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The local day begins at 05:00 clock time.
The last day of the month is the Universal Time (UT) day starting at 05:00 that contains the Dark Moon conjunction. This means that the month begins with the day whose 05:00 UT follows Dark Moon.
Months can have either 29 or 30 days, depending on the timing of Dark Moon, and the days are numbered sequentially from 1 to 29 or 30.
There are twelve of these months in most years, and they 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.
Birch Moon has to contain the moment of northern winter solstice, Alder Moon has to contain the moment of northern spring equinox, Oak Moon has to contain the moment of northern summer solstice and Vine Moon has to contain the moment of northern autumn equinox.
In order to maintain the above linkage, extra months are brought in as necessary. It is also just possible for there to be only one month between Birch and Alder, and in this case it is called Rowan and Ash Moon, though it doesn't happen until at least 2050.
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.
The reason for the last rule is that Vine, Ivy, Reed and Elder is the order of months before the solstice in Robert Graves’ The White Goddess. Keeping Vine as the equinox moon means there isn't normally room for all of these, and in a sense, Yew takes over from both Reed and Elder in most years. The revision from Lunar Tree Calendar to Moonwise Calendar was partly an attempt to get closer to Graves’ sequence: not completely possible, as Graves has 13 months of 28 days, plus a day. Lunar months, as measured dark to dark or full to full, are 29 or 30 days. If the northern summer solstice moon is Oak, as it needs to be, the other solstice moon has to be Birch, if there is to be room for the sequence Birch, Rowan, Ash, Alder, Willow, Hawthorn and Oak. In years with no Gorse Moon, it is understood that gorse is a second “tree” of Alder Moon. In years with no Heather Moon, heather is a second “tree” of Oak Moon, and in years with no Apple Moon, apple is the second tree of Hazel Moon. Mistletoe is an additional “tree” of Birch Moon.
The four main Pagan festivals: Samhain, Imbolc, Beltane and Lúnasa / Lammas are shown at their traditional times: the last night of the month, leading in to the first day. Dark Moon and/or Full Moon within 12 days of the festival also takes on the name: for example, Samhain Dark Moon. The moons are linked in a similar way to the solstices and equinoxes, so that there is Midwinter Full Moon, and so on.
On the printed calendar, each night there is a moon phase, moonsign and a pair of moon words. They are intended to apply to the whole night, but they are all calculated for 23:00 UT. The moonsign is the tropical zodiac sign the moon is in at 23:00 UT, and the words are related to the moon’s position and relationship to the sun and planets. See more about that here.
Best astronomical predictions are used. If a Dark Moon (or indeed a solstice or equinox near Dark Moon) is predicted to be very close to 05:00 UT, then best efforts will be made to find whether it falls before 05:00, and in the end, an educated judgement might have to be made. This doesn’t happen for the foreseeable future.