Here we have Mishnah 1 in chapter 5 of Maseches Sukkah. The Mishnah opens by addressing one of the items taught in the first Mishnah of chapter 4: that the flute, the musical instrument, was played for five or six of the days of Sukkos. What does this refer to?
"Zehu he'chalil shel Beis Hasho'evah":
This musical instrument, which was played along with many other instruments, belongs to Simchas Beis Hasho'evah - the celebration held in honor of the drawing of the water that was to be poured on the mizbe'ach.
"She'eino docheh lo es haShabbos v'lo es Yom Tov":
The playing of the instrument, and with it the entire celebration, does not override Shabbos or Yom Tov, and therefore it can take place only on Chol Hamoed.
From this comes the difference between five days and six:
When Shabbos and Yom Tov fall on the same day: six days of Chol Hamoed remain, and the flute can be played on all of them.
When Shabbos and Yom Tov are two separate days: even though there are still six days of Chol Hamoed, one of them is Shabbos, and the flute is not played on it - neither on Shabbos nor on Yom Tov. Therefore in such a case there will be only five days.
"Amru: kol mi shelo ra'ah simchas Beis Hasho'evah, lo ra'ah simchah b'yamav":
So said Chazal: whoever did not see Simchas Beis Hasho'evah never saw true joy in his life. That is how extraordinary the joy of Simchas Beis Hasho'evah was.