While it is difficult to give any one definitive reason for the achievements of individuals in the Bach family, there certainly are many factors which worked in their favor. Most notably they all possessed a determined spirit that thrived on adversity. And so it happened that during one of the most tragic periods of German history that the Bachs were able to accomplish outstanding musical achievements. The strong solidarity of their family ties were incredibly helpful as well. Also, they often married into families who were themselves musically inclined, mixing the creative talent of many sources.
Among J. S. Bach's close relatives there were some very talented composers. However his own father did not exhibit any creative gifts, and this was probably true of his grandfather, Christoph Bach, as well. Therefore it seems as though the musical talent inherited from his father's side accumulated in two generations before bursting forth.
Although Bach's mother died when he was 10 years old, she was still a very important influence on him. The importance of his mother's family is shown by one fact in particular. His father, Johann Ambrosius, had a twin brother, Johann Christoph. These twins were so alike that even their own wives could not tell the two of them apart. And as musicians they were said to be nearly identical. Yet their children's musical talent shows no comparison. Christoph's son was an organist of mediocre talent, while J. S. Bach grew into a musical genius.
Bach's children seem to have inherited their parents' passion for music as well. A genius does not necessarily produce children with equally creative gifts. But in spite of J. S. Bach's enormous contributions to music, two of his sons became leaders in the music of their time, while two others themselves seem to have showed substantial musical talent.
Below is a brief family tree for Johann Sebastian Bach. No attempt at completeness has been made, however what is shown has been diligently researched.

Originally published February 5, 2015
Updated October 26, 2017
Researched and Written by: Thomas Acreman
Sources:
The Bach Family by Karl Geiringer
Other Classic History articles on the life of J. S. Bach include:
J. S. Bach - His early life
J. S. Bach - Years of Growth (1702-1706)