Is it still possible for mathematicians to contribute to the theory of music? Is it still possible that mathematicians contribute to the theory of music? Is the mathematical foundation of music still an area of research? If yes, what new researches have been done regarding that?
 A: I know that a member of Mathoverflow Tobias Schlemmer works in this topic, you can consult with him.
A: Key reference:

Musical Instrument Digital Interface (MIDI)

That set theory is useful with digital music,
especially MIDI, seems not to be widely known.Consider
a sequence consisting of the following chords: $F^1 , C^2 , G7^4$ .

Changing the chords means that
the set of notes $F^1 = \{F,A,C\}$ is replaced by the set of notes
$C^2 = \{E,G,C\}$ is replaced by the set of notes $G7^4 = \{F,G,B\}$ .
Mind the notes in common; these are joined by bows in the score.
In MIDI, first the notes in $F^1$ are set On (Note On).
After a duration of four beats, the notes in $C^2$ should sound.
In order to accomplish smooth transition of the chords,
this should be done by first hitting the chord $C^2$ before (immediately)
releasing the chord $F^1$. More in detail, apply a Note On event to the
elements in $\{E,G,C\}$ minus $\{F,A,C\} = \{E,G\}$ immediately followed
(timestep $0$) by a Note Off event applied to the elements in $\{F,A,C\}$
minus $\{E,G,C\} = \{F,A\}$ . Note that nothing happens with the note $C$ .
After a duration of four beats again, the notes in $G7^4$ should sound.
This should be done by first hitting the chord $G7^4$ before (immediately)
releasing the chord $C^2$. More in detail, apply a Note On event to the
members of the set $G7^4 \setminus C^2$ immediately followed
by a Note Off event applied to the members of the set $C^2 \setminus G7^4$ .
This is in a nutshell how chord transition works - or rather should work
- in MIDI. It's implemented in my personal mathematical contribution to music : MidiDoos .
A: Yes, specifically in the area of musical tuning theory: the xenharmonic wiki is a good place to read about this. Harmonic entropy and xenharmonic temperament theory are two relatively new topics with a lot of current (albeit somewhat obscure) research going on in them.
A: Aside from the mentioned set theory, many 20th century composers used various atonal techniques called 12-tone composition where each of the 12 notes between an octave were used sequentially before the sequence could be repeated. They would then apply various aspects of set theory to this which included transposition of the sequence, retrogrades of the sequence, and combinations thereof. Further 20th century techniques explored the overtone series and actually orchestrated overtone series which had some very interesting sonic effects. This could be equated to a sonic exploration of a Fourier series. I experimented with that myself back at university in a symphony that I never completed writing. The university orchestra did a reading of the incomplete work and it was quite a stunning sound. I had very strong bass C notes and basically orchestrated the instruments up the overtone series in decreasing dynamic intensity along with upper wind instruments basically scaling up and down the series. Despite having what should have been dissonant notes within the soundscape, they all served to reinforce the fundamental C pitch and gave it a rather striking and unique timbre!
As someone studying engineering mathematics and holding a music composition degree, I'm extremely curious about applying mathematical concepts to future compositions once I complete my engineering degree! (no time for that really at the moment unfortunately whilst I study).
