Shir du jour -- Demystifying the Psalm of the Day

1 Today is Sunday, when the Levites would chant in the Temple:
Of David, a song. The land and the fullness thereof are the Lord's; the world and those who dwell therein.
2 For He founded it upon seas and established it upon rivers.
3 Who will ascend upon the Lord's mount and who will stand in His Holy place?
4 He who has clean hands and a pure heart, who has not taken My name in vain and has not sworn deceitfully.
5 He shall receive a blessing from the Lord and charity from the God of his salvation.
6 This is the generation of those who seek Him, who seek Your presence-Jacob, forever.
7 You] gates, lift your heads and be uplifted, [you] everlasting portals, so that the King of Glory may enter.
8 Who is this King of Glory? The Lord, Who is strong and mighty, the Lord Who is a mighty warrior.
9 [You] gates, lift your heads and lift up, [you] everlasting portals, so that the King of Glory may enter.
10 Who is this King of Glory? The Lord of Hosts-He is the King of Glory forever.

Today is holy Sabbath when the Levites would chant in the Temple:

1 A song with musical accompaniment for the Sabbath day.
2 It is good to give thanks to the Lord, and to sing to Your name, O Most High.
3 To declare in the morning Your kindness and Your faith at night.
4 Upon a ten-stringed harp and upon a psaltery, with speech upon a harp.
5 For You have made me happy O Lord, with Your work; with the work of Your hands I shall exult.
6 How great are Your works, O Lord! Your thoughts are very deep.
7 A boorish man does not know; neither does a fool understand this.
8 When the wicked flourish like grass, and all workers of violence blossom, only to be destroyed to eternity.
9 But You remain on high forever, O Lord.
10 For behold Your enemies, O Lord, for behold Your enemies will perish; all workers of violence will scatter.
11 But You have raised my horn like that of a wild ox; to soak me with fresh oil.
12 My eye has gazed upon those who stare at me [with envy]; when evildoers rise up against me, my ears hear [them].
13 The righteous one flourishes like the palm; as a cedar in Lebanon he grows.
14 Planted in the house of the Lord, in the courts of our God they will flourish.
15 They will yet grow in old age; fat and fresh will they be.
16 To declare that the Lord is upright, my rock in Whom there is no injustice.

The Gemara rejects this argument. It is different there, as in any case “Sing aloud” is the psalm of the day, either because it was an ordinary Thursday or because it was Rosh HaShana. However, there is no proof from here that in all uncertain cases they would recite the psalm for an ordinary weekday, as it is possible that they did not recite any psalm at all. § The Gemara expands on the topic of the daily psalms recited by the Levites. It is taught in a baraita that Rabbi Yehuda said in the name of Rabbi Akiva: On the first day of the week, Sunday, what psalm would the Levites recite? The psalm beginning with the phrase: “The earth is the Lord’s, and its fullness” (Psalms 24:1), in commemoration of the first day of Creation, because on that day He acquired the world and transferred it to man, and He was the only ruler in His world, as the angels were not created until the second day. On the second day of the week what psalm would the Levites recite? The psalm that begins: “Great is the Lord, and highly to be praised in the city of our God, His sacred mountain” (Psalms 48:2). This is because on the second day of Creation He separated His works, dividing between the upper waters and the lower waters, and ruled over them as King; and this psalm speaks of Jerusalem as “The city of a great King” (Psalms 48:3). On the third day of the week they would recite the psalm beginning: “God stands in the congregation of God” (Psalms 82:1), because on the third day of Creation He revealed the land in His wisdom and thereby prepared the world for His assembly that could now live on the dry land. On the fourth day of the week they would recite the psalm beginning: “O Lord God, to Whom vengeance belongs” (Psalms 94:1), because on the fourth day of Creation He created the sun and the moon, and in the future He will punish and take vengeance upon those who worship them. On the fifth day of the week the Levites would recite the psalm beginning: “Sing aloud to God our strength” (Psalms 81:2), because on the fifth day of Creation He created birds and fish to praise His name. On the sixth day of the week they would recite the psalm beginning: “The Lord reigns, He is clothed with majesty” (Psalms 93:1), because on that day He completed His labor and ruled over all of creation in full glory. On the seventh day of the week, Shabbat, they would recite the psalm beginning: “A psalm, a song for the day of Shabbat” (Psalms 92:1), as the future world will be a day that is all Shabbat. Rabbi Neḥemya said: What did the Sages see that led them to distinguish between these chapters, as they interpret the psalms recited on the six weekdays as referring to the past, whereas the psalm recited on Shabbat is referring to the future. Rather, all of the psalms refer to the past. The first six are as explained above: On the first day, the reason is that He acquired the world and transferred it to man, and He was the only ruler in His world; on the second day, the reason is that He separated His works and ruled over them as King; on the third day, the reason is that He revealed the land in His wisdom and thereby prepared the world for His assembly. On the fourth day, the reason is that He created the sun and the moon, and in the future He will punish those who worship them; on the fifth day, the reason is that He created birds and fish to praise His name; on the sixth day, the reason is that He completed His labor and ruled over all of creation. However, on the seventh day, the reason is that He rested from His work, as the phrase “A psalm, a song for the day of Shabbat” is referring to the first Shabbat of Creation. The Gemara comments: And these tanna’im disagree with regard to a statement of Rav Ketina, as Rav Ketina said: The world will exist for six thousand years, and for one thousand years it will be destroyed, as it is stated: “And the Lord alone shall be exalted on that day” (Isaiah 2:11), and one day for God is a thousand years, as indicated in the verse: “For a thousand years in Your sight are but as yesterday when it is past” (Psalms 90:4). Rav Ketina’s statement is in accordance with the opinion of Rabbi Akiva. Conversely, Abaye said: The world will be destroyed for two thousand years, as it is stated: “After two days He will revive us” (Hosea 6:2). According to the opinion of Abaye that the destruction will be for two days, there is no connection between the future world and the day of Shabbat, which is only one day. § The Gemara further asks: When it came to the additional offerings of Shabbat, what would the Levites recite? Rav Anan bar Rava said that Rav said: They would recite in accordance with the mnemonic hei, zayin, yod, vav, lamed, kaf. They would divide the song of Ha’azinu into six sections, each of which began with a letter of the mnemonic: “Give ear [ha’azinu], you heavens” (Deuteronomy 32:1); “Remember [zekhor] the days of old” (Deuteronomy 32:7); “He made him ride [yarkivehu] on the high places of the earth” (Deuteronomy 32:13); “The Lord saw it [vayar] and spurned” (Deuteronomy 32:19); “Were it not [lulei] that I dread the enemy’s provocation” (Deuteronomy 32:27); “For [ki] the Lord will judge His people” (Deuteronomy 32:36). And Rav Ḥanan bar Rava said that Rav said: In the manner that the verses of the song of Ha’azinu are divided here for the recitation of the additional offerings of Shabbat in the Temple, so too are they divided when they are read in the synagogue on Shabbat. The Gemara asks another question: When it came to the daily afternoon offering on Shabbat, what would the Levites recite? Rabbi Yoḥanan said: “Then sang Moses” (Exodus 15:1), and: “Who is like You” (Exodus 15:11), the two halves of the Song of the Sea, and: “Then Israel sang this song” (Numbers 21:17), the entire Song of the Well.

R. Yossi says the day the Temple was destroyed was Saturday night and motzei [the year after] Shmita on the 9th of Av and the watch of Yehoyariv was serving on the 9th of Av, and similarly, in the destruction of the 2nd [Temple]. In both [Temples at the time of the destruction] the Levites were standing on the Duchan and saying "va'Yashev Aleihem Et Onam uv'Ra'atam Yatzmitem Yatzmitem Hashem Eloheinu." (Even though this is the song for Wednesday, G-d caused that they said it at the time.)
Do we sing the song of the day with Nesachim [libations of wine poured onto the altar]? (R. Yochanan): We learn from this [Beraita, that we do] - in both, the Levites were standing on the Duchan and saying "va'Yashev Aleihem Et Onam..."
(R. Avahu): R. Yochanan and Reish Lakish argued about this. R. Yochanan said that we sing the song of the day with Nesachim, and Reish Lakish said that we do not sing the song of the day with Nesachim.
(R. Yochanan, against Reish Lakish): It was taught that in both, the Levites were standing on the Duchan and saying "va'Yashev Aleihem Et Onam..."
(Reish Lakish): What do you learn from this? (R. Yochanan): Since these [that they offered now] were Nesachim [from last Wednesday's offering, which they were not able to bring until now], this is the [proper] time for the song [of Wednesday]. (for Reish Lakish): R. Yochanan says that [they sang] the song for the [correct] day. Reish Lakish says [that they sang] the song of yesterday (i.e. G-d caused them to sing the wrong song, so we cannot learn from there).