Which texture is characterized by independent melodic interest in multiple parts?

The texture characterized by independent melodic interest in multiple parts is contrapuntal music. In a contrapuntal texture, each melodic line, or voice, holds its own distinct musical interest and often develops independently of the others. This results in a complex interweaving of melodies that can create a rich and intricate sound.

This contrasts with homophonic texture, where there is a dominant melody accompanied by chords, making the harmony more important than the individual lines. Monophonic texture consists of a single melody line without any harmonic support or additional melodies. Chordal texture, while often homophonic, focuses predominantly on the chords rather than independent melodic lines.

Thus, the defining feature of contrapuntal texture is the interplay of multiple independent melodies that collectively create a fuller musical texture, making it the correct choice for this question.