Source For This Article
 
 

Leviticus 23: Verses 33 - 36
 
       And the Lord spake unto Moses, saying, Speak unto the children of Israel,
       saying, The fifteenth day of this seventh month shall be the feast of
       tabernacles for seven days unto the Lord.  On the first day shall be an holy
       convocation:  ye shall do no servile work therein.  Seven days ye shall
       offer an offering made by fire unto the Lord:  on the eighth day shall be an
       holy convocation unto you; and ye shall offer an offering made by fire unto
       the Lord:  it is a solemn assembly; and ye shall do no servile work therein.

Numbers 10: Verse 10
 
       Also in the day of your gladness, and in your solemn days, and in the
       beginnings of your months, ye shall blow with the trumpets over your burnt
       offerings, and over the sacrifices of your peace offerings; that they may be
       to you for a memorial before your God:  I am the Lord your God.

Nisan 1 begins the ecclesiastical year and Tishri 1 begins the civil.  But the correspondence doesn't end there.  On Nisan 10 The Lamb was inspected and on Tishri 10 the second set of tablets was given to the children of Israel (Yom Kippur).  Nisan 15 - 21 Passover is celebrated and Tishri 15 - 21 Sukkot is celebrated, both commencing on the full moon.
 
The spring feasts are a time of expansion and growth culminating seven weeks later on Shavuot, Sivan 6.  It was on Shavuot that the first set of tablets was given to the children of Israel.  These were smashed by Moses though on Tammuz 17 because of worshipping the golden calf.
 
But the fall feasts herald a period of contraction known as the rainy season.  So instead of a feast 7 weeks and a day after Sukkot the feast is 7 days and a day after Sukkot.  Read what the Rabbis have to say about this day:

From this article we get a deeper insight into Shmini Atzeret:
 
       The name "Atzeret" is actually used most in the Talmud to refer to the
       holiday of Shavuot.  Shavuot can be seen as the "completion" of Pesach
       - for Pesach, commemorating the Exodus, represents our "physical" birth
       as a nation, while Shavuot, commemorating our receiving of the Torah,
       represents our "spiritual" birth...
 
       Shmini Atzeret should really have been placed seven weeks after Sukkot
       as Shavuot, the "closure" of Pesach, follows it by seven weeks, but,
       according to the Midrash, Hashem had mercy on the Jewish People.  For
       Pesach is in the Spring and Shavuot is in the Summer, both pleasant
       times for travel (these holidays are all "Regalim," Pilgrim Festivals, on
       which Jewish males are obligated to travel to the Temple in Yerushalayim),
       but seven weeks after Sukkot would already be into the rainy season in
       Israel, and travel would not be pleasant then.  Therefore, Hashem allowed
       the closure of Sukkot, Shmini Atzeret, to be celebrated right after Sukkot.

Shmini Atzeret is to Tabernacles as Pentecost is to Passover.  As you know the Christianized Pentecost is the birthday of the Church.  A Shmini Atzeret Rapture (September 26, 2013) would certainly put closure on the Church Age and would mirror the fulfillment of Pentecost.
 
Sivan 6 is neither a full moon nor a new moon and so the same for Tishri 22.  The Leviticus verses quoted above indicate that Shmini Atzeret is a holy day in its own right and the quote from Numbers indicates that trumps are blown on all the holy days.  So since Shmini Atzeret is the last holy day till spring the last trump is blown on that day.