I don't understand why nonperiodic signals still has the concept of frequency/frequency domain?
Any absolutely integrable function can be represented as a sum of infinitely many periodic functions even though it is not periodic. These periodic functions are the frequencies making up the frequency spectrum of that function. You can compute them using the Fourier Transform.
Since nonperiodic signal doesn't have cycle(does't form a pattern), therefore it shouldn't have the concept of Frequency neither?
Musical notes are not periodic - periodic functions are infinitely long and all music notes have a finite duration - so would you say that music does not contain frequencies?
You don't actually have to have a period to be associated with a frequency. You just have to contain a periodic function. When you pluck a string or strike a drum the note eventually dies out and is therefore not periodic, but it does contain a series of periodic signals, including the frequency of the note, which sum to the (finite and thus not periodic) tone of the note.