The Torah gives at least 1 reason for all other parties and even up to 2 reasons for some. The feast of Matzoth commemorates the Exodus from Egypt. It also celebrates the beginning of the barley harvest. The Feast of Shavuot is a celebration of the harvest of Wheat.
Yom Ha-Kippurim is a national day of repentance as described in detail at the end of Leviticus 16. Finally, the feast of Sukkot celebrates agricultural harvesting. In contrast to all these festivities, Yom Teruah has no clear purpose other than to rest.
Yet its very name Yom Teruah gives us a clue of its purpose. Teruah means making a loud noise. This word can describe the sound of a trumpet. In Leviticus 23:24, Yom Teruah is also referred to as Zichron Teruah.
Yom Kippur completes the period of ten days that begin with the New Year of Rosh Hashanah. The transformation of Yom Teruah to Rosh Hashana is due to the Babylonian influence. The first point of the transformation was the adoption of the Babylonian names of the months.
This is like the influence of the Christmas on the Jewish holiday Hannukah. This influence began with the inoffensive habit of giving gifts in Hanukkah. Until the Jews arrived in North America this custom was unknown and is still a rarity in Israel.
Hanukkah does not need to compete with Christmas for the hearts and minds of Jewish youth. Once Hanukkah assumed this trivial aspect it was ripe for more significant influences.
Today many Jews have established the habit of erecting a Hanukkah bush. It is a Jewish alternative to the Christmas tree. These Jews judaized the Christmas tree and incorporated it in Hanukkah. Hanukkah often falls around the same time of Christmas. It made it easier to incorporate Christmas into their observance of the Hannukah.
The ancient Rabbis got influenced by the pagan religion of Babylon. Although many Jews returned when the Exile ended in 516 BCE, the chief Rabbis remained in Babylon. The rabbinical Judaism took shape. Babylon remained as the heartland of Rabbinic Judaism until the 11th century CE. The Babylonian Talmud abounds in influences of Babylonian paganism. Indeed, pagan deities even appear in the Talmud recycled as genuine angels and demons.
A Babylonian influence was in the observance of Yom Teruah as a New Year celebration. From ancient times the Babylonians had a lunar calendar like the biblical calendar. The result was that Yom Teruah often fell on the same day as the Babylonian New Year holiday known as Akitu.
Akitu fell on the first day of Tishrei that coincided with Yom Teruah on the first day of Seventh Month. At the same time the Rabbis did not want to adopt Akitu. So they judaized it by changing the name of Yom Teruah to Rosh Hashanah.
In the Torah, the middle of the Seventh Month is actually the end of the agricultural cycle. In the Land of Israel, grains are sown in the Fall and harvested in the Spring. The new agricultural cycle does not begin until the plowing of the fields. This is not done until the first rains moisten the soil enough to plow. In the Land of Israel, this could be as early as the middle of the Seventh Month but is usually in the Eighth Month or later.