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Metonic calendars include the calendar used in the Antikythera Mechanism about 2000 years ago, and the Hebrew calendar.
The complexity required in an accurate lunisolar calendar may explain why solar calendars (which have months which no longer relate to the phase of the Moon, but are based only on the motion of the Sun relative to the equinoxes and solstices) have generally replaced lunar calendars for civil use in most societies.
The nones always occur 8 days (one Roman week) before the ides, i.e., on the fifth or the seventh.
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.
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.