Most appropriate spectral analysis method for a recording of a saxophone note?
The plot OP shared looks similar to log magnitude spectrum(black line) plotted with its envelope(red line). so from what OP says, OP need to plot log magnitude spectrum of recorded audio/signal. One thing to remember for this analysis is that same note should be played for entire duration of the signal to be analysed. if not, one should use proper windowing methods to ensure that signal remains stationary(in same note) for the entire duration of window under analysis. I am assuming that a note corresponds to a few particular frequencies. 

that is the way if you want to plot/analyse a graph similar to the one shared. are you also looking for other methods for determining the peaks corresponding to a particular note at a time?