The date and time of this actual observation depends on the exact geographical longitude as well as latitude, atmospheric conditions, the visual acuity of the observers, etc.Therefore, the beginning and lengths of months defined by observation cannot be accurately predicted.
Therefore, the Moon takes a little longer to return to perigee than to return to the same star.
A synodic month is longer than a sidereal month because the Earth-Moon system is orbiting the Sun in the same direction as the Moon is orbiting the Earth.
The Sun moves eastward with respect to the stars (as does the Moon) and it takes about 2.2 days longer for the Moon to return to the same apparent position with respect to the Sun.
All months landing on a knuckle are 31 days long and those landing between them are not.
When the knuckle of the index finger is reached (July), go back to the first knuckle (or over to the first knuckle on the other fist, held next to the first) and continue with August.
At the simplest level, most well-known lunar calendars are based on the initial approximation that 2 lunations last 59 days: a 30-day full month followed by a 29-day hollow month — but this is only roughly accurate, and eventually needs correction by using larger cycles, or the equivalent of leap days.