The most commonly used birthstone system (based on the birth month)
has nothing to do with astrology and signs of the Zodiac.

Since the signs of the Zodiac change anywhere from the 19^th  to
the 22^nd of each month (depending on the source consulted),
a list of Zodiac birth stones cannot be based on the months of the calendar year.

According to the system of astrology most used in the western world,
Zodiac (or birth) signs are determined by the position of the Sun in one's horoscope.

In the Sidereal system, the birth sign is determined by the position of the Moon.

But in either system, it is the Zodiac sign that is important, not the birth month.
Each sign of the Zodiac is ruled by one of the planets and each planet
in turn rules over certain gemstones according to their color.
In Asia, this group of birthstones are known as the nine astral gems.

  The following list of birthstones is based on the planet ruling one's Zodiac sign;
it has been known and accepted throughout Asia since ancient times:

Jewelry Metal Facts