Given what we know, we can confirm that the theatrical performance being described by the question that was invented by the Romans is the pantomime.
It is as the question describes, a performance in which one person will act out a tale without the use of any vocal performance, whilst a singer or musician accompanies the tale through music. This is an adaptation of traditional greek theatre but is Roman in origin.
Therefore, we can confirm that the theatrical performance being described by the question that was invented by the Romans is the pantomime.
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