Zac M. answered  04/09/20
Eastside Local | Composer, Audio Designer, & Writer
Absolutely! However, it does depend on what you mean by 'notes we have currently'. For example, in college, I studied non-Western music (focusing extensively on Gamelan and Indian music); whereas Western music has 12 notes per octave (A - G, with enharmonic sharps and flats to match), an important division of pitch in Indian classical music is the shruti (I highly recommend at least starting with the wiki on shruti here: https://en.wikipedia.org/wiki/Shruti_(music)).
Short answer to your question - Western music can use microtones (usually half-of-a-half tone), but this is just an extension of the host note (e.g. A 1/2b, or G3/4#). Displaying this type of notation on a computer usually requires some kind of score-building software (like Finale or Sibelius).
If you can hear it, you can definitely find a way to notate it - it just depends on the style of music and the notes available to you from that music.
I hope this helps! Please feel free to add a comment if you would like more information or would like to clarify anything.