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 http://www.ou.org/chagim/shmini-simchat/realatzeret.htm 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 28, 2021) 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.