Each of the musical snippets is part of the melody of "Take Me Out to the Ballgame" except that the lyrics have been replaced by a crossword-style clue and the music has been transposed into some minor key. Each of the answers to the clues are the name of a AAA minor league baseball team.
So now you have a bunch of minor league teams and minor keys. Take the corresponding major league team and corresponding relative major key. For example, "Message from MIT" is a ZEPHYR which is the minor league team for the Marlins and it is sung in C# minor whose relative major is E major.
Each of the major keys has A natural as one of its tones; this is the note labeled "scale degree" at the bottom of the puzzle. So you take the nth letter of the major league team where n is the scale degree of A in the corresponding major key. E.g., A is the 4th note in E major so you would take the 4th letter of MARLINS.
The full set of clue answers, keys, and extracted letters are as follows, sorted into the order in which the clue melodies constitute "Take Me Out to the Ball Game":
"Message from MIT": ZEPHYR in C# minor --> marLins in E major
"Detergent brand": TIDE in A minor --> oriolEs in C major
"Competition": BEE in F# minor --> Angels in A major
"Nuclear family member": ISOTOPE in C# minor --> dodGers in E major
"Multiple of 17": 51 in D minor --> blUejays in F major
"Communicate": EXPRESS in B minor --> rangErs in D major
"Prince William, for example": SOUND in C# minor --> breWers in E major
"Crackerjack": ACE in E minor --> dIamondbacks in G major
"Plains grazer": BISON in D minor --> meTs in F major
"Sydney ferry boat": RIVERCAT in D minor --> atHletics in F major
"Be up": BAT in D minor --> reDs in F major
"Civilization unit": KNIGHT in E minor --> wHitesox in G major
The letters extracted by the scale degrees spell out LEAGUE WITH DH, which clues AMERICAN, the answer to the puzzle.