The centerpiece of Rosh Hashana is the Misva of Shofar. In Parashat Nisavim, there is a Remez (allusion) to this Misva. The Pasuk warns the Jewish people, "Pen Yesh Bachem Shoresh Poreh Rosh V'La'Anah"- Perhaps you will have amongst you a root of someone who wants to throw off Hashem's yoke and rebel. In the Hebrew text, these words form the acronym, Sh'O'Fa'R. This means that the Shofar has the spiritual ability to remove the bad root ingrained in us. After Adam HaRishon ate from the Tree of Knowledge, the evil became an integral part of humanity. Before that, good and evil were separate entities. It was much easier to avoid evil. Now that they are intermingled, it is harder to do good. The Shofar has the ability to engender a spiritual awakening to subjugate the evil root within us.
The Halacha requires blowing thirty Shofar blasts in the first stage. This is derived as follows. The Torah uses the word "Teruah" (broken sound) three times. This indicates that three Teruot are required. From other Pesukim, it is learnt that each Teruah blast requires a "Tekiah" (solid sound) both before and after it. This creates a sub-total of nine mandatory blasts. The Gemara records three possibilities as to what is the sound of the Teruah referred to by the Pasuk. It is either what we today call "Teruah" or "Shevarim" or both together-"Shevarim Teruah." Therefore, the Halacha requires to take all three possibilities into account and blow one set of nine blast with each of the three variations. These are known as TaSHRaT (with the Shevarim Teruah), TaRaT (with the Teruah) and TaSHaT (with the Shevarim). Together, this equal the thirty mandatory blasts, which are the main Misva. The Shofar is sounded again during the Amidah, but that is for a different reason.
Each type of blast has a minimum measure of length in order to be valid. The Gemara states that minimum length of the Teruah is nine "Tromitin"-the minimal sound possibly emitted. The proper way to blow Teruah is a sequence of nine miniscule sounds, and not like some people, who blow the last sound with more length, creating a "tail". The measure of the Tekiah is the same as the Teruah, but instead of staccato, it is blown solidly. One should avoid the practice of some people who end the sound of their Tekiah by raising the pitch. The Shevarim also has a minimal length of nine Tromitin. Therefore, each of its three blasts, is three Tromitin long.
Thus, each of the thirty blasts of shofar is comprised of nine Tromitin, creating a total of 270 Tromitin. This number is significant because it is the Gematria of "RA" (evil). This means that the 270 Tromitin that we hear have the ability to subjugate the "RA"-evil within us. Therefore, one should pay very close attention to the Shofar blowing.
While our primary intent is to fulfil the Misva because Hashem commanded it, there are many deep secrets in this Misva, and we pray to Hashem that He count it as though we had all of the other mystical intents.