Suggested by Thomas J. Mathiesen, whose own listed book contains a large bibliography:
 
Anderson, W. D.  Music and Musicians in Ancient Greece (Ithaca, NY:  Cornell University Press, 1994).

Belis, A. Les musiciens dans l'antiquite (Paris: Hachette, 1999).

Comotti, G. Music in Greek and Roman Culture (Baltimore: Johns Hopkins University Press, 1989).

Landels, J. G.  Music in Ancient Greece and Rome (London: Routledge, 1999).

Mathiesen, Thomas J. Apollo's Lyre: Greek Music and Music Theory in Antiquity and the Middle Ages (Lincoln: University of Nebraska Press, 1999).

Riethmueller,  A. and F. Zaminer. Musik des Altertums (Laaber: Laaber-Verlag, 1989).

West, M. L.  Ancient Greek Music (Oxford: Clarendon, 1992).
 

Suggested by Peter JVD Bryant:

Barker,  Andrew. Greek Musical Writings, Volume I: The Musician and His Art ( Cambridge:Cambridge University Press,1984).

Barker, Andrew. Greek Musical Writings, Volume II: Harmonic and Acoustic Theory  (Cambridge:Cambridge University Press,1989).

McKinnon, James.  Music in Early Christian Literature (Cambridge:Cambridge University Press,1987).
 

Suggested by Paul Halsall, who has a site on ancient and medieval music at http://www.unf.edu/~phalsall/medieval/medievalmusic.htm:

Abraham, Gerald. Concise Oxford History of Music (London: Oxford University Press, 1985).

Anderson, Warren  D. Music and Musicians in Ancient Greece (Ithaca NY: Cornell University Press, 1995).

Conomos, Dimitri E. Byzantine Hymnography and Byzantine Chant  (Brookline MA: Holy Cross Orthodox Press, 1984).

Farmer, Henry George. A History of Arabian Music to the XIIIth Century (London: Luzac Oriental, 1967, 1994, repr.
Weatherhill, 1996).

Haik-Vantoura, Suzanne. The Music of the Bible Revealed: The Deciphering of a Millenary Notation. Translated by
Dennis Weber  (Berkeley CA: BibalPress; San Francisco CA: King David's Harp, 1991. 2nd rev, edition. D & F Scott 1991).

Hiley, David. Western Plainchant: A Handbook (London: Oxford University Press, 1993).

Hoppin, Richard H. Medieval Music. The Norton Introduction to Music History (New York: W.W. Norton, 1978).

Jefferys, Peter. Re-envisioning Past Musical Cultures (Chicago: University of Chicago Press, 1992).

Levy, Kenneth. Gregorian Chant and the Carolingians (Princeton, NJ: Princeton University Press, 1998).

Manniche, Lise. Music and Musicians in Ancient Egypt (London: British Museum Press, 1991).

McGee, Timothy J. The Sound of Medieval Song: Ornamentation and Vocal Style According to the Treatises.
Latin translations by Randall A. Rosenfeld (London: Oxford University Press, 1998).

McGuickin, John A. At the Lighting of the Lamps : Hymns of the Ancient Church (Harrisburg PA: Morehouse, 1997).

McKinnon, James, ed. Antiquity and the Middle Ages:  From Ancient Greece to the 15th Century. Music and Society
Series, Vol. 1 (Englewood Cliffs NJ: Prentice Hall, 1990).

Schueller, Herbert M. Idea of Music: An Introduction to Musical Aesthetics in Antiquity and the Middle Ages. Early
Drama, Art, and Music Monograph Series, 9 (Kalamazoo MI: Western Michigan University, 1988).

Stolba, K Marie. The Development of Western Music: A History. Brief 3rd edition. New York: McGraw-Hill, 1998.
With Recorded Examples to Accompany The Development of Western Music: An Anthology, Volume I (From Ancient
Times through the Classical Era) 3rd edition. (New York: McGraw-Hill, 1998).

Tardo, Lorenzos. L'Antica melurgia bizantina nell'interpretazione della Scuola Monastica di Grottaferrata (Grottaferrata, 1938).

Tillyard, Henry Julius Wetenhall. Byzantine Music and Hymnography (London: The Faith Press, 1923. repr. AMS Press, 1976).

Wellesz, Egon. A History of Byzantine Music and Hymnography. 2nd. ed. (Oxford: Clarendon Press, 1961).

Wellesz, Egon. Ancient and Oriental Music. New Oxford History of Music, Vol.1 (London: Oxford University Press,
1957, repr. 1979).