Since we have an overview for Medieval music, it is natural to ask about even earlier music. In fact, this question has drawn quite a bit of interest here. To remain analogous to the rest of the FAQ, this section will only treat Western cultures. Whereas there is some argument that certain repertories in other cultures date back to ancient times, the topic has also been less studied under the standards of Western academia. We will not debate the merits of such distinctions here, only acknowledge them.
Of course, it must also be noted that performances of this music are even more provisionary than they are for medieval music. Everything is more distant from us, in terms of notation and its interpretation, but perhaps more significantly the volume of material is so small that it is more difficult to be buoyed by internal consistencies. The volume of textual and iconographic evidence is also vastly smaller, but fortunately not nonexistent.
The ancient repertory which has drawn the most attention is that from Ancient Greece. A rather broad range of fragmentary sources do survive, including both vocal & instrumental music, although not enough to fill more than a single volume. Of course, the effect of musical ideas on such ancient philosophers as Pythagoras is well known, and so stimulates interest. A recent interpretation of scholarly merit:
Information regarding other recordings of this music may be found linked to the above citation.
Besides music from classical Greece and through the Hellenistic phase, some material from the Roman imperial period survives with attribution to Greeks and is also included broadly in this repertory.
Beyond that, there was a definite musical activity in the later Roman Empire. Ample evidence survives for instruments, and even a good deal of theory. A series devoted to this music is now appearing, based on authentic instruments but not on surviving musical notation. The first volume:
There is a body of surviving Byzantine music which has not yet been thoroughly studied, but has appeared on some records (a few such items are off of the Greek citation above). This leads seamlessly into the Medieval era, and presumably into the Ottoman court styles outside of the focus of this site. However, any truly notable Byzantine releases will be mentioned here as they appear.
Finally, the earliest known songs of what might be broadly termed Western culture are the Biblical Psalms. There is a notation accompanying the entire Old Testament, although its claims to antiquity (more than a thousand years or so) are disputed. Various attempts to interpret these signs have been made in different eras, but one modern solution seems to be catching on. A recording devoted to these reconstructions:
Some clay tablets from Ancient Mesopotamia indicating a musical notation also exist, and attempts have been made to perform them. Some attempts have also been made for Ancient Egyptian music. This survey must conclude at this point, without a journey farther East, as it becomes more & more removed from the focus of the FAQ. The natural continuation for even music of Egypt or Sumeria is the classical Arabic tradition which formed in medieval times, and this is not treated by the FAQ.
Returning to Western Europe, some scattered Roman chant manuscripts do survive from the 7th & 9th centuries alongside some similar Milanese repertory, but these are in styles similar to the Gregorian chant of the 10th century and beyond. Plainchant survives in quantity from c.890 onward.
Whether it remains a possibility that other surviving ancient music styles will come to light, I cannot really say, but suspect not, at least for Europe. While some try to place Celtic music in this category, like all folk music, it is not written and hence a different artform. The demonstrable roots of Celtic music are not especially old, and what preceded it is unknown.