"Roman music" covers an astonishingly large era (over two millenia), from about 750 B.C.E. through the fall of the Western Roman Empire in 476 C.E. and then the fall of the Eastern empire in 1453 C.E. Obviously, there are no audio recordings from this time so must of what we know comes from limited written descriptions of music (such as from Pliny the Younger) and artistic depictions of instruments.
For the purpose of answering your question, I'm going to focus primarily on my area of expertise, the late Republic and early Empire period from about 100 B.C.E. to 79 C.E. (Caesar's birth to the eruption of Vesuvius and death of Vespasian). I'm also going to provide a somewhat abbreviated reverse chronology of modern musical notation in order to present relevant context.
Modern musical notation owes its existence to Guido of Arezzo, a Benectine monk who lived circa 992 C.E. to approximately 1033 C.E. Before he published the Micrologus, there was no standard musical notation; Guido also invented solfege (do-re-mi-fa-so-la-ti) as a means of aiding the teaching and memorization of religious chants. Historians such as Henry Lunn acknowledge Guido as "one of the most earnest workers in the cause of musical advancement^(2)."
Before Guido's text, there was not really a standard way of writing or transcribing music. Boethius (who is famous for De Consolatione Philosophiae) published De Institutione Musica in the decades following Odoacer's sack of Rome in 476. This work discusses "the Pythagorian unity of mathematics and music, and discussed the Platonic concept of the relationship between music and society^(1)."
However, in the early empire period, musical notation would have been limited, at best, and there was no standard way of writing music. Unfortunately, as historian William Johnson notes, "literary sources . . . in general tell us little specific about the content or style of musical entertainment in the Roman era^(3)."
Because there are no recordings of music, and precious little written music of any sort, we sadly know very little about what music would have sounded like. What we do know comes from several papyrus fragments, which indicate that musical performance was "florid and dramatic [with] an unusually dramatic array of poetic and melodict adornment^(4)."
We do, however, have a decent knowledge of Roman musical instruments. The Romans had metal wind instruments (analogous to today's "brass instruments" but primarily made of other alloys) wood wind instruments, percussion instruments, and stringed instruments.
The metal wind instruments include the tuba, which was a long, uncurved bugle- or trumpet-like instrument, and the cornu, a curved horn which wrapped around the body like a modern sousaphone and was shaped like the letter "G", for example^(5).
Both of these instruments were used as signal instruments by the Roman legions, as was the buccina, an instrument which was very similar to the cornu. In battle, cornicens played the cornu to relay orders, for example.
Woodwind instruments include the tibiae, which were somewhat similar in appearance to an oboe. According to historian Maria Ginsberg-Klar, these were "a double shawn with double reeds . . . its nasal timbre may have resembled modern bagpipes. Both pipes were blown by one player at the same time^(6)." Boethius notes that this instrument "utters its mournful note from a curving horn^(7)."
Stringed instruments included the harp and the lyre, as well as the lute, which is similar in appearance to a mandolin and guitar, and the cithara, which was similar in appearance to a lyre with two sets of strings^(8).
The Romans also had percussion instruments including cymbals and shakers, as well as bells. Perhaps most interestingly, they had piped instruments such as the fistula (panpipe) similar to modern organs; the most advanced of these was the hydraulic organ.
Maria Ginsberg-Klar regards the hydraulic organ as "one of the most significant technological and musical achievements of antiquity^(9)." This instrument used hydraulic pressure to produce sound; Ginsberg-Klar notes an example which had twenty-nine pipes. Such organs were used "in amphitheatres, in the theatre, and even at home^(10)."
I'm sorry I can't tell you more about what the music played by these instruments sounded like in concert, but unfortunately there just isn't much evidence to show. As far as musical notation goes, the basic answer is that there was no standard form of written notation, and such a form did not exist until some centuries later. There are several examples of papyrus fragments which contain some musical notation, and I'd particularly recommend reading William Johnson's article for further information on the topic.
Notes:
Bower, Calvin Martin. "Boethius' The Principles of Music, an Introduction, Translation, and Commentary," Ph.D. Dissertation, George Peabody College for Teachers, 1967, pp, 31-44. University Microfilms no. 67-15,005.
Lunn, Henry C. "The History of Musical Notation (Concluded)." The Musical Times and Singing Class Circular 12, no. 279 (1866): 285-87. Page 285.
William A. Johnson. "Musical Evenings in the Early Empire: New Evidence from a Greek Papyrus with Musical Notation." The Journal of Hellenic Studies 120 (2000): 57-85. Page 57.
William A. Johnson. "Musical Evenings in the Early Empire." Page 59.
Alexandrescu, Cristina-Georgeta. "The Iconography of Wind Instruments in Ancient Rome: Cornu, Bucina, Tuba, and Lituus." Music in Art 32, no. 1/2 (2007): 33-46. Pages 33-38.
Ginsberg-Klar, Maria E. "The Archaeology of Musical Instruments in Germany during the Roman Period." World Archaeology 12, no. 3 (1981): 313-20. Page 315.
Boethius. De Institutione Musica, Book 1.
Ginsberg-Klar, Maria E. "The Archaeology of Musical Instruments." Page 317-320.
Ginsberg-Klar, Maria E. "The Archaeology of Musical Instruments." Page 316.
Ginsberg-Klar, Maria E. "The Archaeology of Musical Instruments." Page 316.
Sources:
Alexandrescu, Cristina-Georgeta. "The Iconography of Wind Instruments in Ancient Rome: Cornu, Bucina, Tuba, and Lituus." Music in Art 32, no. 1/2 (2007): 33-46. www.jstor.org/stable/41818803.
Boethius. De Institutione Musica, Book 1. http://cmed.faculty.ku.edu/private/boethius.html
D'Arezzo, Guido. Micrologus. (Note: Unfortunately, the only .pdf version of this I could find for free was only available in Italian)
Ginsberg-Klar, Maria E. "The Archaeology of Musical Instruments in Germany during the Roman Period." World Archaeology 12, no. 3 (1981): 313-20. www.jstor.org/stable/124243.
Johnson, William A. "Musical Evenings in the Early Empire: New Evidence from a Greek Papyrus with Musical Notation." The Journal of Hellenic Studies 120 (2000): 57-85. www.jstor.org/stable/632481.
Lunn, Henry C. "The History of Musical Notation (Concluded)." The Musical Times and Singing Class Circular 12, no. 279 (1866): 285-87. www.jstor.org/stable/3351906.
Van Waesberghe, Jos. Smits. "The Musical Notation of Guido of Arezzo." Musica Disciplina 5 (1951): 15-53. www.jstor.org/stable/20531824.
Some what related, would the Romans music sound similar to Dr Armand d Angour's Ancient Greek's work? Or would the Roman's use different different instruments etc?
Sorry if this is against the rules Mods