Jazz and blues contribute to inform the way Cane was written. There is a particular attention to the sound and flow of words, repetitions of lines and concepts (see the beginning and end of "Becky" and "Carma"), the Biblical quality of the tone. These formal characteristics make passages in the book structured as a popular chant or melody, which resembles the technique of call and response, typical of blues. The beginning of "Becky" and "Carma", for example, can be seen as a call for a response, the central part can be seen as the response and the repetition of the initial lines can be read as a return to the first voice, much like a preacher/community interaction. In addition, the different narrating voices share a communal dimension and heavily rely on symbolism, recurring elements in blues. The third link below takes you to Barbara Bowen's landmark article "Untroubled Voices: Call and Response in Cain", which analyses this call and response mechanism in detail.
In the sketch "Seventh Street", the narrator explicitly refers to jazz in his description of the street:
A crude-boned, soft-skinned wedge of nigger life breathing its loafer air, jazz songs and love, thrusting unconscious rhythms, black reddish blood into the white and whitewashed wood of Washington. Stale soggy wood of Washington.
Jazz is here considered as an African American creation that has to be put into the wider American context and ultimately defies racial boundaries: jazz thrusts its "unconscious rhythms" in "the white and whitewashed wood of Washington".