Moon phases

http://tycho.usno.navy.mil/vphase.html

http://users.ameritech.net/paulcarlisle/MoonCalendar.html

http://aa.usno.navy.mil/cgi-bin/aa_moonphases.pl?year=2005&ZZZ=END



http://www.hermetic.ch/cal_stud/lunarcal/lunarcal.htm
It is possible to devise a lunar calendar which is simple to use (only elementary arithmetic is required for its use — a computer is not needed) and which may be expected to stay in sync with the lunar phases over a period of several millennia. This article presents such a calendar, more specifically, a rule-based solar-count lunar calendar. (For an explanation of types of calendars, with a brief discussion of lunar calendars in general, see Types of Calendars.) In this calendar, years have 12 or 13 months, which are named after goddesses from various cultures, and so this calendar is called the Goddess Lunar Calendar (or Goddess Calendar for short).
I shall now define this calendar, then I shall discuss the accuracy of it with respect to the lunar and solar cycles. Then I shall relate the Goddess Calendar to the Common Era Calendar (which is the same as the Gregorian Calendar except that instead of the A.D./B.C. year-numbering system the astronomical year-numbering sytem is used), and finally I shall provide software to allow conversion between dates in these two calendars.

The Goddess Lunar Calendar DefinedAccording to the Goddess Calendar, time, although not strictly cyclic, is measured in cycles — cycles of 1689 calendar years. For the purpose of viewing time as linear, and assigning each day a unique date, each 1689-year cycle is associated with an integer -2, -1, 0, 1, 2, ... Years within each cycle are numbered 1 through 1689.
Years usually have twelve months, but sometimes thirteen (such years are known as 'long years'). A year has a 13th month if and only if the number of that year is divisible by 3 or if the digits of the year sum to 2, 22 or 23.
Months are numbered from 1 through 13. Odd-numbered months have 29 days and even-numbered months have 30 days.
The months are named after thirteen goddesses, as follows:
Monthnumber
Month name
Number of days
1
Astarte
29 usually
2
Bast
30
3
Cybele
29
4
Diana
30
5
Eris
29
6
Freya
30
7
Gaia
29
8
Hathor
30
9
Isis
29
10
Juno
30
11
Kali
29
12
Lakshmi
30
13
Maat
29 always
The names of the months begin with the letters A, B, ... M, making it easier to remember their order.
The thirteen goddess names are taken from eight different cultures. There are four Egyptian goddesses, two Greek, two Roman, two Hindu and one goddess from each of the, Canaanite, Phrygian and Norse cultures. This is thus a multi-cultural calendar.
As stated above, the 1689-year cycles are numbered -2, -1, 0, 1, 2, 3 ...., the months are numbered 1 through 13 and the days are numbered 1 through 31 (in various months). Thus a date in the Goddess Calendar has the form "cycle-year-month-day", for example, "3-0825-02-27".
We will see later how dates in Goddess Calendar correspond to dates in the Common Era Calendar.
A year is a "long" year if it contains a thirteenth month. The rule for when a year is a long year was given above, and here more formally:
A year is a long year (and so has a 13th month) if and only if 1.1 The year is divisible by 3 or1.2 The digits of the year sum to 2, 22 or 23.
1.1 and 1.2 are mutually exclusive possibilities (that is, a year is a long year by virtue of 1.1 or 1.2 but not both).
Examples:
The year 0-567 in the Goddess Calendar is a long year because 567 is divisible by 3.
The year 1-779 is a long year because the digits of the year sum to 23 (= 7 + 7 + 9).
The year 2-1003 is not a long year because the digits of the year sum to 4 (= 1 + 0 + 0 + 3) and 1003 is not divisible by 3.
In any year (whether or not long) odd-numbered months (1, 3, ..., 13) have 29 days and even-numbered months (2, 4, ..., 12) have 30 days, except that in a long year the length of the first month is determined by the following rule:
2.1 If the digits of the year sum to 2 then the first month has 28 days.2.2 If the digits of the year sum to 22 then the first month has 29 days.2.3 If the digits of the year sum to 23 then the first month has 30 days.2.4 If the year is divisible by 9 then the first month has 30 days.2.5 If the year is divisible by 3 but not by 9 then the first month has 31 days.
These are mutually exclusive possibilities, so in a long year only one of 2.1 - 2.5 can be used to determine the number of days in the first month.
The regular structure of the Goddess Calendar (twelve months of alternating lengths 29 and 30 days, with an occasional 13th month of 29 days, and a variable length for the first month in long years), and the rule for when a year is a long year, are sufficiently simple to be remembered easily, and the rule for the length of the first month is not particularly difficult to remember.
To reckon time according to this calendar it is necessary only to know the Goddess date for a particular day and to be able to calculate the date which follows any given date. This requires only the ability to remember the rules of the calendar and to add four single-digit numbers and to divide a four-digit number by 3 and by 9, all of which can be done mentally.
For example, is the year 1686 (in any cycle) a long year? And if so, how many days are in the first month? Well, 1686 divided by 3 is 562 exactly, and divided by 9 is 187 with 3 over, so by rule 1.1 the year 1686 is a long year, and by rule 2.5 Astarte's month has 31 days (in this year).