Yet already in the Mishna we find some statements that clearly place this day at the beginning of the year:
"באחד בתשרי ראש השנה לשנים ולשמיטין וליובלות, לנטיעה ולירקות"
"בראש השנה כל באי העולם עוברים לפניו כבני מרון, שנאמר: 'היוצר יחד לבם המבין אל כל מעשיהם'" Etc...
How are these to be reconciled? Here, like in most places, it is the job of the Talmud and the Rishonim to display the congruity between these two works. Already in the Torah itself it is written regarding Succot that it takes place "at the exiting of the previous year", so it's clear from here as well as many other verses that the Torah's calander is loyal to the agricultural calander.
Rabbi Moshe ben Nah'man (as well as other commentators) attempt to bridge the gap: He points out that the general year starts in the "seventh month" of the Torah but out of respect of the Exodus it counts from the spring, and even the general world counted the new year from the beginning of the fall at that time. He also says that it's clear that the very reason for the horn soundings and why it should be a day of "remembrance" (which in biblical Hebrew always means remembrance to a judge for a positive verdict) and the reason it's a time of "holy gathering" is because the Torah acknowledges this season as the beginning of the year and some sort of judgement seems to be taking place, in which our participation is a key aspect, hence Mishnaic statements such as the one likening it to a sheep-herd count.
Yet what of the horn-blowing? What does this act that is generally considered in the bible one of aggression or announcement have to do with our being examined by G-d on the new year, or with "making G-d king over us"? I once saw a simplistic and innovative explanation for this: The two words to describe the sounds that are to be created on this day (and by horns in general for that matter) are "לתקוע\תקיעה" and "תרועה". The first word comes from the word "nailed into place" (תקוע) and is a strong constant sound. It is used when an army is confidently marching forward. The second word, which comes from the word "unstable" (רעוע) is used when the group is scattering in a guerrilla-warfare-like battle. It is evident, therefore, that the intention of the Torah's command to utilize these war instruments on this day of “remembrance” and the intention behind these confidence and alarm-soundings is, of course, a representation of the confidence/alarm that should be traversing our minds on this day.....