As a preliminary observation I wish you to make note that these Days are MEMORIALS,--every one of them!  Now God instituted some memorials just for the Israelites only, as Circumcision, for that  rite memorializes a Covenant between Him and the Israelites only.  BUT THE DAYS WE NOW CONSIDER MEMORIALIZE GOD'S GREAT CONTACTS WITH ALL MANKIND, AND THEREFORE ARE TO BE OBSERVED BY ALL RACES, THE WORLD OVER.  These Days memorialize some tremendous contact of God with all mankind and so are to be observed by all races, everywhere.
The Trumpets and Atonement Sabbaths memorialize GOD'S MESSAGES TO MANKIND--TO ALL RACES--as the Bible; The Divine or real message on Repentance, on Regeneration, so forth and so on, on the one hand, and OUR DIRE NEED OF REDEMPTION on the other (Atonement)....While the Feast of Tabernacles the Scriptures show plainly is an eight-day THANKSGIVING SEASON, the first and last days of which are Sabbaths regardless of on what days of the week they may fall.  Is gratitude merely a Jewish virtue?  Then this Thanksgiving Week is of worldwide significance.  We now pass to a more detailed study of these last three, which three constitute the autumnal cluster.

Relative to our redemption, God sets apart two Memorial Days, separated far apart in the year:  PASSOVER DAY, coming in the spring, and ATONEMENT DAY, coming in the fall.
Why two days, so different and so far separated?  Because one is to set forth God's part in our redemption (Passover, with the Lord's Supper "showing forth the manner of His death"), and the other sets forth REDEMPTION FROM HUMANITY'S VIEWPOINT,--OUR DIRE NEED THEREFORE!
"Ye shall afflict your soul", it is written of Atonement Day.  The Hebrew word that is translated "afflict" is anah and means lower, or, humble!  Read Lev. 16:29,31; 23:27,32; and Numbers 29:7.  It is A DAY FOR HUMBLING ONESELF:  for realizing that we have no atoning quality PER SE, that we are  destitute of redeeming power within ourselves; that left to our own natures, we are wholly lost and for ever undone!
This is an annual Sabbath Divinely constituted for CONTEMPLATING OUR UTTER NEED OF GOD, THE ABSOLUTE NEED OF DIVINE INTERVENTION AND A DIVINE REGENERATION.  It is then that we especially pray, "Create a clean heart in me, O God!" Psa. 51:10.
On the Day of Atonement we are to make "an offering by fire", which means an offering free from all carnal purposes.