The Gemara brings a tradition that when the first Temple was destroyed on the ninth day of the month of Av, it was the day after Shabbat, i.e., Sunday, at the conclusion of the Sabbatical year. The priests and Levites were in the midst of singing Chapter 94 of Tehillim, which includes the passage “And He hath brought upon them their iniquity, and will cut them off in their evil” (verse 23). Before they were able to complete the verse and say “the LORD our God will cut them off” the Temple was captured. Similar events took place at the time of the destruction of the second Temple, supporting Rabbi Yose’s teaching that “Good things are brought about on an auspicious day, and evil ones on a bad day.”
The Gemara quotes Rava – some say it was Rav Ashi – as pointing out a difficulty in the story. Tehillim 94 was sung in the Temple (and, indeed, is recited at the end of the daily morning service) on Wednesdays, while the tradition quoted indicates that the destruction took place on a Sunday, whose official song is Tehillim 24!
In answer, the Gemara suggests that the chapter of Tehillim was not being sung to accompany a sacrifice, rather it was being recited as a lamentation text, given the recognition that the attackers were approaching the Temple. This is true, even though the singers were described as standing on the dukhan – “on the platform” – in the Temple, in accordance with Resh Lakish who ruled that the song may be sung even if no attending sacrifice was brought together with it.
According to most commentaries, the dukhan was part of the Court of the Sanctuary, in a place that was sanctified for use during the sacrificial service. For this reason, the Gemara assumes at first that the song must have been connected with a sacrifice that was being brought.