Why did David’s harp playing relieve King Saul's illness?

Question

David's harp playing brought relief to King Saul from his mental illness. (Source: I Samuel 16.)

Why did the harp playing provide relief?

Answer 1

The spirit of H' departed from Shaul and there began to terrify him an evil spirit, Ruach Raah ... 1 Samuel 16:14

"Evil spirit" specifically. So what is music exactly? It's the Berur, selection and refinement, of good Ruach (spirit, notes) from out of the bad notes, as Rabbi Nachman explained.

A Kosher musician is able to draw holy Music from its source which is from the two Holy Birds. Thus Chazal teach, Why does the Metzora bring an offering of birds? etc. But opposite to everything in holiness there's the Other Side, Sitra Achra and there's such a thing as bad music. This is all explained in Likutei Moharan, Torah #3.

So David, in whom there was a strong revelation of Mashiach and of course being able to draw holy music, was able to play and to heal Shaul.

There are many connections to this matter, such as Mitzva Gedolah Lihyot b'Simcha Tamid, It's a big Mitzva to be happy, it's a constant Mitzva (that takes devotion), finding good points in your fellow Yisrael (Likutei Moharan #282), the Repair and refinement from Adam's fail, the Aspect of Shabbat (when good can be taken from bad but not vice versa which is Uvda d'Chol profane labor) etc. but that's a start.

Answer 2

Simcha is poretz gadarim, it breaks walls. See Tanya Ch. 5.

What this means is that, our soul has various powers and facilities, with the highest being ta'anug; delight. Delight is often synonymous with life itself, and everything about life has some sort of pleasure in it, even just breathing.

Lack of innocence (guilt, distrust/suspiciousness), depression and bad middot all create gedarim (walls) to the pleasure of these powers and facilities in our soul, to the point we end up only experiencing them in a deadened, numb and incomplete way.

Simcha, joy, breaks down the walls and allows the full pleasure of the emotions, intellect and even higher faculties like the faculty of Ruach HaKodesh and prophecy (see Derech Hashem 3), allowing them to function properly.

It is well known from common experience (and Nissim Nanach's answer deals with some technicalities) that music is able to cheer us up! So is good food (see narrative of Yiztchak Avinu requesting us favourite dish before giving a prophetic beracha, and commentary there, such as Rabbeinu Bachya's).

Thus while Shaul was depressed he couldn't have real Ruach HaKodesh, and, just like Yaacov Avinu upon finding out that Yosef his son was still alive (see Rashi, who makes the connection with the Shechina), simcha brought it back.

See this exquisite explanation and fascinating shiur, or this one for more in depth analysis.