Thinking about the black moon as being the third of four new moons in a astronomical quarter (between an equinox and solstice), has lead me to think about William Morris's Moonwise calendar whose rules are at
Each month begins at 05:00 UT after a new moon.
The month names are based mainly on the Celtic Tree calendar on which the Ogam Wheel is based.
In a 12 month year the months are called Birch Moon, Rowan Moon, Ash Moon, Alder Moon, Willow Moon, Hawthorn Moon, Oak Moon, Holly Moon, Hazel Moon, Vine Moon, Ivy Moon and Yew Moon.
The months are scheduled so that the solstices and equinoxes occur thus:
December Solstice always in Birch Moon
March Equinox always in Alder Moon
June Solstice always in Oak Moon and
September Equinox always in Vine Moon.
Some times one of these 4 months will occur 4 months after the previous one. Then an intercalary month must occur in between. They are determined as follows:
If there are three months between Birch and Alder, they are Rowan, Ash and Gorse.
If there are three months between Alder and Oak, they are Willow, Hawthorn and Heather.
If there are three months between Oak and Vine, they are Holly, Hazel and Apple.
If there are three months between Vine and Birch, they are Ivy, Reed and Elder. In this last case, the usual Yew Moon is suppressed.
This intercalary months are very similar to the black moons. One important difference arises from the fact that the months begin at 05:00 UT after the new moon rather than the new moon. This would cause some of the intercalary months to occur later than the corresponding black moons.