What historical and cultural influences evolve jazz?

The Jazz Age consisted of prohibition and speakeasies. These helped to grow the popularity of jazz music. Since there was an increase in popularity, recording industries made it possible for people to have access to jazz that could not before. As it spread, new jazz sounds were forming. The Stock Market Crash and The Great Depression caused a change in music since nobody could afford it. Swing music then was created and caused a huge boom in sales and popularity. This also helped to bring different races and cultures together.  

Explanation:

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i'm not sure i completely understand your question, but i will do my best.

i think the answer you are looking for is sequence. (see below)

a repeated melodic fragment would be best defined as repetition, while repeated chords (different or the same) could be a chord progression. however, you then mentioned the chord and melodic fragment beginning on higher or lower pitches, in that case, the answer could be best defined as a sequence. the definition of a sequence is as follows: "the restatement of a motif or longer melodic (or harmonic) passage at a higher or lower pitch in the same voice."

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