's dated to be from 200 BC - 100 AD, with first century AD being the more plausible guess. You can Google it to listen to many different versions.
Also refer to the older musical compositions section of the same article which states that there might be a few other older compositions, but they are not complete or not proven to be older than Seikilos epitaph. Most notable of them are the Hurrian songs which are a collection of incomplete songs from around 1400 BC.
(Or of course there is a remote possibility that the few inscriptions and writings that we have not yet deciphered actually turn out to be musical compositions, of all things)