Final answer:
The musical composition genre with roots in the Medieval period is Gregorian chant, characterized by monophonic, unaccompanied vocal music. A twentieth-century example reflective of medieval influences is Carl Orff's 'Carmina Burana'. Modernism in the early twentieth century often referenced historical styles for new musical expressions.
Step-by-step explanation:
Medieval Music and its Modern Echoes:
The type of musical composition that has notated examples going back to the Medieval period and can trace its origins back to the Roman Empire through pictorial representations is Gregorian chant. This genre is characterized by its use of monophonic, unaccompanied vocal music, set to Latin liturgical texts. These chants were foundational in the development of Western music, influencing the creation of more complex polyphonic forms.
A modern example of music that reflects the influence of Medieval styles would be compositions in the neo-medieval genre, which draws upon the musical idioms of the past. One such twentieth-century example is the work of composers like Carl Orff, whose 'Carmina Burana' is a collection of cantatas that use medieval texts and somewhat echo the modal melodies and rhythmic patterns of Medieval music.
Modernism in music during the early twentieth century also showed composers striving for new expressions in music, some of which took inspiration from historical styles.