We have now seen roughly how the positions of the planets are calculated for the ephemeris and a little about their astrological usages. Let us next explore the making of the calendar, which entails the five inclusive calendar features.

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**Lunar Weekday and Date **

The first and second of these five features are the lunar weekday (*gza’*) and date of the lunar month (*tshe*). These are involved in the mechanism through which the lunar and solar calendars are brought into harmony. As mentioned above in the discussion of the three types of days, lunar date days (*tshe-zhag*) are the period of time it takes for the moon to travel one-thirtieth the distance between new moon positions in each successive sign in the zodiac. They are thus correlated with the phases of the moon. They are counted in a cycle of seven lunar weekdays named for the days of the week which, as has been noted, are also the names of seven of the planets.

Since the exact new moon does not occur at precisely the same time of day each month, these lunar weekdays start at a different time each solar day. The solar day is from dawn to dawn. In other words, the moon can start to travel one of these little distances of one-thirtieth of its cycle at any time of the solar day, and that period of time it takes for it to travel that one-thirtieth the distance of its cycle is called by the day of the week. Thus, the day of the week is starting at different times during the solar day (*nyin-zhag*). The problem, then, is how to make these lunar weekdays fit in with the solar days.

There is one further complication. It takes the moon a different amount of time to cover each of these little one-thirtieth distances since its speed will vary with the position of both itself and the sun in the zodiac. In other words, as was noted just before, the position of the moon is calculated from that of the sun, and both the sun and the moon move at different speeds at different points in the zodiac. Therefore, once the hour of the mean lunar weekday (*gza’-bar*) occurring at dawn is derived from the progression of the lunar weekday’s daily passage-of-time constant (*gza’-rtag*), it must be corrected by using quadrant-motion constants for both the moon and the sun. In this way one arrives at the hour of the corrected lunar weekday (*gza’-dag*) occurring at the dawn of the solar day. As a consequence of all this, the amount of a lunar weekday that passes between the dawn of two successive solar days is variable because the length of a lunar weekday is likewise variable.

Dates of the lunar month, which is the second inclusive calendar feature, are numbered 1 to 30 and last from dawn to dawn in the manner of solar days. The problem is to determine which date is to be assigned to each day of the week. The solution is not so obvious because the lunar weekdays, which are what determine the day of the week, since they are called Sunday, Monday, etc., start at and last for different lengths of time.

The rule is that the day of the week is decided according to which lunar weekday is occurring at the dawn of the lunar date. For instance, a lunar weekday, such as Monday, may start in the afternoon of the second date of a month and end in the afternoon of the third. Since at the dawn of the third, which here is taken standardly to be at 5 A.M., the lunar weekday is still Monday, then the third will be considered a Monday.

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**Doubled and Omitted Dates**

A day of the week can never be repeated or skipped. Directly after a Sunday, there must be a Monday, not a second Sunday or a Tuesday. But sometimes the dawns of two successive dates occur within the same lunar weekday. For instance, the lunar weekday Monday may begin five minutes before the dawn of the 3rd, and the next day, the Tuesday, may begin five minutes after the dawn of the 4th. This would make both the 3rd and the 4th Mondays! But there cannot be two Mondays in a row. One of these dates must be omitted. This is why in the Tibetan calendar there are omitted lunar dates (*chad*) in some months.

On the other hand, sometimes the beginning of two lunar weekdays occur before the dawn of the next date. For example, if the lunar weekday Monday begins five minutes after the dawn of the 3rd and ends five minutes before the dawn of the 4th, then, by the first rule, the 3rd should be a Sunday and the 4th a Tuesday, and there would be no Monday. But since it is not possible for it to go from a Sunday to a Tuesday without an intervening Monday, one of these dates will have to be doubled in order for one of them to be the Monday. This is why sometimes there are two 8ths or two 25ths, for instance, as added lunar dates (*lhag*) in a Tibetan month.

The Hindu religious calendar systems, such as the one based on the *Sun System of Tenets*, also share this feature of omitted and doubled dates. The lunar weekdays, or “tithi” in Sanskrit, are simply numbered 1 through 15 for both the waning and waxing phases of the moon, known as the black and white phases respectively. In the North Indian calendar, since the month begins with the day after the full moon, the black phase precedes the white. The number of the lunar weekday occurring at the dawn of a solar day determines the date within that phase. Thus, if a lunar weekday begins after dawn one day and ends before the dawn of the next day, the date with its number is omitted. If the dawns of two successive dates both occur within the same lunar weekday, the dates of both of them take the same number. The Hindu Vikrami and Shaka civil calendars are strictly solar and do not have doubled or omitted dates.

The early Tsurpu tradition used primarily the precis system of mathematical calculations, while the Pugpa developed the full tenet system two centuries later. The two mathematical systems differ very slightly over the daily motion constants for both the sun and the lunar weekday. During the course of centuries, however, the discrepancy between the two systems became larger, and the leftover positions of the planets and lunar weekday at the end of the 60-year cycles started to differ considerably. Because of these factors, the doubled and omitted dates are totally different in the two systems, with often the number of them in each month being different. As a result, the number of days in a particular corresponding month in the two systems will be different so that sometimes there is a discrepancy of one day concerning the start of the next lunar month.

In the present-day Pugpa almanac, the calendar is arranged according to the full tenet system, but the data from the precis system is included, since it is used for calculating the occurrence of eclipses.

The Tsurpu tradition was reformulated during the first half of the 20th century by Ozer Rabten (*’**Od-zer rab-brtan*). In the present-day Tsurpu tradition, the full tenet system is used for calculating the lunar weekday rather than the traditionally used precis system. A few of the mathematical constants, however, are different from those used in the Pugpa tradition of the full tenet system. As a result, some of the omitted and doubled dates in the present-day Tsurpu calendar are the same as those in the Pugpa one, while some differ by one or two days. The modern-day Tsurpu tradition, however, still uses the precis, not the full tenet system rules for adding extra doubled months. This will be explained below.

In the new Geden system of Mongolia and Buryatia, the rules for doubling and omitting dates are the same as in the Pugpa system. But, because the starting point for the calculations within a 60-year cycle is different, it works out that the doubled and omitted dates are different.

In the yellow calculation system of Inner Mongolia, the doubled and omitted dates are calculated as in the Pugpa system, so that it is determined how many days there are in a month. The days of that Chinese-style month are then numbered consecutively without any of these dates being doubled or omitted. In this system, then, the Chinese-style months always correspond in length and initial date with the Pugpa full tenet system months.

In the yellow calculations found in the Pugpa system itself in Tibet, the length of the Chinese-style months is calculated differently, according to a particular formula, without any reference to the doubled and omitted days calculated for the Mongolian month. As a result, there is sometimes a discrepancy in the length and start of these two systems of months, Mongolian and Chinese style. But, as in the yellow calculation system of Inner Mongolia, there are no doubled or omitted dates.

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**Doubled Months in Non-Tibetan Systems**

To make the lunar calendar further correspond with the solar, a thirteenth month must occasionally be added to the year in the form of an extra doubled or leap-month (*zla-zhol*). For instance, sometimes a year will have two 5th months. To understand how this is done in the various lineages of the Tibetan astro system, let us first examine how it is done in some of the other systems.

In the classical Chinese calendar, the doubled month is added 7 times in a 19-year cycle, namely every 3rd, 6th, 9th, 11th, 14th, 17th and 19th year. The Chinese and ancient Greeks seem to have developed this rule independently of each other. Cross-cultural contact seems highly improbable. It appeared in China about a century before the Greek astronomer Meton discovered in the 5th century BCE what is known as the Metonic cycle of 19 years. It was not standardized in China until the 3rd century BCE. The theoretical basis for the Chinese system, however, is quite different from that of the Greek.

Meton discovered that it takes 19 years for a particular phase of the moon to occur once more at a particular point in the solar calendar. For instance, if the new moon of a first lunar month occurs on the winter solstice, it takes 19 years for another new moon to coincide with this point. Seven doubled months must be added during those 19 years if it is wished for the seasons to stay correlated with the lunar months, otherwise after 19 years it will not be the first lunar month that occurs at the winter solstice, but rather the 8th. The years in which these leap-months are to be added over the 19 years are the same as those found in the Chinese custom, except that the 8th year of the cycle is specified rather than the 9th.

The ancient Greek city states never actually adopted this rule of Meton for adding 7 doubled months every 19 years in their rather individualistic calendars that added extra months rather haphazardly. It was adopted, however, in the Babylonian calendar in the early 4th century BCE and, from there, into the Hebrew calendar. The Muslim calendar, on the other hand, although lunar, is not correlated with the solar calendar. Thus, doubled months are not added and the New Year is not at a fixed season of the year.

When the Chinese standardized their system for adding extra months, it was based on a meteorological cycle of 24 mini-seasons in a year. The sequence starts either with the one called “winter solstice,” or with the one three later, “establishment of spring.” The former corresponds with Shao Yong’s *I Ching* system of seasons in which the astrological new year starts with the prelude month of spring right after the winter solstice. The latter is consistent with Zou Yan’s correlation of the five seasons with the five elements, in which case the new year begins with the build-up towards spring at the beginning of the month prior to the one in which the vernal equinox occurs.

Each mini-season begins when the sun conjuncts one of 24 equidistant points along its orbit. Thus, unlike the twenty-eight Chinese lunar constellations which are arranged along the celestial equator, these 24 meteorological points are arranged along the ecliptic, the pathway of the sun’s revolution within the zodiac. This, then, is the closest the classical Chinese system comes to having an equivalent of a zodiac. It will be recalled that in the Greek and European sidereal zodiac, zero degrees Aries is taken to be the position of the sun on the vernal equinox in the northern hemisphere. The 24 Chinese meteorological points include the two solstices and two equinoxes, and thus at the start of the mini-season “winter solstice,” the sun is approximately at the beginning of the European sign Capricorn. At the start of the mini-season “establishment of spring,” it is approximately at the mid-point of Aquarius.

It was observed that it takes the sun slightly longer than a lunar month to cross two of these points. Thus, although most lunar months will contain two of these points, some will fail to do so. The rule was that those months in which only one of these points is found are to be doubled. Sometimes the points are put into pairs, with a pair consisting of the two mini-seasons that would correspond to the sun’s passing from the mid-point of one European zodiac sign to the mid-point of the next. In such cases, it is only those months in which the second of a pair of points is not found that are to be doubled. This works out to 7 doubled months in 19 years, in most cases at the same intervals as with the Metonic cycle.

I do not know what Meton himself suggested, but in the Babylonian version, 6 out of 7 times during 19 years the doubled month was the last of the year, just prior to the vernal equinox. During the 17th year, the month prior to the autumnal equinox was doubled. In the Hebrew version, it is always the month prior to the vernal equinox that is doubled. In the Chinese system, the doubled month could be any month of the year, according to the rule just mentioned, except for the 11th, 12th and 1st months, the prelude, middle and finale months of spring in Shao Yong’s *I Ching* system. If it works out that one of those three months should be doubled, the following 3rd month is doubled instead. The current 19-year cycle in the Chinese calendar began in 1985.

In the most ancient Indian Hindu calendars, doubled months were added once every five years and it was generally the 4th or 5th month that was doubled. In the later, more standard *Sun System of Tenets*, the number of added months for an entire world age is calculated and then distributed over that world age. Any month can be doubled, but further research must be done to ascertain at which intervals they were added. The Indian Vikrami and Shaka civil calendars, being solar, do not have doubled months. In the Kalachakra system as used in Tibetan astronomy, two theoretical bases seem to have been combined, one reminiscent of the Chinese system and one more strictly Indian.

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**Tibetan Systems of Doubled Months**

As noted earlier, according to the Kalachakra description of the universe, the orbit of the sun is an oblique eccentric circle around Mount Meru. It is described by a series of oblique or slanting epicycles, each of which has its highest or “hill-ascending” zenith-point (*sgang-’char*) closest to Mount Meru, and its lowest or “breath-catching” nadir-point (*dbugs-thob*) furthest away from it. There are 12 such epicycles, and each has 12 divisions along its circumference for the 12 signs of the zodiac. Although they are not in the shape of cogs in a wheel, it may be useful to describe them as such.

The 12 epicycles are arranged around the circumference of the larger circle which is the mean orbit of the sun around Mount Meru, and the center of each epicycle is on that circumference. Each epicycle has a different cog touching the circumference of the larger circle and these proceed in consecutive order. In other words, epicycle one will have cog one, Capricorn, touching the larger circle and, proceeding clockwise, epicycle 2 is rotated one cog clockwise so that cog 2, Aquarius, is touching the circle and cog 1 has moved one position up, which is higher in elevation and closer to Mount Meru. Epicycle 3 has cog 3, Pisces, touching the circle and cog 1 is two positions up, and so on. When the sun is in Capricorn, its orbit is described by the oblique eccentric circle derived by connecting all the Capricorn “cogs” on the 12 epicycles.

The zenith-point of the sign is conjuncted when the sun is four-fifteenths of the distance of that sign past where the sign begins, i.e. 36 degrees past the cusp of the sign in the system in which each sign has 135 degrees in a 1620-degree zodiac. The nadir-point is reached when the sun is 23-thirtieths or 103.5 degrees past the cusp. The 24 zenith- and nadir-points are thus equidistant from each other, with 67.5 degrees separating them, as is the case with the 24 meteorological points in the classical Chinese calendar. Although the distance between the points is the same, the location of the points is different and thus they do not coincide.

According to the Kalachakra system, a doubled month is added regularly once every 32.5 months or, putting it the way it is calculated, once every 65 half-months. The number of half-months passed since the beginning of the 60-year cycle is calculated. To this is added the leftover number of half-months since the end of the last 65 half-month period of the previous 60-year cycle. The sum is divided by 65 and according to the remainder, the doubled month is added. The remainders increase by two at a time.

In the full tenet calculations of the Pugpa system, the month in which the remainder from the division by 65 is 48 or 49 is doubled. In the precis system within the Pugpa, it is doubled when the remainder is zero or one. But one must take into consideration that the leftover number of half-months from the previous 60-year cycle is different from that given in the full tenet system. The Pugpa calendar doubles the month according to the full tenet calculation, while the Tsurpu from the precis one.

This way of calculating the added month is considered the quick, short-cut method. The more complete technique in the full tenet system is to calculate it from the zenith-point. The sign-cusp, nadir- and zenith-points are calculated from this remainder of the division by 65. Thus, they are not conceptualized as being the three points of a particular zodiac sign, but rather are considered the three points of a specific lunar month. Consequently, the sequence is that first the nadir-point, for instance of the calculated month eleven, is met, then the cusp and finally the zenith-point. The calculated month number of whichever zenith-point the sun conjuncts during the course of a Mongolian lunar month is assigned as the number of that Mongolian month in which it occurs. Thus, the peculiarity arises that the nadir-point and sign-cusp of calculated month number eleven may be met during Mongolian month ten.

If no zenith-point is conjuncted during a Mongolian lunar month, then according to the full tenet system within the Pugpa tradition, that lunar month is given the same number as the preceding month, with one being called, for instance, the 11th month and the other the latter 11th month. Such a year will thus have 13 months. Since this calculation of the zenith-point is derived from the remainder of the division by 65, it always works out that no zenith-point is conjuncted during the lunar month in which that remainder is 50 or 51. Thus, those months are given the same number as the previous months, and consequently, as was stated in the rule, the months with the remainders of 48 and 49 are doubled. This is why the zenith-point rule is followed in the full tenet system within the Pugpa lineage.

In the precis system, as calculated in the Pugpa, but actually followed in making the calendar in the Tsurpu, since the extra month is added when the remainder of the division by 65 is zero or one, this does not correlate with the zenith-point rule. Therefore, in these systems, no reference is made to the zenith-point.

The new Geden system used in Mongolia follows the rule of the full tenet system of the Pugpa. But since the starting point of the 60-year cycle is different, it works out that the doubled months will be different.

Several observations can be made here. First of all, this zenith-point rule is reminiscent of the classical Chinese custom concerning the 24 meteorological points. The 24 are similar to the Kalachakra nadir- and zenith-points, but they are not at the same location in the zodiac. Thus, the doubled months in the classical Chinese calendar do not coincide with those in any of the Tibetan calendars. Secondly, in the Chinese system, the month is doubled whenever any of these 24 are missed, and in some systems only when the second of a pair of them is missed, whereas in the Tibetan, only when a zenith-point is missed, which is the second of a pair for a calculated month. Thirdly, although conceptually the nadir- and zenith-point descriptions are given within the framework of the Kalachakra cosmology without reference to the 24 Chinese mini-seasons, still the fact that these points are not used in the earlier precis system, but only in the later reconstructed full tenet system makes one suspect a mixing and reworking of Indian and Chinese ideas.

In the yellow calculation system within the Pugpa tradition in Tibet, broad Chinese-style nadir- and zenith-points are calculated for each Chinese-style month, but within the context of Kalachakra 60-year cycles and Kalachakra-style methods of calculation. These Chinese-style months do not coincide with either the classical Chinese months or Mongolian months, and these nadir- and zenith-points do not correspond to the Kalachakra-derived ones. They do, however, appear to coincide with the classical Chinese 24 meteorological points, since they include the two equinoxes and two solstices. Thus, the Chinese-style nadir- and zenith-points of a particular Chinese-style month correspond respectively to the mid-point of a sign of the European sidereal zodiac and the cusp of the next sign, as explained before. These are approximately 34 degrees of a 360-degree system ahead of the corresponding mid-point and cusp-points of the signs of the fixed-star zodiac as used in the Kalachakra system. As with Kalachakra nadir- and zenith-points, the nadir-point of a particular Chinese-style month precedes the zenith-point and can actually occur during the prior Chinese-style month.

The Chinese-style nadir- and zenith-points are not linked with a remainder of a division by 65 and thus are not linked with the rule of a doubled month being regularly added every 65 half-months. But the rule for doubling the month is the usual zenith-point rule. The Chinese style-month in which there is no such zenith-point is the doubled month, receiving the same number as the month before.

In the yellow calculation system of Inner Mongolia, the doubled Chinese-style month is calculated according to the Pugpa full tenet system rules. But then the rule from the classical Chinese calendar is followed according to which only certain months are allowed to be doubled, while others may not. Thus, if the month to be doubled works out to be the 11th, 12th or 1st month, the following 3rd month is doubled instead.