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#1 Guitar
The strings of a guitar keep increasing in diameter from the first string (bottom) to the last string (top). This is crucial to produce a pleasant sound. Moreover, placing your finger on each fret gives rise to a new set of sounds. For these to sound good, the Pythogoras ratio of Fret is important.
#2 Piano
In music, notes are expressed using the first seven letters of the Alphabet: A,B,C,D,E,F,G. Moving towards the right increases the pitch by an octave while moving left decreases the pitch by an octave. Group theory is at play here: a-b (mod 12). After every interval of twelve notes (including the sharps and flats, i.e both white and black keys) the same note repeats itself.
Not just in the guitar and the piano, but in every musical instrument in general, there is always a mathematical concept at work. Remember, even a slight change and it's going to sound completely off key. To maintain harmony in rhythm, tone, pitch and interval, these concepts greatly help!
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