Psalm 81: 4 reads: תקעו בחדשׁ שׁופר בכסה ליום חגנו׃
Blast the shofar on [at] the hodesh; on [at] the keiseh, for the day of our hag.
Hodesh and keiseh were speaking of the same day- the first of the month. The prepositional phrases “on the hodesh” and “on the keiseh” identified the time when the shofar was to be blasted; the final prepositional clause identified the reason for such a blast “for the day of our hag.”
This was explained:
כִּ֤י חֹ֣ק לְיִשְׂרָאֵ֣ל ה֑וּא מִ֝שְׁפָּ֗ט לֵאלֹהֵ֥י יַעֲקֹֽב׃
For it is an ordinance [hoq] of Israel; a judgment of the Deity of Jacob. Psalm 81:5
The preposition לְ can indicate a dative or a genitive relation- that is, it can be either to, for, or of. The hoq mentioned is most definitely the hoq of Pesah as it was mentioned at the origin of this institution
וְהָיָה֩ הַיֹּ֨ום הַזֶּ֤ה לָכֶם֙ לְזִכָּרֹ֔ון וְחַגֹּתֶ֥ם אֹתֹ֖ו חַ֣ג לַֽיהוָ֑ה לְדֹרֹ֣תֵיכֶ֔ם חֻקַּ֥ת עֹולָ֖ם תְּחָגֻּֽהוּ׃
And this day shall be for you as a memorial, and you shall celebrate it, as YHWH’s celebration; throughout your generations, an eternal statute [huqqath ‘olam], you shall celebrate it. Exodus 12:14
וַיֹּ֤אמֶר יְהוָה֙ אֶל־מֹשֶׁ֣ה וְאַהֲרֹ֔ן זֹ֖את חֻקַּ֣ת הַפָּ֑סַח כָּל־בֶּנ־נֵכָ֖ר לֹא־יֹ֥אכַל בֹּֽו׃
Then YHWH said to Mosheh and Aharon, “This is the ordinance of Pesah; no foreign son may eat of it.” Exodus 12:43
וְשָׁמַרְתָּ֛ אֶת־הַחֻקָּ֥ה הַזֹּ֖את לְמֹועֲדָ֑הּ מִיָּמִ֖ים יָמִֽימָה׃
Thus you shall observe this ordinance- at its appointed time; from days to days. Exodus 13:10
At the giving of the Torah, this ordinance was included among the mishpatim given
וְאֵ֙לֶּה֙ הַמִּשְׁפָּטִ֔ים אֲשֶׁ֥ר תָּשִׂ֖ים לִפְנֵיהֶֽם׃
So these are the judgments which you shall set before them. Exodus 21:1
אֶת־חַ֣ג הַמַּצֹּות֮ תִּשְׁמֹר֒ שִׁבְעַ֣ת יָמִים֩ תֹּאכַ֨ל מַצֹּ֜ות כַּֽאֲשֶׁ֣ר צִוִּיתִ֗ךָ לְמֹועֵד֙ חֹ֣דֶשׁ הָֽאָבִ֔יב כִּי־בֹ֖ו יָצָ֣אתָ מִמִּצְרָ֑יִם וְלֹא־יֵרָא֥וּ פָנַ֖י רֵיקָֽם׃
The celebration of the Matstsoth you shall observe- seven days you shall eat matstsoth as I commanded you, in the appointed month of the Aviv, for in it you came out of Egypt; and you shall not appear before me destitute. Exodus 23:15
It was the final clause of Psalm 81:4 [ליום הגנו leyom haggeinu- for the day of our celebration] which was intended as the hoq and mishpat which was given by the Deity of Jacob to Israel when they left out of Egypt- as well as the cause to blow the shofar in the month, at the new moon [the time when the moon is nearly completely covered with darkness]. It was the month of the Aviv which we were commanded to observe in its proper time- along with its ordinances and judgments
וְיַעֲשׂוּ בְנֵי-יִשְׂרָאֵל אֶת-הַפָּסַח בְּמוֹעֲדוֹ
בְּאַרְבָּעָה עָשָׂר-יוֹם בַּחֹדֶשׁ הַזֶּה בֵּין הָעֲרְבַּיִם תַּעֲשׂוּ אֹתוֹ בְּמוֹעֲדוֹ כְּכָל-חֻקֹּתָיו וּכְכָל-מִשְׁפָּטָיו תַּעֲשׂוּ אֹתוֹ
So let the children of Israel do the Pesah in its appointed time. On the fourteenth day of this month, between the two evenings, you shall do it in its appointed time- according to its ordinances and according to its judgments shall you do it. Numbers 9:2-3
The blowing of the shofar, at the hodesh of the Aviv, ensured that all of Israel knew when the month began and could observe the ordinance, judgment, and commandment of the Pesah and celebrations of the days of matstsoth.
That the word keseh indicated the new moon can be determined from the use of the word in other ancient Semitic societies surrounding Israel. Kusuh was an ancient Hurrian moon deity whose number was thirty- corresponding with the number of days in a lunar month; in Hittat hieroglyphics, his determinate was the sickle moon.
In early Semitic writing, the root KSH meant to cover, to conceal. In the early lands of the Semitic peoples- Akkad, Sumer, etc., there was a lunar deity called KuSuH. His day was the 30th of each month- when the moon was covered nearly completely. His determinate sign was the sickle moon- the crescent moon. This means the day known as KuSuH was the day the moon was covered except for the smallest sliver- the crescent. The early Hurrian name for the month is KuSuH- named after the lunar deity.
One final comment; in 81:5 the words hoq and mishpat are qualified by the masculine pronoun הוא. This indicated that the subject was a masculine word. In the preceding verse [81:4], the verbal form for blowing a shofar is T-Q-A’- the substantive would be תְּקִיעָה ; this demonstrated that it was the hag- a masculine noun- and not the blowing of the shofar which was qualified by the words hoq and mishpat.