The Hermetic Lunar Week Calendar

A timezone-independent lunar calendar which tracks the Moon
and avoids the problem of rescheduling events every year


There are plenty of calendars which have been proposed as a replacement for the Gregorian Calendar (a.k.a. the Common Era Calendar) now in use worldwide. That is because that calendar has only one good feature: It stays in sync rather well with the vernal equinox year (the period from one vernal equinox to the next). Its two worst features are:

  1. It has a year made up of so-called “months” which consist of anything from 28 to 31 days and which have no connection to the actual lunar cycle (since a lunation averages 29.53 days).
  2. In combination with the usual 7-day week, there are never an integral number of weeks in a year, and consequently any given day of a given month occurs on one day of the week in one year and a different day of the week in the next year, making it impossible to draw up schedules which stay the same from year to year.

The Hermetic Lunar Week Calendar (HLWC) possesses neither of these flaws, and still manages to stay more-or-less in sync with the vernal equinox year.

This lunar calendar specifies days by means of dates consisting of four numbers:

year – month – week – daywhere the day number ranges from 1 through 9, the week number ranges from 1 through 4, the month number ranges from 1 through 13, and the year number is an integer (-2, -1, 0, 1, 2, 3, …). Each year has 12 or 13 months, and each month has 29 or 30 days (but are not numbered in that way). Most weeks have 7 days, some have 8 days, and a few have 6 or 9 days.

The calendar is called a “lunar week” calendar because the four quarters of the Moon (dark moon, full moon and the two half moons) occur either on, or close to, the last day of the week. Thus each month has exactly four weeks and those weeks stay in sync fairly closely with the lunar phases.

The HLWC is a calendar based on astronomical calculation (in this respect it is in the same class as the Chinese Calendar). Accordingly its definition depends on the definition of astronomical terms, specifically, vernal equinox, dark moon, full moonand half moon.

A dark moon occurs when the Sun and the Moon are astronomically conjunct (or more exactly, when either the Moon’s center lies on the line joining the centers of the Earth and the Sun or the plane defined by the Sun, Earth and Moon is perpendicular to the Earth’s orbital plane). The terms full moon, half-moon and vernal equinox have their usual astronomical meaning.

A formal definition of the calendar will now be given, followed by a discussion of some of its properties, showing its superiority to the Common Era Calendar.

Definition of the Calendar

The Hermetic Lunar Week Calendar (HLWC) is defined as follows, where defined terms are shown in bold face. The first few defined terms allow us to state, in a well-defined manner, that days in the HLWC are divided into hours and minutes in the usual way, beginning at 00:00, but that days begin at approximately dawn rather than at midnight.

A time is a pair of numbers (h,m), where h may range from 0 through 23 and is interpreted as a number of elapsed hours andm may range from 0 through 59 and is interpreted as a number of elapsed minutes. Between any two times there is a time difference, calculated in the usual way, e.g., 09:03 to 12:22 is 3:19 and 22:50 to 04:10 is 5:20.

A clock is an instrument with a dynamic, cyclic display of hours and minutes. A clock displaying HLWC time is a clock such that a correctly-set clock located at Greenwich, England, always displays 18:00 at local midnight. (In contrast, clocks at Greenwich displaying conventional time always display 00:00 at local midnight.)

A timezone is a region on the Earth such that, at the instant when a clock in Greenwich displaying HLWC time displays 18:00, all correctly-set clocks displaying HLWC time in that region display a time which differs from 18:00 by a certain fixed amount (in hours and minutes). That amount is called the timezone offset. E.g., Vietnam is part of the +7 hours timezone. The +0 hours timezone is the same as the GMT timezone in the conventional system of times (although in this timezone a GMT time and a HLWC time differ by six hours.)

A day in the HLWC, when used in a particular timezone, is the period which begins with the event of a correctly-set clock displaying HLWC time (in that timezone) changing its display from 23:59 to 00:00 and ends at the next such event. Thus the Sun generally rises at about the start of a day and sets about halfway through the day (this, of course, varies with the season and geographical location). Dates in the HLWC denote these days.

A lunar quarter is the event, and the time of the event, of a dark moon, a full moon or a half moon (waxing or waning). A lunar quarter occurs at a point in time which can be calculated exactly and expressed as hours:minutes (to the nearest minute) in the local timezone (in either HLWC time or in conventional time).

An offset lunar quarter is a point in local time which is an offset from the local time of a lunar quarter, the offset being equal to the negative of the timezone offset. For example if this calendar is in use in Hanoi, which is at timezone +7 hours, and a full moon occurs at 16:00 local time, then the offset full moon occurs at 09:00 local time, i.e., at 16:00 – 7 hours. Another example (still in Hanoi): If a dark moon occurs at 05:00 local time then the offset dark moon occurs at 22:00 local time on the previous day.

An offset quarter day is a day in which an offset lunar quarter occurs, i.e., a day in which an offset dark moon, an offset half moon or an offset full moon, occurs. It is assumed that the offset used is equal to the timezone offset in the timezone where the calendar is being used.

A week consists of a sequence of consecutive days beginning with a day following an offset quarter day and running up to and including the next offset quarter day. The days of a week are numbered ‘1’, ‘2’, etc. The days of the week also have names (these are given below).

A week terminated by a day on which an offset waxing half moon occurs is numbered ‘1’. Subsequent weeks are numbered ‘2’, ‘3’ and ‘4’

A month consists of a sequence of four consecutive weeks beginning with a week numbered ‘1’.

It follows from the above that an offset dark moon always occurs on the last day of a month.

An offset vernal equinox is a point in local time which is an offset from the local time of a vernal equinox, the offset (as above) being equal to the negative of the timezone difference. In other words, the timezone difference is subtracted from the local time of the vernal equinox to get the local time of the offset vernal equinox.

An end-of-year offset dark moon is an offset dark moon which either (a) immediately follows an offset vernal equinox and is closer to it than the preceding offset dark moon or (b) immediately precedes an offset vernal equinox and is closer to it than the following offset dark moon.

An end-of-year month is a month which is terminated by an end-of-year offset dark moon.

A month which immediately succeeds an end-of-year month is numbered ‘1’. Succeeding months are numbered ‘2’, ‘3’, etc., up to and including the next end-of-year month (which will be numbered either ’12’ or ’13’).

A year consists of a sequence of consecutive months beginning with a month numbered ‘1’ up to but not including the next month numbered ‘1’.

A date in the HLWC is a sequence of four numbers: a year-number, a month-number, a week-number and a day-number (in that order, and separated by hyphens). The acronym ‘HLW’ is used to indicate that a date is a date in this calendar. A date denotes a unique day in the timezone where the calendar is in use.

The Year 5003 is the year which ended with the most recent end-of-year offset dark moon (most recent from the day of the publication of this calendar, 2005-02-13 CE) which occurred on the same day as an offset vernal equinox. For the +0 hours timezone this is the dark moon which occurred at 2004-03-20 CE 22:41 GMT, since there was a vernal equinox at 2004-03-20 CE 06:49 GMT. (It will be shown below that the identification of this offset dark moon is independent of timezone.) Thus year 5004 began on the following day, so we have the following correlation between the HLWC and the CE Calendar: 5004-01-1-1 HLW = 2004-03-21 CE.

This completes the definition of the Hermetic Lunar Week Calendar, except for the names given to the months and days.

The Lunar Week

The lunar cycle, or lunation (from one dark moon or full moon to the next), varies in length from about 29.27 to about 29.83 days, with the average length of a lunation (a.k.a. the synodic month) being 29.53059 days, but it is not true that each quarter-lunation (e.g., from dark moon to 1st quarter, a.k.a. waxing half moon) is always close to one quarter of this period (i.e., from about 7.32 to about 7.46 days).

Here are two examples of quarter-lunations which are far from the average of 29.53059/4 = 7.383 days:

  • A 3rd quarter occurs at 07:17 GMT on 2006-02-21 CE, and is followed by a dark moon at 00:31 GMT on 2006-02-28 CE. The time from 3rd quarter to dark moon is 6 days, 17 hours and 14 minutes, i.e., 6.718 days (22.75% of a synodic month).
  • A 1st quarter occurs at 05:50 GMT on 2010-09-15 CE, and is followed by a full moon at 09:17 GMT on 2010-09-23 CE. The time from 1st quarter to full moon is 8 days, 3 hours and 27 minutes, i.e., 8.014 days (27.58% of a synodic month).

The reason for the fact that quarter-lunations vary in length more than lunations is that the Moon does not rotate about the center of the Earth, but rather about the barycenter of the Earth-Moon system. Moreover, the orbit of the Moon is significantly ellipical, and it travels faster when it is closest to the barycenter and to Earth, when it is said to be at its perigee. Furthermore, the position of the Moon’s perigee itself rotates slowly about the Earth (as explained in detail here), leading to great irregularity in the time that the Moon spends in successive quarters of its orbit.

In this calendar a week is always terminated by a day within which an offset lunar quarter occurs. The actual lunar quarter usually occurs in the same day (as is always the case if the timezone offset is zero) but may occur on the next day (if the timezone offset is positive) or on the previous day (if the timezone offset is negative). Due to the irregularity described in the previous paragraph a lunar week in the HLWC may vary in length from six days to nine days. It usually has seven, and less often eight. Six- and nine-day weeks are less common.

The days of the week are named as follows: The first five days are named Dayone, Daytwo, Daythree, Dayfour and Dayfive. The last day of the week is named Moonday. If the week has seven or more days then the second-last day is named Freeday. If the week has eight or more days then the third-last day is named Herday. If the week has nine days then the fourth-last day is named Nineday (because it occurs only in weeks which have nine days, and on average only about one month in nine has a Nineday). The names of the days in all four possible weeks are thus:

1 2 3 4 5 6 7 8 9
Dayone Daytwo Daythree Dayfour Dayfive Moonday
Dayone Daytwo Daythree Dayfour Dayfive Freeday Moonday
Dayone Daytwo Daythree Dayfour Dayfive Herday Freeday Moonday
Dayone Daytwo Daythree Dayfour Dayfive Nineday Herday Freeday Moonday

Parties are best scheduled for the evening of Freeday, allowing Moonday for recovery before the next week begins.

For the days Herday, Freeday and Moonday it may sometimes be desirable to add a number which shows how many days occur in the weekend of which that day is a part. E.g., Moonday-3 would be a Moonday in a weekend having three days (rather than 1, 2 or 4), and Herday-4 would be a Herday in a weekend having four days. Nineday can occur only in a weekend having four days, so never needs a number.

A Moonday may also be named Waxingmoon, Fullmoon, Waningmoon and Darkmoon if it occurs in the 1st, 2nd, 3rd or 4th week respectively. A Freeday may be named Freeday before Fullmoon (the best day of the month for parties) and so on.

The names of the weeks are: Weekone, Weektwo, Weekthree and Weekfour.

There are exactly 84 quadruples of the numbers 6, 7, 8 and 9 whose elements sum to either 29 or 30, so a month always has one of 84 possible structures. Some, such as (7,8,7,7), are common. Some, such as (6,9,9,6), may either be very rare or may never occur at all.

Some calendars which have been proposed as a replacement for the Gregorian (a.k.a. Common Era) Calendar make a virtue of preserving the 7-day week, either on the grounds simply of tradition or because such a week “fully respects the Fourth Commandment of the Bible” (as if it were important for some reason to do this). Tradition is fine if it enhances life, but not if it amounts to mere superstition, contributing nothing in the way of art or science. As for the Bible, it seems strange to suggest in support of a proposed calendar that it conforms to the alleged commands of some supposed deity, especially when belief in that deity conditions the minds of only a minority of the Earth’s population. Moreover, to claim that a calendar is acceptable only if it conforms with some religious injunction comes close to an attempt to impose a particular religion upon all, whether they want it or not. Accordingly an enlightened mind has no problem with a calendar which has weeks which do not always have seven days.

Regularity of the Calendar

The main objection to the Common Era Calendar, as used in combination with the usual 7-day week (composed of five “workdays” and two days at the “weekend”), is that the weekdays and days of the month match up differently from one year to the next. E.g., a professor might set out a schedule of classes, tutorials, assignment due dates, examination dates, etc., with his lecture on, say, the US Constitution, being scheduled for Friday, February 28th. But next year February 28th is not a Friday (maybe it’s a Saturday), so he has to change the date, perhaps to February 27th, or even to March 2nd. In fact, he has to re-do the entire schedule at the start of every year. Moreover, days are numbered as days in each month, and in only one of those months is there an integral number of 7-day weeks, and even that happens only in three years out of four. In short, from the point of view of scheduling, the Common Era Calendar is a disaster.

The HLWC solves this problem quite simply. Every month has exactly four weeks, and there are always at least six days in a week. The first five days of any week are always named Dayone, Daytwo, Daythree, Dayfour and Dayfive. Thus our hypothetical professor can state that he will deliver his lecture on the US Constitution on Dayfive in the second week of Benjamin, and he can say the same thing next year, and every year. The HLWC guarantees our professor at least 48 5-day periods each year, with the days in each period numbered 1 through 5 and named Dayone through Dayfive, with 1-4 day “weekends” between these 5-day periods. Completely regular, except for the variable number of days in the weekends.

The only irregularity that affects our professor’s scheduling is that occasionally there are 52 weeks in a year, not just 48. Such “long” years occur on average seven times in every 19 years. But our professor doesn’t have to use these extra four 5-day periods. Mahakala can be a month of R&R for everyone.

The HLWC exhibits a second kind of regularity, that of staying in sync with the lunar cycle. In the GMT timezone a lunar quarter always occurs on the last day of a week, and a dark moon always occurs on the last day of a month. In places in timezones that are a considerable distance from the +0 hours timezone (e.g., California, -8 hours, and Queensland, +10 hours) the lunar quarters may or may not occur on the last day of a week, but will always occur either on that day or in the first half of the day after (for timezones East of Greenwich) or the last half of the day before (for timezones West of Greenwich).

Thus this calendar restores to humanity a calendar whose months are observably related to the phases of the Moon, a connection which was destroyed by Julius Caesar in 46 BCE when he ordered the adoption of a calendar whose months had 28, 29, 30 or 31 days, and thus totally lacked any correlation with the lunar cycle, encouraging an alienation from Nature which was reinforced by other historical factors in succeeding centuries, and which has reached an extreme degree in the modern world.

Timezone Independence

The Chinese Lunar Calendar is an admirable device whose months stay in sync with the lunar cycles and whose years stay in sync with the seasons. In this respect it resembles the HLWC. But it has one major flaw, namely, that the structure of the calendar depends on the meridian of longitude used to define local midnight (since days in the Chinese Calendar begin at midnight). The modern Chinese lunar calendar uses longitude 120 degrees East. When this calendar is used in a place, a different longitude may be used to define local midnight (e.g., longitude 105 degrees East in Vietnam, and 135 degrees East in Korea). Months in this calendar always begin with a day (the period from midnight to midnight) in which a dark moon occurs. If a dark moon occurs at 23:30 local time in Vietnam then it occurs at 00:30 local time in China, and thus the dark moon is assigned to different days in the two calendars. The numbering of months depends on the number of “solar terms” occurring in each month, so if the start of a month is delayed by one day, a day in which a solar term occurs, then the number of solar terms in that month may change, and so the number of the month itself may change. Thus it happens that New Years Day in the Vietnamese Calendar occasionally differs by one month from New Years Day in the Chinese Calendar.

Clearly it is better to have a calendar whose structure does not depend on the geographical location of the place where it is used. The HLWC is such a calendar, or in other words, its structure is independent of the timezone in which is is used. To demonstrate this it is sufficient to show that, for any date d in the HLWC, and for any two timezones A and B, date d in the HLWC as used in timezone A denotes a day which is an offset quarter day if and only if d in the HLWC as used in timezone B also denotes such a day.

Suppose the HLWC is in use in timezone A, whose timezone offset is n hours (and possibly some minutes), and also in timezone B, whose timezone offset is m hours (and possibly some minutes), where m >= n. (These timezone offsets may be positive or negative.) Suppose d is a date which denotes a day DA in timezone A and also a day DB in timezone B. (These two days, considered as physical time periods, will not coincide if the timezones are different, but they will normally overlap.)

Suppose DA is an offset quarter day, then some offset lunar quarter occurs in DA. Let the time (in timezone A) of this offset lunar quarter, to the nearest minute, be [d,t]A, where t is in the range 00:00 through 23:59. Then the time of the corresponding actual lunar quarter is [d,t+n]A. (If n is large and positive then this may actually denote a time in the day following DA.) The time difference between timezone A and timezone B is m-n hours (with m-n >= 0), so the time of the actual lunar quarter in timezone B is [d,t+n+(m-n)]B = [d,t+m]B, so the time of the offset lunar quarter in timezone B is [d,(t+m)-m]B = [d,t]B. Since t is in the range 00:00 through 23:59, [d,t]B is a time within day DB, so an offset lunar quarter occurs in DB, so DB is an offset quarter day.

Conversely, suppose DB is an offset quarter day, then similar reasoning shows that DA is an offset quarter day.

Thus the number of days in each lunar week in the sequence of lunar weeks is independent of the timezone where the calendar is used.

We have also shown that the time of an offset lunar quarter is independent of the timezone. E.g., if an offset full moon occurs at 12:00 in one timezone then it also occurs at 12:00 in all other timezones. Thus the time of any offset lunar quarter in any timezone is the same as the time in the +0 hours timezone (a.k.a. the GMT timezone) of the actual lunar quarter.

The only other question is whether the timezone affects whether a year has 12 or 13 months.

Reasoning similar to that above shows that the time of an offset vernal equinox is also independent of the timezone, and is the same as the time of the actual vernal equinox in the +0 hours timezone. Thus a year in the HLWC as used in any given timezone has the same number of months (12 or 13) as a year in the HLWC as used in the +0 hours timezone.

We can thus see that the structure of the HLWC as used in any given timezone can be ascertained by assuming that the actual lunar quarters and the actual vernal equinoxes occur at the same time in that timezone as they do in the +0 hours timezone, even though they do not. They occur later for timezones with positive timezone offset, and earlier for timezones with negative timezone offset, but the difference does not exceed 12 hours (except for wacky places with a timezone difference of greater than +/- 12 hours).

Sample Month

To illustrate the concepts presented above we give here a sample month. This is the month during which this calendar was published, namely, Lilly in the year 5004 HLW. Since all HLW dates are in the year 5004 and all CE dates are in the year 2005, the year numbers have been omitted from the dates. The date on the first line in each cell is the HLW date (e.g., 12-1-1 = 5004-12-1-1) and the date on the second line is the CE date (e.g., 02-09 = 2005-02-09). The times given are those of the actual lunar quarters, and are HLWC times (which differ from GMT times by six hours). For use in timezones other than the +0 timezone the times of the actual lunar quarters would be given in local time.

The 30-day month of Lilly in the year 5004 HLW
02-09 We
02-10 Th
02-11 Fr
02-12 Sa
02-13 Su
02-14 Mo
12-1-7 18:16
02-15 Tu
02-16 We
02-17 Th
02-18 Fr
02-19 Sa
02-20 Su
02-21 Mo
02-22 Tu
12-2-8 22:54
02-23 We
02-24 Th
02-25 Fr
02-26 Sa
02-27 Su
02-28 Mo
03-01 Tu
03-02 We
12-3-8 11:36
03-03 Th
03-04 Fr
03-05 Sa
03-06 Su
03-07 Mo
03-08 Tu
03-09 We
12-4-7 03:10
03-10 Th

Some explanation may be helpful. Remember that days in the HLWC and days in the CE Calender do not exactly coincide, but overlap by 18 elapsed hours, with a day in the HLWC beginning six elapsed hours later than the corresponding day in the CE Calendar (even though both days start at 00:00).

  • There is a dark moon at 2005-02-08 CE 22:28 GMT, so the next day, 2005-02-09, is the start of the HLWC month.
  • The first quarter (which is the same as the offset first quarter, because the timezone is +0 hours) occurs at 2005-02-16 CE 00:16 GMT. Thus this dark moon occurs in the HLWC day associated with 2005-02-15 CE, so the start of week 2 occurs on 2005-02-16 CE. Thus the first week has seven days.
  • The full moon occurs at 2005-02-24 CE 04:54 GMT. The situation is the same as previously, so week 3 begins on 2005-02-24 CE, and so the second week has eight days.
  • The third quarter occurs at 2005-03-03 CE 17:36 GMT, so the next day, 2005-03-04 CE, is the first day of week 4, and thus the third week also has eight days.
  • The dark moon occurs at 2005-03-10 CE 09:10 GMT, so the next day, 2005-03-11 CE, is the start of the next month (and also of the new year, 5006), and week 4 thus has seven days.

Since there are two 7-day weeks and two 8-day weeks, there are 30 days in the month.

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