I found a fairly simple lunar calendar, which can be used with a solar calendar that has exactly 52 weeks in a year, such as the World Calendar or the International Fixed Calendar.
Each of these calendar reforms has 364 days in each year that are a normal day of the calendar week plus one or two days that are not a normal day of the calendar week. I’ll call the days that are a normal day of the calendar week regular
days. Each year then has 364 regular days.
The months of the lunar calendar are grouped into dwarf-years of normally 7 months, so that the odd-numbered months of the dwarf-year have 29 regular days and the even-numbered months of the dwarf-year have 30 regular days. Also every month
begins with a regular day.
Every 13th dwarf-year has 9 months and all other dwarf-years have 7 months.
The resulting cycle has 93 months equal to 391 calendar weeks. This cycle is one calendar week longer than seven and a half years. Every 9-month dwarf year begins on the same day of the calendar week.
52 of these cycles form a 391-year cycle of 4836 months = 403 lunar years = 676 dwarf-years.
For the World Calendar, International Fixed Calendar or any other such calendar that uses the Gregorian year the mean month =
(391/4836)*365.2425 = 29.530566… days.
The calendar could be made more accurate by occasionally having a 9-month dwarf-year occur 6 dwarf-years after the previous 9-month dwarf-year instead of the usual 13 dwarf-years. Every 9-month dwarf year would still begin on the same day
of the calendar week. Having every 50th 9-month dwarf-year occur 6 dwarf-years after, would create a 372-year cycle and mean month of 29.53058248… days. Having every 24th would produce a 353-year cycle with mean month of 29.53060… days.
What ever is the expectation of 'this' exercise, I recall having discussed some ideas during my projecting 'Use of Tropico-sidereal aspect of
using Sidereal Days for civil calendar, like the Gregorian calendar in my published contribution "Metric, Sidreal or Decimal calendar; V26 No.2&5; April 1992 - March 1993; The Standards Enginee; Bureau of Indian Standards, New Delhi; pp 44-47 wherein I pointed:
Why was it that man never thought of using duration of Sidreal Day for use in a civil calendar?
My recent exposition of 896-year/11082 Moons/327257+1 day Harappa Tithi Calendar having 326919 'tithi' using Tithi=1 338/336919 day (calculations
- the format of which is already known to list members, on removing 13th date in every month of Gregorian calendar of (12x29 1/2=354 Tithi). This needs 68 1947/4779 cycles of 162-moons, each of 4779 tithi.
5 1/2 lunar years make up for 1947 tithi; to complete one cycle of 11082 moons in 68 66/162 cycles - each of 13.5 lunar years. This combination results in Mean Year= 365.2421875 days; and Mean Lunation
= 327258/11082=29.530590146183 days (29d 12h 44m 2s.98863).