With a specific musical, how can I compare/contrast the music with the book and lyrics to conclude how the two combine to drive the musical as a whole?
Full question I'm working with: Pick a musical of your choice and use it as the basis to compare and contrast the music with the book and lyrics. How do these two sides combine to drive the musical as a whole? What element is the most important—plot, music, or character? What can you conclude about the relations between the musical and literary components?
The answer to your question is going to depend heavily on the particular musical that you choose, but the process of analysis will be the same no matter what in a general sense. Choose a work that you are highly familiar with, one which you have access to as far as the book (in this situation, that means the script). Copies of those can be hard to find. I'll first go through the analysis process in general and then apply it to a specific case study to give you an idea of what you're looking for.
This task requires you to be able to analyze text (the lyrics of the songs and the lines the actors say) for characterization, theme, and tone. Think of this just as you might analyze a poem in English class. Overall, what do the words of the musical show you about the main character's personality, emotions, and goals? What type of overall message about the world (theme!) is sent by the story? What "feel" does the viewer get emotionally because of the words, and which words specifically make it that way? The question also asks you to figure out what's most prominent: the plot, the music, or the character. I'd suggest working with character, as it lends itself to the most possibilities to say something intelligent.
Then, you've got to listen to the score. Once you've analyzed the words, you have an idea of what your main character is going through, feeling, and learning throughout the course of the story. What you need to do now is see how the music itself (including components like major/minor keys, tempo, genres of music, instrumentation, volume, number and type of voices, etc.) reflects those ideas. We all recognize a sad song when we hear one, right? The trick is identifying the details that tell us that it is "sad"--normally a slow tempo, a single singer, minor key, and a build in intensity throughout the piece. The music will reflect the character's journey, so pay attention to shifts in the sounds as well as in the words.
This is an extremely detailed process, but I'll go through a simplified version quickly. To see how I did it, you can read the synopsis of the show, and the novel it's based on, using the links below. The popular musical Wickedis a very character-based story, focusing on the journey of Elphaba, a talented green girl who starts life feeling rejected and awkward, finds empowerment through her rage at injustice, and ultimately makes the choice to live a life being percieved as a "wicked witch" in exchange for following her heart. A couple specific examples of ways in which the words and music help tell this story are below:
-In the early song "Not that Girl," Elphaba sings, "Hands touch, eyes meet, sudden silence, sudden heat, hearts meet in a dizzy whirl/he could be that boy/but I'm not that girl." The words reflect her longing for acceptance and her despair at not fitting the description of one who is loved. The slow, stirring tempo and haunting, sparse interpretation reinforce the somber mood and lonely outlook of the character.
-Later in the story, when she has become more empowered and ready to accept her true self, Elphaba sings the showstopper "Defying Gravity." She cries, "Too long I've been afraid of losing love. I guess I've lost. Well, if that's love, it comes at much too high a cost!" The lyrics paired with the uptempo, full, soaring instrumentation and powerful vocals indicate that our character has truly, magnificently changed.
These are just two examples, but hopefully you get the idea!