Leviticus 23:24

Speak with the children of Israel, saying: In Tishri, which is the seven month, shall be to you a festival of seven days, a memorial of trumpets, a holy convocation.

the seventh month The months are counted from Nisan, since that was when we left Egypt. Also, the barley ripening begins a new agricultural year, followed by “the first-fruits of the wheat harvest” [Exodus 34:22], and followed in turn by “when you gather your products” [Exodus 23:16]. Each of these festivals must be celebrated on its given day.

Since we normally blow trumpets every month [Numbers 10:10], this day is called a remembrance of sounding the horn . Elsewhere it is called “a day of sounding the horn” [Numbers 29:1], because on this day (as on Yom Kippur of the jubilee year [25:9]) there is a separate commandment to blow the ram’s horn. Attend carefully now, and perhaps you shall understand the mysteries which I now reveal. The Exegetes have said that Rosh HaShana is the Day of Judgment, and that the ram’s horn reminds us of God’s majesty. Although every new month is heralded by the blowing of trumpets, the month of Nisan is preeminent over them all. On the first of Nisan the Tabernacle was erected; so it will also be in the future, as Ezekiel says: “in the first month, on the first day of the month” [Ezekiel 45:18]. He also says, “and so shall you do on the seventh day of the month” [Ezekiel 45:20] and on that day the sun and the moon are at right angles to one another. The festival on the fifteenth of Nisan occurs when the sun and the moon are directly opposite one another. The seventh day of the festival occurs when the sun and moon are again at right angles to one another. Accordingly, Rosh HaShana is the most important day of the year. Although it is a Day of Judgment, one may not fast on it, as Ezra showed us [Nehemiah 8:10]. Yom Kippur comes next, when the Moon is in conjunction with Aquarius, and then comes Sukkot, which is just like Passover except it has eight days instead of seven. Thus the mystery of the festivals and the mystery of the Sabbath are now clarified [not quite — Translator].

IT SHALL BE UNTO YOU ‘SHABBATHON’ (A SOLEMN REST). This means that it is to be a day of rest, to repose thereon [from work]. And our Rabbis have said: “The word shabbathon (a solemn rest) amounts to a positive commandment.” Thus one who does work on a festival day transgresses both a negative commandment and a positive commandment, and one who rests thereon fulfills a positive commandment. And in their opinion [that of the Rabbis], all the festivals have been likened one to another, (above, Verse 4), thus teaching that all festivals are alike with respect to a positive commandment [to rest thereon] and a negative commandment [not to do work], even though in the case of some festivals, [as the text continues], the positive commandment is not clearly set forth in Scripture. for the word shabbathon is not mentioned in connection with the Festival of Unleavened Bread, nor in connection with the Festival of Weeks.
Now in the Mechilta on the section of Hachodesh I have seen the following text: 9. “And ye shall observe this day. Why is this said? Has it not already been stated, no manner of work shall be done in them?, Verse 16. From this I would know only about matters which are considered m’lachah [‘work’ on the Sabbath, and thus forbidden by the Torah]. How do I know about activities which are termed sh’vuth [as explained later on]? From the verse which states, and ye shall observe this day, which includes those activities which are termed sh’vuth. I might then think that on the intermediate days of the festival those activities known as sh’vuth are likewise forbidden, and the following reasoning would favor it etc; is stated both with reference to the first and seventh days of Passover (ibid., Verse 16), and the intermediate days of the festival are similarly termed [in the verse, These are the appointed seasons of the Eternal, even holy convocations — Verse 4 in the section before us], therefore just as activities called sh’vuth are forbidden on the first and seventh days, so are they also forbidden on the intermediate days of the festival.” — So it could be argued. Scripture therefore states [with reference to the Festival of Tabernacles], On the first day shall be a solemn rest” [but not on the following intermediate days of the festival].
Thus the Sages interpreted the word shabbathon as indicating a complete rest on that day, …

זכרון תרועה, you will be remembered by G’d as a result of your blowing the t’ruah from the shofar. This is spelled out more clearly in Numbers 10,9 והרעותם בחצוצרות...ונזכרתם...וגו', “when you will blow the sound of the t’ruah…. You will be remembered… etc.

זכרון תרועה, a remembrance of the Royal t’ruah. When jubilating toward one’s king, one employs these blasts on the trumpets or shofars to demonstrate such regard for one’s king. Compare Psalms 81,2 הרניני לאלוקים עוזנו, הרנינו!, “stir up jubilation to G’d, our strength, raise a shout!” The expression זכרון, is an allusion to the fact that on this date G’d sits on the throne of Justice remembering the deeds of each one of His subjects and evaluating them as a judge. This is why the Talmud in Rosh Hashanah 8 calls on us to recite verse 4 in the above mentioned chapter of Psalms which reads תקעו בחודש שפר בכסא ליום חגנו, כי חוק לישראל הוא משפט לאלוקי יעקב, “blow the horn on the new moon on the day the moon is veiled, on our feast day, for it is a statute for Israel a ruling of the G’d of Yaakov.” On such days we have additional reason to rejoice in the fact that He is our King, who inclines toward leniency, and is likely to find us as possessing sufficient merit to pass this examination of the way we led our lives in the year just concluded. Isaiah expresses these sentiments in referring to G’d as כי ה' שופטנו, ה' מחוקקנו, ה' מלכנו, “for the Lord is our Judge, the Lord is our lawgiver, the Lord is our King” (Isaiah 33,22)