Intercalary months in the Kālacakra and Tibetan calendars |
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Intercalary months in the Kālacakra and Tibetan calendars |
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One aspect of Tibetan calendars that has caused problems both for Tibetan calendar makers and
later analysts is the determination of intercalary months. Because the length of the mean lunar month (tshes zla) is less than the length of the mean solar or zodiacal month (khyim zla), in order for the months to stay in step with the seasons, just over every two and a half years, an extra, or intercalary, month (zla shol, zla ba lhag pa) is required. For the original Kālacakra calendar, an approximate relationship is used to determine which months are intercalary, but as this is approximate, it only holds for about 40 years and then has to be reset. Each month in this system is named after the zodiac sign that the Sun enters during the month, and the definition of an intercalary month is one in which the Sun does not change zodiac sign. Of course this can be determined simply by inspecting the numbers once a calendar has been calculated. The approximate relationship used is that 67 mean lunar months = 65 mean solar months. The calculation used is straightforward. The number of years since the epoch is multiplied by 12 to yield zodiacal months, and then any additional months are added as required. This number of zodiacal months is then multiplied by 1 2⁄65. This converts the count of solar months to a count of (shorter) lunar months. When the fractional part carries over, this indicates an intercalary month, one during which the Sun does not change sign. This calculation will be demonstrated later. In the description of the Kālacakra calendar as given in the Vimalaprabhā, we are told that the calendar needs to be adjusted in the light of observation. In particular, the longitude of the Sun has to be corrected, by means of detemining the time of winter solstice, when the longitude will be set to 0° Capricorn (20;15 in lunar mansions and nāḍī). It follows that the conversion of zodiacal into lunar months and the check for intercalaries would also need to be adjusted. In the description that follows, bear in mind that the principles in later Tibetan calendars are basically the same as with the Kālacakra system, but in the Kālacakra the 67:65 relationship is an approximation, whereas in the later Tibetan calendars, it is mistakenly taken as precise. Tibetan calendar makers such as Phugpa Norzang Gyatso (phug pa nor bzang rgya mtsho) and Zhonnu Pal (gzhon nu dpal) came to the view that they could construct a perfectly accurate calendar in which the longitude of the Sun would not need correction. They took the relationship of 67 mean lunar months = 65 mean solar months to be exact, and from this determined a new mean change of longitude of the Sun each lunar month. This introduced significant error in their calculations. At the same time, they took a relationship given in the Kālacakra Tantra for converting lunar days into solar days, and used this to create a new figure for the mean length of a lunar month in terms of solar days – this actually produced a small improvement in the accuracy of that particular relationship. One effect of this was to create a repeating structure to the calendar, which must have been very compelling. Exactly the same 24 intercalary months repeat every 65 years. The numbers used in the zodiacal to lunar month conversion have meanings that can be clearly defined. The calculation is done by writing the zodiacal month number in two places. The lower is multiplied by 2, and then divided by 65. The quotient from this division is then added to the top line, the result being the number of lunar months. Take as an example, 101. This means that we are calculating for a new Moon, 101 zodiacal months after an epoch: 101 + 3 = 104 101 x 2 = 202 ÷ 65 = 3 rem. 7 This tells us that 104 lunar months have elapsed since the epoch. The remainder of 7 can be called the intercalation index. There is one thing missing from this – usually there will be a similar remainder (rtsis 'phro) when the equivalent calculation is performed for the epoch, from some previous epoch. This remainder would also need to be added in here. Take as an example a previous remainder from the epoch of 16, and the calculation becomes: 101 + 3 = 104 101 x 2 + 16 = 218 ÷ 65 = 3 rem. 23
101 + 3 = 104 101 x 4 + 32 = 436 ÷ 130 = 3 rem. 46 The result is still the same. An intercalary month is identified when the quotient on the bottom line increases by one, and the remainder (dividing by 65) is either 0 or 1. As an example, take the two months calculating for 97 and 98 zodiacal months, and ignoring any previous epoch remainder: 97 + 2 = 99 97 x 2 = 194 ÷ 65 = 2 rem. 64 98 + 3 = 101 98 x 2 = 196 ÷ 65 = 3 rem. 1 Notice that with an increase of just one zodiacal month, the count of lunar months has increased by two. The month that falls between these two is the intercalary; considering the three months that are respectively 99, 100 and 101 lunar months after the epoch, the middle one is intercalary, and in the original Kālacakra tradition would be considered to be a repeat of the month before. A couple of years later, the calculation would yield the next intercalary: 129 + 3 = 132 129 x 2 = 258 ÷ 65 = 3 rem. 63 130 + 4 = 134 130 x 2 = 260 ÷ 65 = 4 rem. 0 This tells us that the lunar month that is 133 months after the epoch is an intercalary. Notice that for the first intercalary described here (after 100 months) the remainder on the bottom line of the calculation when the month count increases is 1; for the next intercalary (after 133 months) that remainder is 0. These remainders alternate, and when such a remainder of 0 or 1 occurs in the calculation, this indicates an intercalary month. Of course, in the Kālacakra way of doing things (dividing by 130 instead of 65), this remainder would have the values 0 or 2. Tibetan writers often mention that the calculation for an intercalary arrives either at the end of the month, or in the middle; of course, an intercalary month has to be a whole month, nothing else, so just what is intended here? The answer to this lies in the point made by Tsuglag Threngwa that the division on the bottom line should really be by 32 1⁄2. This is just a mathematical curiosity – all intercalary months are the same. For each one, just before the new Moon at the beginning of the month, the Sun enters a new zodiac sign, and immediately after the new Moon at the end of the month, the Sun enters the next sign. The concept of the calculation taking us to the middle or end of the months depends on just where the counting is considered to start, and is best explained by the method described by Zhonnu Pal. He considers an idealised structure to the calendar,and states: "Regarding the appearance of intercalary months, starting with the first white lunar day of the month of Caitra in Prabhava, with the Sun just entering Aries, 32 and a half months after the start of that month there are two months of Mārgaśīrṣa". The table below can illustrate this. Consider counting from the very end of an intercalary month of Phalguna; the month of Caitra is just about to start, and, immediately after the new Moon, the Sun enters Aries, defining that month of Caitra. Just such a situation obtained at the Kālacakra epoch in 806 CE (at least, according to the data given in the Tantra). The count of 1 takes us to the very end of the month of Caitra – going forwards, and a couple of years later, the count of 32 takes us to the end of a month of Kārtikka, and half a month later we are therefore, arithmetically, in the very middle of Mārgaśīrṣa. This is dependant simply on where we started counting – from the very end of the previous intercalary. Mārgaśīrṣa is duplicated &ndash the calculation brings us to the middle of the normal month of Mārgaśīrṣa, and another (duplicated) Mārgaśīrṣa follows it. The counting then continues from that intercalary month &ndash in the calculation, the true lunar month count increases by 2, as above. Intercalaries are in bold:
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