Final answer:
Ragtime is the musical idiom that combines regular formal structure with piano works and syncopated rhythms influenced by African music, blending polyrhythms and syncopation with European harmonies.
Step-by-step explanation:
The idiom that combines regular, sectional formal structure, emphasis on works for piano, and syncopated rhythms influenced by African music is known as ragtime. Ragtime is a musical style that emerged during the late 19th and early 20th centuries, representing a blend of African rhythmic structures, such as polyrhythms and syncopation, with European harmonic structures. This genre is typically recognizable by its syncopated melodies where the rhythm is intentionally placed off the regular beats, and was often composed for solo piano. The ragtime style was a precursor to jazz and played a significant role in its development. Originating in African American communities, ragtime played a pivotal role in influencing later musical genres with its complex rhythms and lively tempo, becoming a part of American popular culture.