Tibetan astronomical calendar calculation

Tibetan calendar: Beginning of the 3rd Hor month in the Tibetan calendar from Lhasa for the water pig year 1923/24

The Tibetan astronomical calendar calculation ( Tib . : lnga bsdus or yan lag lnga bsdus ) is a branch of Tibetan astronomy (Tib .: skar rtsis ). The aim of this astronomical time calculation is the construction of the Tibetan calendar as an aid to the order of time and the orientation in time.

For the Tibetan calendar, world events are chronologically ordered through time cycles, i.e. through periodically recurring time units such as “age of the world”, “sixty-year cycle”, year, month and day. The time units of these cycles are usually defined by astronomical phenomena:

The age of the world begins and ends with the meeting of all planets at the zero point of the ecliptic , the year results from the completion of the apparent rotation of the sun around the earth, the month describes the time span between two new moons and the calendar day is the natural day.

In the context of the calculation of time, Tibetan astronomy deals, among other things, with the apparent movement of the sun and the calculation of the ecliptical length of the moon .

The astronomical calculations were carried out with the Tibetan sand tobacco .

The Tibetan calendar calculation is still practiced today both in Tibet and outside of Tibet to create the annual calendar.

Historically, the Tibetan calendar calculation is based on the teachings of Chapter 1 of the Kālacakratantra and is therefore of Indian origin.

Cyclical time units

World age

The four world ages (Tib .: dus bzhi ) form the largest time cycle . The length of a world age is defined as the period between two successive conjunctions of all known, movable celestial bodies or planets in Tibet at the zero point of the ecliptic. In addition, this event is always characterized by the fact that the astronomical calendar year begins and a certain sixty year annual cycle begins. In Tibetan this event of a great conjunction is referred to as stong 'jug "entry into the void".

The Kālacakratantra names four world ages, which have different lengths, which are supposed to fulfill these astronomical conditions and which have followed one another, namely that

Sanskrit : kṛtyuga, tib .: rdzogs ldan gyi dus with 1,728,000 years, that
Sanskrit: tretāyuga, Tib .: gsum ldan gyi dus. With 1,269,000 years, the
Sanskrit: dvāparayuga, tib .: gnyis ldan gyu dus with 864,000 years and that
Sanskrit: kaliyuga, tib .: rtsod pa'i dus with 432,000 years.

The Tibetan astronomers of the 15th century now found out that with the numerical material of chronological sizes and initial values of the planets available in the Kālacakratantra, a large conjunction was neither possible in these time intervals nor at all.

The Kālacakratantra uses the beginning of the month nag zla ba (Sanskrit: Caitra) of the year 806, which corresponds to March 23, 806, as the epoch of its calendar . At the beginning of this day, according to the Kālacakratantra, the sun and moon are at the zero point of the ecliptic, which coincides with the beginning of the zodiac sign Aries (Tib .: lug ) or the lunar house tha skar . The year 806 is the 20th year of the sixty-year cycle, which begins in the year 787.

Because a major conjunction has to take place with the beginning of the 1st year of a sixty-year cycle and because the conversion ratio from solar month to synodic month generally also takes place

{\ displaystyle A = 1 + {\ frac {2} {65}} = {\ frac {67} {65}}}

is stated, this means, according to the verifiable calculations of the Tibetan astronomers, that at the beginning of a 20th year of a sixty-year cycle, the beginning of a synodic month can never coincide with the beginning of a solar month.

Conversely, this showed that if the corresponding values of the Kālacakratantra were accepted, a major conjunction at the beginning of a sixty-year cycle could not take place.

With this and further criticism of the initial values, the astronomers of the Phugpa school calculated a value for the period of the occurrence of large conjunctions, which in the white beryl of the regent Sanggye Gyatsho was put at 279 623 511 548 502 090 600 years. Expressed in number names, these are 279 trillion, 623 quadrillion, 511 trillion, 548 billion, 502 million, 90 thousand and 6 hundred years.

The cycle of 60 years

The next smaller cyclical unit of time is a period of 60 years. In astronomy, the sixty-year cycle always means the so-called Rab byung cycle of Indian origin. The cycles are counted with ordinal numbers . The individual years have individual names.

The year

The following smaller cyclical time unit is called the year (Tib .: lo ). On the one hand, it is defined as the time it takes for the sun to go through the 12 signs of the zodiac. This is the tropical year or solar year. On the other hand, the Tibetan calendar year exists as a further time unit with different lengths of 12 or 13 synodic months . The size of the solar year, however, served as a guideline to regulate the different lengths of the year in the calendar.

Months

The next smaller cyclical time unit is the month (Tib .: zla ba ). To this end, astronomers differentiate between three types of months:

Solar month (Tib .: khyim zla ), that is, the length of time that the mean sun needs to pass through a zodiac sign.

Lunar month (Tib .: tshes zla ), that is the length of time it takes for the moon to change its elongation by 360 degrees.

Calendar month (tib .: zla ba ), d i. a period of 29 or 30 natural days. It begins one day after the natural day on which the previous lunar month ends. It ends with the natural day the current lunar month ends.

Days

There are three types of days in Tibetan astronomy (Tib .: zhag gsum ):

Zodiac day (Tib .: khyim zhag ), ie a solar month. The zodiac day is a time value that is only used for astronomical calculations and cannot be experienced. ${\ displaystyle {\ frac {1} {30}}}$

Lunar day (Tib .: tshes zhag ), that is, the period of time that the moon needs for an elongation (angular distance between the sun and moon) of 12 degrees. A lunar month has 30 lunar days. The lunar day is also an astronomical quantity of time that cannot actually be experienced. However, it has a special meaning for the time order as it provides the counting variables for numbering the natural days within a month. When calculating the length of a lunar day, the irregularity of the movement of the sun and moon is taken into account by means of so-called mid - point equations , the calculation of the lunar days is one of the most difficult parts of the Tibetan calendar calculation.

Natural day (Tib .: nyin zhag ), the time span between two successive dawns. The unit of time, natural day, is usually denoted in modern science with the symbol "d".

Astronomical division of the three types of day

For astronomical calculations, all three types of day are divided as follows:

1 day = 60 chu tshod ,
1 chu tshod = 60 chu srang ,
1 chu srang = 6 dbugs .

It should be noted that the absolute size of these time sizes varies depending on the type of day.

Implementation of the calendar calculation

Goal setting

The astronomer Pelgön Thrinle (15th – 16th centuries) reckons with the sand abacus

The focus is on the calculation of the date (lunar day) within a month with which a natural day or weekday is counted within a month. Furthermore, the ecliptical lengths of the sun and moon as well as two other astrologically significant quantities are calculated for each day of the week, which are called byed pa (Sanskrit: karaṇa) and sbyor ba (Sanskrit: yoga). Since the calculations boil down to the determination of these five components, the calendar calculation is also referred to as yan lag lnga bsdus “summary of five components”, often abbreviated to lnga bsdus .

Mathematical procedure

Tibetan calendar calculations are program texts, i.e. a series of arithmetic rules to build up a time order. The calculations themselves are carried out on the sand abacus.

For the special notation of multi-digit numbers of the sexagesimal system used in the following cf. Calculating with numbers in the sexagesimal system . The result of dividing two whole numbers without a remainder and the remainder of this division is referred to as a special notation ${\ displaystyle \ left \ lceil {\ frac {a} {b}} \ right \ rceil}$ ${\ displaystyle \ left \ lfloor {\ frac {a} {b}} \ right \ rfloor}$

Time management structures

Basically, the basis of the Tibetan calendar is the sequence of natural days (Tib .: nyin zhag ). The subdivision of the natural day into 21,600 dbugs , that is, breaths of 4 seconds in length, forms a physical basis for the measure of this time unit.

The natural days can be distinguished by their designation as days of the week . The seven days of the week bear the names of the seven most important planets in a cyclical sequence . They are counted from 0 to 6 in the astronomical calculations. The weekday counted with 0 is always Saturday (Tib .: spen pa ). The week itself is also known as the cycle or wheel of the planets (Tib .: gza '' khor ). The wheel of the days of the week or planets runs continuously and into the past and future without end.

The month as the next larger time segment to classify the natural days is ultimately defined by the time span between two new moons, which is also a time value that can be easily experienced. Basically, however, the length of synodal months varies between 29.272 d and 29.833 d. As a result, a whole number of natural days do not fit into the cycle of the lunar months. Adjustments must therefore be made so that the two cycles fit together. Ultimately, this boils down to the fact that the number of natural days in a month varies.

The classification of the natural days in the monthly cycle is not done by simply counting them with natural numbers. Rather, the individual natural days are counted with the number of the lunar day that ends in the relevant natural day.

Because of the different lengths of the lunar days, this means that two lunar days can end in certain days. In this case, the number of the second lunar day is not assigned. For example, the weekday Monday with the date number nine is followed by a Tuesday with the date number 11. The date 10, which does not occur, is called chad "omitted date".

Furthermore, it can happen that no lunar day ends on a certain weekday. In this case, this weekday is assigned the date of the following weekday. As an example, a Wednesday with the date number 12 is followed by a Thursday with the date 13 and a Friday with the same date number 13. The first of these two days with the same date number is referred to as lhag "additional".

When classifying the lunar months, usually 12, in the cycle of the tropical years, comparable adjustment problems arise as when inserting the natural days into the monthly cycle. 12 lunar months are shorter than a tropical year. The compensation is made by inserting so-called leap months (tib .: zla lhag or zla bshol ).

Epoch and initial values

A calculation of time requires a certain starting point from which the temporal structures are built. This point in time is commonly referred to as the epoch. In the astronomical calendar, this is usually the beginning of the first lunar day of the first astronomical month ( nag zla ) of the first year (called rab byung ) of one of the sixty-year cycles. One of the few exceptions is the kālacakratantra, whose epoch falls in the year 806, which is counted as the 20th year of a sixty-year cycle. For the Tibetan calendar itself, epochs are to be set at the beginning of the first natural day of the first civil month of the year.

The problem with these epochs is that at these times the beginnings of the various time cycles are mostly shifted from one another. Although the astronomical year always begins with the 1st lunar day, this usually does not correspond to the beginning of the 1st weekday, which is counted with 0, namely Saturday. The beginning of the year usually does not correspond to the beginning of the solar year. The ecliptical lengths of the sun, moon and the other planets are of course not zero. The ideal epoch would therefore be the point in time of a great conjunction, but calculating with such large numbers on the sandabacus is completely impractical.

In calendar calculations and astronomy, this is taken into account using initial values (Tib .: rtsis' phro ). With regard to the day of the week, such an initial value then records the day of the week and the time of day on which the first lunar day begins. With regard to the sun and moon, the initial values capture the ecliptical lengths of both celestial bodies at the beginning of the year.

Calculation of the 5 components of a day

The exact task is: In what day of the week W and at what time of day does the lunar day T of the Tibetan lunar month M end in a Tibetan year that is counted as year J since epoch? What are the ecliptical lengths of the sun and moon? How are the two astrological components calculated?

In the following, the Kalacakratantra, i.e. the beginning of the Tibetan year, which falls in the year 806, is chosen as the epoch.

Please note that Tibetan dates list the year with the number of the Rab byung cycle (Z) and a year name. In this respect, it is necessary to count individually which year (JZ) it is numerically. In order to find J for the epoch of the year 826, one calculates

J = (Z -1) · 60 + (JZ - 1) + 221, since the 1st year of the 1st Rab byung cycle falls in the year 1027.

Examples:

It is the 5th year in the 8th Rab byung cycle. Then J = (8-1) * 60 + (5-1) + 221 = 645
It is the 58th year in the 16th Rab byung cycle. Then J = (16-1) * 60 + (58-1) + 221 = 1178

In the Tibetan calendar calculation one naturally begins with the 1st day of the 1st month of the new year and calculates the five components one after the other for all days of the year. The result is a calendar in which the necessary information is recorded for all days of the 12 or 13 months of a year.

Number of the past lunar months ( tshes zla rnam par dag pa or zla dag )

The starting point for all calculations is initially to determine the number of solar months that have passed since the epoch (Tib .: khyim zla ). To do this, multiply the number of "past years" (J-1) by 12 and add the number of "past months" (M-1) of the current year:

SOL (J, M) = (J-1) * 12 + M-1.

With the kālacakratantra, the relationship between the solar month and the lunar month became the value.

{\ displaystyle A = [1,2] / (-, 65) = 1 + {\ frac {2} {65}} = {\ frac {67} {65}}}

handed down. This conversion factor has never been questioned in Tibet.

For the calculation of the number of the past lunar months (Tib .: tshes zla ) it should be noted that if you choose any epoch, the beginning of the 1st solar year and the 1st lunar month are usually not the same. In this respect, an initial value must be added, which is referred to here as R (m).

These initial values differ for the various astronomical schools of Tibet, which are designated here with m. In relation to the epoch of Kālacakratantra, the majority of these schools calculate with R = 0. The most important exception is the Phugpa school (m = 1), which calculates with the value R (1) = 61. The reasons for this result from the considerations given above for the calculation of the major conjunction.

For the calculation of the number of the past lunar months L at the beginning of the month M the following results:

{\ displaystyle L (J, M) = \ lceil A \ cdot SOL (J, M) \ rceil = \ left \ lceil ((J-1) \ cdot 12+ (M-1)) \ cdot \ left (1 + {\ frac {2 + R (m)} {65}} \ right) \ right \ rceil}

The result is referred to as the “exact number of the past lunar months” (Tib .: tshes zla rnam par dag pa ).

The formula above corresponds to the representation on the sand abacus:

[(J-1) x 12 + (M-1), ((J-1) x 12 + (M-1)) x 2 + R (m)] / (-, 65) = [L (J, M), r (M)] / (-, 65).

When creating a calendar, the above calculation is only carried out for the beginning of the year, i.e. for M = 1. When changing from one month within a year to the next month, do not repeat the above calculation, but add the amount 1 to L (J, M) and the amount 2 to r (M), thus calculating L (J, M + 1 ) = L (J, M) + 1 and r (M + 1) = r (M) + 2.

Leap months

The remainder r (M) of the above calculation, that is , is the size of the shift of the beginning of the solar month compared to the lunar month. It is called zla bshol rtsis ´phro "calculation remainder for the leap month". ${\ displaystyle \ lfloor A \ cdot SOL (J, M) \ rfloor}$

This designation was chosen for the following reason: If r (M) = 0 or r (M) = 1, then the shift between the two types of month exceeds the length of one month and the value L (JM) increases by 2. As a result, the deviation of the beginning of the calendar year from that of the solar year is greater than one month. To correct this, a leap month (Tib .: zla bshol or zla lhag ) is added.

According to Dragpa Gyeltshen (1147–1216), when r (M) = 0 or r (M) = 1 occurred, the previous month was counted twice. This upstream activation means that z. E.g. a previous Sa ga month in which r (M) = 63 or r (M) = 64 was present, followed by a second Sa ga month, which was then also regarded as a leap month.

In addition, one can observe the use of a subsequent activation in which the following month was counted twice. The Sa ga month, for which r (M) = 63 or r (M) = 64, was followed by a 1st and a 2nd sNron month, the 1st sNron month being the leap month.

Furthermore, one can observe the usage that alternately with r (M) = 0 the leap month was assigned to the previous month and with r (M) = 1 the leap month was placed in front of or assigned to the currently invoiced month.

A momentous change in the calculation of the leap months was brought about by the Phugpa school in that this school, calculated back to the epoch of the Kālacakratantra (beginning of the year in 806), changed the initial value for converting the past solar months into synodic months from 0 to 61. As a result, leap months were inserted three months later than the method described above. Of course, it cannot be ruled out that a subordinate, upstream and alternating counting of leap months was occasionally in use.

Another radical change in how leap months are calculated took place in the 17th century. This new method was based on the calculation of the so-called Ch'i centers (Tib .: sgang ) of Chinese astronomy. This new switching method was introduced in the administrative area of the central Tibetan government in 1696 after the death of the 5th Dalai Lama and is still used today.

The day of the week for the beginning of the middle lunar month M ( gza 'yi dhru va )

The aim of this calculation is to determine the day of the week for the beginning of the middle lunar month M.

The conversion value of the mean lunar day into the time value of the natural day ascribed to the proclamation by the Buddha becomes with

{\ displaystyle B = 1- [0,1,1] / (-, 64,707) = 1 - {\ frac {1 + {\ frac {1} {707}}} {64}}}

specified. If you multiply 30 (lunar days) by B, you get the amount as the length of a lunar month in natural days

[29,31,50, 0, 480] / (-, 60,60,6,707) d.

The amount exceeding 28d, i.e. the 4 whole weeks, is thus:

W (1,1) = [1,31,50,0,480] / (7,60,60,6,707) d.

In order to determine the day of the week in which the lunar month M begins, it should also be noted that the epoch usually does not begin with day 0. In this respect, an initial value WA (m) must be taken into account.

As can be seen from the following table, the various Tibetan schools of astronomy that have become known so far use different values for W and WA and also for R in some cases. All initial values refer to the epoch of Kālacakratantra.

Table 1:

school	W (m, 1): amount exceeding 28d of length of a middle lunar month	W (m, 2): When changing from one month to the next value to be added for the day of the week	WA (m): Initial values for the day of the week	R (m): initial values for the lunar month
m = 1: Phugpa school	W (1.1) = [1,31,50,0,480] / (7,60,60,6,707)	W (1,2) = [1,31,50,0,480] / (7,60,60,6,707)	WA (1) = [0.50.44.2.38] / (7.60,60,6,707)	R (1) = 61
m = 2: so-called exact byed rtsis	W (2.1) = [1.31.50] / (7.60.60)	W (2.2) = [1.31.50] / (7.60.60)	WA (2) = [2.30.0] / (7.60.60)	R (2) = 0
m = 3: Calendar calculation of the Kālacakratantra	W (3.1) = [1.31.50] / (7.60.60)	W (3.2) = [1.32.0] / (7.60.60)	WA (3) = [2.30.0] / (7.60.60)	R (3) = 0
m = 4: Calendar calculation by Chögyel Phagpa	W (4.1) = [1.31.50] / (7.60.60)	W (4.2) = [1.31.50] / (7.60.60)	WA (4) = [2.30.0] / (7.60.60)	R (4) = 0
m = 5: Calendar calculation of the Kālacakrāvatāra	W (5.1) = [1.31.50] / (7.60.60)	W (5.2) = [1.32.0] / (7.60.60)	WA (5) = [2.30.0] / (7.60.60)	R (5) = 0
m = 6: 1. different calendar calculation of the Kālacakrāvatāra	W (6.1) = [1.31.50] / (7.60.60)	W (6.2) = [1.32.0] / (7.60.60)	WA (6) = [2.30.0] / (7.60.60)	R (6) = 0
m = 7: 2nd deviating calendar calculation of the Kālacakrāvatāra	W (7.1) = [1.31.50] / (7.60.60)	W (7.2) = [1.32.0] / (7.60.60)	WA (7) = [2.30.0] / (7.60.60)	R (7) = 0
m = 8: Calendar calculation of the Tshurphu school	W (8.1) = [1,31,50,0,480] / (7,60,60,6,707)	W (8.2) = [1,31,50,0,480] / (7,60,60,6,707)	WA (8) = [2,25,20,2,352] / (7,60,60,6,707)	R (8) = 0
m = 9: Calendar calculation of the Jamgön Kongtrül Lodrö Thaye	W (9.1) = [1,31,50,0,30] / (7,60,60,6,44)	W (9.2) = [1,31,50,0,30] / (7,60,60,6,44)	WA (9) = [2,21,43,5,2] / (7,60,60,6,44)	R (9) = 0
m = 10: New calendar calculation for the Ganden Monastery	W (10.1) = [1,31,50,0,480] / (7,60,60,6,707)	W (10.2) = [1,31,50,0,480] / (7,60,60,6,707)	WA (10) = [2,26,50,4,352] / (7,60,60,6,707)	R (10) = 0

If one now multiplies the amount W (m, 1) by the number of the past lunar months L (J, M) and adds the initial value WA (m), one obtains the sought-after mean if the integer multiple of 7 ( mod 7) is omitted Day of the week and the time of day, for the beginning of the month M, or if M = 1, for the beginning of the year, according to the calendar calculation m.

{\ displaystyle WO (M, m) _ {1} = (L (J, M) \ cdot W (m, 1) + WA (m)) \ mod 7}

.

This value is called gza 'yi dhru va in Tibetan .

When moving from one month to the next, one does not repeat the entire calculation, but instead adds to the amount W (m, 2) recorded in Table 1 above, a value that should not actually differ from W (m, 1), which but for reasons of rounding up it is different for some calendar calculations. This invoice is used to create a calendar ${\ displaystyle WHERE (M, m) _ {1}}$

{\ displaystyle WO (m, M) _ {2} = ((L (J, 1) \ cdot W (m, 1) + WA (m)) \ mod 7 + W (m, 2) \ cdot (M -1)) \ mod 7}

out.

The results and do not differ for the calendar calculations m = 1, 2, 4, 8, 9 and 10. For all other calendar invoices, the invoice must be carried out later. ${\ displaystyle WHERE (m, M) _ {1}}$ ${\ displaystyle WHERE (m, M) _ {2}}$ ${\ displaystyle WHERE (m, M) _ {2}}$

Transition to the middle lunar day ( gza 'yi bar ba )

Excerpt from a table of the temporal lengths of 1 to 30 mean lunar days in d mod 7 and the change in the mean ecliptical length of the sun per lunar day according to the Phugpa school.

Since a lunar month consists of 30 lunar days ( tshes zhag ), one divides the length of a lunar month in d by 30, e.g. B. for the Phugpa School

W (1; 3) = (28 + W (1,1)): 30 = [0,59,3,4,16] / (7,60,60,6,707),

and gets the length of a lunar day in natural days d.

To simplify the further calculation, not all calendar calculations use this value from the Phugpa school, as the following table shows.

Table 2, temporal length of a mean lunar day in natural days d:

school	W (m, 3): Temporal length of a mean lunar day in d
m = 1: Phugpa school	W (1,3) = [0,59,3,4,16] / (7,60,60,6,707)
m = 2: so-called exact byed rtsis	W (2,3) = [0,59,3,4] / (7,60,60,6)
m = 3: Calendar calculation of the Kālacakratantra	W (3.3) = [0.59] / (7.60)
m = 4: Calendar calculation by Chögyel Phagpa	W (4.3) = [0.59] / (7.60)
m = 5: Calendar calculation of the Kālacakrāvatāra	W (5.3) = [0.59] / (7.60)
m = 6: 1. different calendar calculation of the Kālacakrāvatāra	W (6.3) = [0.59] / (7.60)
m = 7: 2nd deviating calendar calculation of the Kālacakrāvatāra	W (7.3) = [0.59] / (7.60)
m = 8: Calendar calculation of the Tshurphu school	W (8.3) = [0.59.3.4.16] / (7.60,60,6,707)
m = 9: Calendar calculation of the Jamgön Kongtrül Lodrö Thaye	W (9.3) = [0.59.3.4.1] / (7.60,60,6,44)
m = 10: New calendar calculation for the Ganden Monastery	W (10.3) = [0.59.3.4.16] / (7.60,60,6,707)

To calculate the day of the week and the time of day for the end of the lunar day T, multiply W (m, 3) by T and add the result , i.e. calculate ${\ displaystyle WHERE (m, M) _ {2}}$

{\ displaystyle WHERE (m, T) = (WHERE (m, M) _ {2} + T \ cdot W (m, 3)) \ mod 7}

.

The result is called gza 'yi bar ba in Tibetan . There is a day of the week and a time of day for the end of the mean lunar day T in month M of year J.

In order to save the calculation T · W (m, 3), more recent Tibetan textbooks on astronomy contain tables in which the result of this multiplication can be read off directly.

Calculation of the sun's mean ecliptical longitude

The task is to calculate the mean ecliptical length of the sun SO (m, T) at the end of the lunar day T according to the calendar calculation m. Angular measurements for determining the length form the division of the ecliptic into lunar houses and their subdivision. Since this task is carried out analogously to the calculation of WO (m, T), the following quantities are required for the various calendar calculations m:

Change in the mean length of the sun per lunar month: S (m, 1),
Change in the mean length of the sun per lunar month in the transition from one month to the following month: S (m, 2),
Change in the mean length of the sun per lunar day: S (m, 3),
The initial value of the ecliptical longitude of the sun at the epoch: SA (m).

In principle, these values, apart from the initial value, can be determined purely arithmetically using the conversion factors A and B. The Phugpa School also follows this, but the values of the other schools differ for different reasons.

Table 3:

school	S (m, 1): Change in the length of the sun per lunar month	S (m, 2): At the transition from one month to the next value to be added for the change in the length of the sun	S (m, 3): Change in the length of the sun per lunar day	SA (m): initial values for the length of the sun
m = 1: Phugpa school	S (1.1) = [2,10,58,1.17] / (27.60,60,6,67)	S (1,2) = [2,10,58,1.17] / (27.60,60,6,67)	S (1,3) = [0,4,21,5,43] / (7,60,60,6,707)	SA (1) = [24,57,5,2,16] / (7,60,60,6,707)
m = 2: so-called exact byed rtsis	S (2.1) = [2,10,58,2,10] / (27.60,60,6,13)	S (2.2) = [2,10,58,2,10] / (27.60,60,6,13)	S (2,3) = [0,4,21,5,9] / (27.60,60,6,13)	SA (2) = [26.58] / (27.60)
m = 3: Calendar calculation of the Kālacakratantra	S (3.1) = [2,10,58,2,10] / (27.60,60,6,13)	S (3.2) = [2.11] / (27.60)	S (3.3) = [0.4.20] / (7.60.60)	SA (3) = [26.58] / (27.60)
m = 4: Calendar calculation by Chögyel Phagpa	S (4.1) = [2,10,58,2,10] / (27.60,60,6,13)	S (4.2) = [2.11] / (27.60)	S (4.3) = [0.4.26] / (7.60.60)	SA (4) = [26.58] / (27.60)
m = 5: Calendar calculation of the Kālacakrāvatāra	S (5.1) = [2,10,58,2,10] / (27.60,60,6,13)	S (5.2) = [2.11] / (27.60)	S (5.3) = [0.4.20] / (7.60.60)	SA (5) = [26.58] / (27.60)
m = 6: 1. different calendar calculation of the Kālacakrāvatāra	S (6.1) = [2,10,58,2,10] / (27.60,60,6,13)	S (6.2) = [2.11] / (27.60)	S (6.3) = [0.4.26] / (7.60.60)	SA (6) = [26.58] / (27.60)
m = 7: 2nd deviating calendar calculation of the Kālacakrāvatāra	S (7.1) = [2,10,58,2,10] / (27.60,60,6,13)	S (7.2) = [2.11] / (27.60)	S (7.3) = [0.4.22] / (7.60.60)	SA (7) = [26.58] / (27.60)
m = 8: Calendar calculation of the Tshurphu school	S (8.1) = [2,10,58,1.17] / (27.60,60,6,67)	S (8.2) = [2,10,58,1.17] / (27.60,60,6,67)	S (8.3) = [0,4,21,5,43] / (7,60,60,6,67)	SA (8) = [0,29,34,5,37] / (7,60,60,6,67)
m = 9: Calendar calculation of the Jamgön Kongtrül Lodrö Thaye	S (9.1) = [2,10,58,2,20] / (27.60,60,6,38)	S (9.2) = [2,10,58,2,20] / (27.60,60,6,38)	S (9.3) = [0,4,21,5,26] / (7,60,60,6,38)	SA (9) = [0,1,54,4,22] / (7,60,60,6,38)
m = 10: New calendar calculation for the Ganden Monastery	S (10.1) = [2,10,58,2,500] / (27.60,60,6,707)	S (10.2) = [2,10,58,2,500] / (27.60,60,6,707)	S (10.3) = [0,4,21,5,488] / (27.60,60,6,707)	SA (10) = [0,18,0,0,135] / (27.60,60,6,707)

The mean length of the sun for the beginning of the mean lunar month M (Tib .: nyi ma'i dhru va ) is thus calculated with

{\ displaystyle SO (m, M) _ {1} = (L (J, M) \ cdot S (m, 1) + SA (m)) \ mod 27}

.

This generally only applies to the beginning of the year, as some calendar calculations add partially shortened values S (m, 2) when moving from one month to the next. To that extent it is generally with

{\ displaystyle SO (m, M) _ {2} = ((L (J, 1) \ cdot S (m, 1) + SA (m)) \ mod 27 + S (m, 2) (M-1 )) \ mod 27}

to be expected.

To calculate the longitude of the mean sun at the end of the mean lunar day T, multiply T by S (m, 3) and add the result to SO (m, M):

{\ displaystyle SO (m, T) = (SO (m, M) + T \ cdot S (m, 3)) \ mod 27}

.

This is the mean ecliptical longitude of the sun at the end of the mean lunar day T (Tib .: nyi ma'i bar ba ) according to the calendar calculation m.

The feet of the moon and the sun: midpoint equations of the moon and sun

Since the moon does not move around the earth on a circular path, but approximately on an ellipse , its angular velocity varies. Its angular velocity is smallest at the point of greatest distance from the earth and greatest at the point of greatest proximity to the earth. The same applies to the apparent path of the sun.

These different angular velocities mean that the observed ecliptical lengths of both celestial bodies deviate from the calculated mean lengths. The mathematical formula used to calculate this deviation from the mean length is called the midpoint equation . In the graphic representation, this equation results in a trigonometric curve.

In Tibetan astronomy this deviation is calculated on the sand abacus using systems of linear equations called feet of the moon (Tib .: zla ba'i rkang pa ) or feet of the sun (Tib .: nyi ma'i rkang pa ).

For the Tibetan calendar calculation, the midpoint equations in connection with the length of a lunar day play a role, as this is defined as the time span that is required to increase the angular distance between the sun and the moon by 12 degrees. If the moon now moves more slowly than the average, this required period of time is correspondingly larger compared to the average length. If the sun moves faster, the same results. With the midpoint equations of the moon and sun, it is therefore possible to calculate the lengthening or shortening of the duration of a lunar day compared to the mean due to the irregularity of the moon and sun movement.

The prerequisite for the calculation with a center equation is the determination of the mean length of a celestial body in the so-called anomalistic orbit , in which the zero point of the angular distance in Tibetan astronomy does not coincide with the point furthest or closest to the earth, but with the point at which the deviation from the mean angular velocity is 0. In addition, this point for the sun and moon is set in such a way that both celestial bodies move from here to the point furthest from the earth.

The feet of the moon

Tibetan table of the equations of the center of the moon

Graphic representation of the Tibetan equations of the moon

For the moon, the full anomalous orbit is divided into 28 parts, which are called ril po . These parts, in turn, are divided into 126 cha-shas . For the sun and the other five planets, the anomalous orbit is divided into 12 parts. The angular distance of the moon in the so-called anomaly from the zero point of the anomalous orbit at a certain point in time is thus with

{\ displaystyle [n, x] / (28,126)}

where and are whole numbers. Here, the order of magnitude ril po is called and is smaller than 28, is called the order of magnitude cha shas and is smaller than 126. ${\ displaystyle n}$ ${\ displaystyle x}$ ${\ displaystyle n}$ ${\ displaystyle x}$

The aim of the calculation is initially to calculate the angular distance of the moon from the zero point of the anomaly at the end of the mean lunar day T of month M in year J since epoch.

The calculation of this anomalistic angle is carried out analogously to the calculation of the ecliptical longitude of the sun. For this you need the following information:

Change in angular distance of the moon per lunar month mod 28: AN (m, 1),
Change in angular distance of the moon in the anomaly when going from one month to the following month: AN (m, 2),
Change in angular distance of the moon in the anomaly per lunar day: AN (m, 3),
The initial value of the angular distance from the moon to the epoch: ANA (m).

Table 3:

school	AN (m, 1): Change in angular distance of the moon in the anomaly per lunar month mod 28	AN (m, 2): Change in the angular distance of the moon in the anomaly during the transition from one month to the following month	AN (m, 3): Change in angular distance of the moon in the anomaly per lunar day	ANA (m): Initial values of the angular distance of the moon in the anomaly
m = 1: Phugpa school	AN (1.1) = [2.1] / (28.126)	AN (1,2) = [2,1] / (28,126)	AN (1.3) = [1.0] / (28.126)	ANA (1) = [3.97] / (28.126)
m = 2: so-called exact byed rtsis	AN (2.1) = [2.1] / (28.126)	AN (2.2) = [2.1] / (28.126)	AN (2.3) = [1.0] / (28.126)	ANA (2) = [5.112] / (28.126)
m = 3: Calendar calculation of the Kālacakratantra	AN (3.1) = [2.1] / (28.126)	AN (3.2) = [2.0] / (28.126)	AN (3.3) = [1.0] / (28.126)	ANA (3) = [5.112] / (28.126)
m = 4: Calendar calculation by Chögyel Phagpa	AN (4.1) = [2.1] / (28.126)	AN (4.2) = [2.1] / (28.126)	AN (4.3) = [1.0] / (28.126)	ANA (4) = [5.112] / (28.126)
m = 5: Calendar calculation of the Kālacakrāvatāra	AN (5.1) = [2.1,] / (28.126)	AN (5.2) = [2.1] / (28.126)	AN (5.3) = [1.0] / (28.126)	ANA (5) = [5.112] / (28.126)
m = 6: 1. different calendar calculation of the Kālacakrāvatāra	AN (6.1) = [2.1,] / (28.126)	AN (6.2) = [2.0] / (28.126)	AN (6.3) = [1.0] / (28.126)	ANA (6) = [5.112] / (28.126)
m = 7: 2nd deviating calendar calculation of the Kālacakrāvatāra	AN (7.1) = [2.1,] / (28.126)	AN (7.2) = [2.0] / (28.126)	AN (7.3) = [1.0] / (28.126)	ANA (7) = [5.112] / (28.126)
m = 8: Calendar calculation of the Tshurphu school	AN (8.1) = [2.1,] / (28.126)	AN (8.2) = [2.1] / (28.126)	AN (8.3) = [1.0] / (28.126)	ANA (8) = [5.112] / (28.126)
m = 9: Calendar calculation of the Jamgön Kongtrül Lodrö Thaye	AN (9.1) = [2.1,] / (28.126)	AN (9.2) = [2.1] / (28.126)	AN (9.3) = [1.0] / (28.126)	ANA (9) = [5.112] / (28.126)
m = 10: New calendar calculation for the Ganden Monastery	AN (10.1) = [2.1,] / (28.126)	AN (10.2) = [2.1] / (28.126)	AN (10.3) = [1.0] / (28.126)	ANA (10) = [5.112] / (28.126)

The angular distance of the moon in the anomalous orbit for the beginning of the middle lunar month M (Tib .: ril cha ) is thus calculated with

{\ displaystyle A (m, M) _ {1} = (L (J, M) \ cdot AN (m, 1) + ANA (m)) \ mod 28}

.

This generally only applies to the beginning of the year, as some calendar calculations add partially shortened values AN (m, 2) when moving from one month to the next. To that extent it is generally with

{\ displaystyle A (m, M) _ {2} = ((L (J, 1) \ cdot AN (m, 1) + ANA (m)) \ mod 28 + AN (m, 2) \ cdot (M -1)) \ mod 28}

to be expected.

To calculate the angular distance of the moon at the end of the mean lunar day T, add T, since AN (m, 3) = 1 for all m, and add the result to : ${\ displaystyle A (m, M) _ {2}}$

{\ displaystyle (A (m, M) _ {2} + T) \ mod 28}

.

This is the angular distance of the moon in the anomalistic orbit at the end of the mean lunar day T according to the calendar calculation m.

The deviations of the actual length of the moon from the mean length resulting from the observation are now recorded in Tibetan astronomy with a table that records the deviation at the beginning of this angle segment for each ril po and notes the change in the deviation within the angular unit ril po .

This table is called in Tibetan with zla rkang re'u mig "table of the feet of the moon", whereby the feet of the moon ( zla ba'i rkang pa ) denote individual calculation rules, which are ultimately linear equations. Since the deviations are symmetrical for 180 degrees within a full anomalous cycle, only 14 values are recorded in the table, since the following 14 values are the same apart from the sign. For the use of the board, one therefore calculates the angular distance for half an anomalous revolution:

A (m, T) = ( + T) mod 14 ${\ displaystyle A (m, M) _ {2}}$

= [n, x] / (14.126) .

Where is the value

{\ displaystyle \ lceil (A (m, M) _ {2} + T): 14 \ rceil}

as it decides whether the values calculated with the midpoint equations are to be added or subtracted.

Table for the equations of the center of the moon:

	n: access numbers of the equations (Tib .: rkang ´dzin )	${\ displaystyle b_ {n}}$ : Slope of the function or multiplier (Tib .: sgyur byed )	${\ displaystyle a_ {n}}$ : Initial values of the functions (Tib .: rkang sdom )
First half of the equations (Tib .: snga rkang ): Amounts to be added.	1	5	5
First half of the equations (Tib .: snga rkang ): Amounts to be added.	2	5	10
First half of the equations (Tib .: snga rkang ): Amounts to be added.	3	5	15th
First half of the equations (Tib .: snga rkang ): Amounts to be added.	4th	4th	19th
First half of the equations (Tib .: snga rkang ): Amounts to be added.	5	3	22nd
First half of the equations (Tib .: snga rkang ): Amounts to be added.	6th	2	24
First half of the equations (Tib .: snga rkang ): Amounts to be added.	7th	1	25th
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	8th	-1	24
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	9	-2	22nd
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	10	-3	19th
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	11	-4	15th
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	12	-5	10
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	13	-5	5
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	0	-5	0

Basically, in the Tibetan calendar calculation, the midpoint equations are used to calculate the change in the length of the lunar day, which results from the deviation of the actual angular speed of the moon from the mean speed. If the moon moves slower than the average, the lunar days become longer. If it moves faster than the average, the lunar days become shorter.

The amounts to be calculated below result in the lengthening or shortening of the duration of a lunar day compared to the mean due to the irregularity of the moon's movement. Since x = 0, this means that the zero point of the angular distance in the anomaly in Tibetan astronomy does not coincide with the point furthest or closest to the earth, but with the point at which the deviation from the mean angular velocity is equal to 0. Since the lunar days become longer after passing through this point, this means that the moon is slowing down, moving towards the point furthest from the earth (apogee) and reaching it at the beginning of the 8th ril po . The point closest to the earth (perigee) coincides with the beginning of the 22nd ril po . ${\ displaystyle \ Delta _ {Mn} (x)}$ ${\ displaystyle \ Delta _ {M0} (0)}$

Using the table of mid-point equations, the corrective quantity for the length of a mean lunar day is calculated using the following equation:

{\ displaystyle \ Delta _ {Mn} (x) = a_ {n} + b_ {n + 1} \ cdot {\ frac {x} {126}}}

.

This amount is added to WO (m, T), i.e. to the value for the day of the week and the time of day for the end of the mean lunar day T in month M of year J, if the result of

{\ displaystyle \ lceil (A (m, M) _ {2} + T): 14 \ rceil}

resulted in the number 0 or 2. If the result of this calculation was 1, the amount is subtracted.

The result is referred to in Tibetan as gza 'phyed dag pa "half-correct weekday for the end of the lunar day". The term “semi-correct” is used because a correction factor caused by the irregularity of the sun's movement has not yet been taken into account.

The feet of the sun

Tibetan table of the equations of the center of the sun

Graphic representation of the Tibetan equations of the sun

To calculate the equation of the center of the sun, the apparent anomalous revolution of the sun is divided into 12 parts, which corresponds to the division into the twelve signs of the zodiac . However, the zero point of the anomalous orbit is shifted by 90 degrees compared to the zero point of the angular dimensions of the zodiac signs or the moon houses.

As with the moon, the zero point of the angular distance in the anomalous orbit of the sun in Tibetan astronomy does not coincide with the point furthest or closest to the earth, but with the point at which the deviation from the mean angular velocity is equal to 0. Since, after passing through this point, the amounts from the equations of the midpoint have to be subtracted to the length of the mean sun, it follows that the angular velocity of the sun becomes smaller, i.e. it moves in the direction of the point furthest from the earth (aphelion). This coincides with the zero point of the orbit in the signs of the zodiac and the lunar houses.

The deviations of the angular velocities of the sun from the mean velocity are represented by a table, as with the moon, to which a corresponding calculation rule belongs.

This table is called in Tibetan with nyi rkang re'u mig "table of the feet of the sun", whereby the feet of the sun ( nyi ma'i rkang pa ) denote individual arithmetic rules that are ultimately linear equations. Since the deviations are symmetrical for 180 degrees within a full anomalistic cycle, only 6 values are recorded in the table, since the following 6 values are the same apart from the sign. To use the table, the angular distance is calculated for half an anomalous revolution.

The starting point of the calculation is SO (m, T), so the mean ecliptical length of the sun at the end of the mean lunar day T (Tib .: nyi ma'i bar ba ) according to the calendar calculation is m. It should be emphasized once again that the angular unit of measurement of this size is given in lunar houses, i.e. the angular measurement is based on the division of the ecliptic into 27 parts.

The value SO (m, T) is first reduced by [6.45] / (27.60) = 90 degrees:

{\ displaystyle SO_ {A1} (m, T)}

= SO (m, T) - [6.45] / (27.60).

This has changed the size specification to the zero point of the anomalous circulation. If the result is greater than [13.30] / (27.60) or equal to this value, this value is subtracted for half a cycle (= [13.30] / (27.60)). It is noted whether this value has been deducted or not. The result is with

{\ displaystyle SO_ {A2} (m, T) = [s_ {1}, s_ {2}, s_ {3}, s_ {4}, s_ {5}] / (27,60,6,67)}

designated. In order to be able to access the midpoint equations of the sun, the number of the respective equation is also calculated

{\ displaystyle n = \ left \ lceil {\ frac {s_ {1} \ cdot 60 + s_ {2}} {135}} \ right \ rceil}

.

To understand this calculation, it should be noted that 12 signs of the zodiac correspond to 27 lunar houses. The conversion factor between these angular dimensions is thus

{\ displaystyle {\ frac {12} {27}} = {\ frac {60} {135}}}

.

The size of the variable x in these midpoint equations is given by:

{\ displaystyle x = \ left [\ left \ lfloor {\ frac {s_ {1} \ cdot 60 + s_ {2}} {135}} \ right \ rfloor, s_ {3}, s_ {4}, s_ { 5} \ right] / (60,60,6,67)}

.

Table for the equations of the center of the sun:

	n: access numbers of the equations (Tib .: rkang ´dzin )	${\ displaystyle c_ {n}}$ : Slope of the function or multiplier (Tib .: sgyur byed )	${\ displaystyle d_ {n}}$ : Initial values of the functions (Tib .: rkang sdom )
First half of the equations (Tib .: snga rkang ): Amounts to be added.	1	6th	6th
First half of the equations (Tib .: snga rkang ): Amounts to be added.	2	4th	10
First half of the equations (Tib .: snga rkang ): Amounts to be added.	3	1	11
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	4th	-1	10
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	5	-4	6th
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	6th	-6	0

Using this table of midpoint equations, the corrective quantity for the mean ecliptical longitude of the sun is calculated using the following equation:

{\ displaystyle \ Delta _ {Sn} (x) = d_ {n} + {\ frac {c_ {n + 1} \ cdot x} {135}} =}

{\ displaystyle d_ {n} + {\ frac {c_ {n + 1} \ cdot \ left [\ left \ lfloor {\ frac {s_ {1} \ cdot 60 + s_ {2}} {135}} \ right \ rfloor, s_ {3}, s_ {4}, s_ {5} \ right] / (60,60,6,67)} {135}}}

.

Correct ecliptical longitude of the sun

To calculate the ecliptical longitude of the sun at the end of the lunar day T, one subtracts the amount from SO (m, T) if the amount SO (m, T) - [6.45] / (27.60) is not by half Circulation was to be reduced. Otherwise, SO (m, T) is added. The result, the exact ecliptical longitude of the sun at the end of the lunar day T, is called nyi ma dag pa in Tibetan . ${\ displaystyle \ Delta _ {Sn} (x)}$ ${\ displaystyle \ Delta _ {Sn} (x)}$

Correct end of the lunar day

To calculate the exact time at which the lunar day T ends in a certain weekday, one also subtracts the amount from the calculated “half-correct weekday for the end of the lunar day” (Tib .: gza 'phyed dag pa ) if the amount SO (m, T) - [6.45] / (27.60) could not be reduced by half a revolution. Otherwise this amount will be added. One ignores the error that arises from the fact that it is a radian measure. ${\ displaystyle \ Delta _ {Sn} (x)}$ ${\ displaystyle \ Delta _ {Sn} (x)}$

The result is called the Tibetan gza'-dag "exact weekday for the end of the lunar day". The amount marked with gza'-dag indicates the day of the week and the time of day at which the lunar day T ends.

This precisely defines the date number T with which a weekday is counted within a month.

Ecliptical longitude of the moon

The calculation of the ecliptical length of the moon is based on the mean angular distance that the moon travels relative to the sun per lunar day. For this, the amount [0.54] / (27.60) lunar houses is taken as a basis in all calendar calculations in Tibet. One multiplies this amount by T and adds the result to the exact ecliptical length of the sun at the end of the lunar day T (Tib .: nyi ma dag pa ). With this one has calculated the ecliptical longitude of the moon at the end of the lunar day T. The equation of the center of the moon is not taken into account in this calculation. The result is known in Tibetan as tshes' khyud zla ba'i skar ma "the star location of the moon wrapping around the lunar day".

If one subtracts the time of day at which the lunar day T ends in the respective weekday, the result is an amount that is called res' grogs zla ba'i skar ma “the star location of the moon assigned to the day of the week” in Tibetan . This is the ecliptical longitude of the moon at the beginning of each day of the week.

The two astrological components of calendar calculation

If you add the length of the sun calculated above and the length of the moon at the beginning of the day of the week, you get a quantity that is called Sanskrit yoga (Tib .: sbyor ba ). In terms of numbers, there are 27 yoga sections, each with its own name.

Another time quantity that is only astrologically significant is called byed pa (Sanskrit: karaṇa) in Tibetan . These time sizes also have their own names and, like the yoga sections, are listed in the almanacs for each calendar day.

Intangible cultural heritage

The Tibetans' astronomical calendar is on the list of the intangible cultural heritage of the People's Republic of China (1028 X-121 Zangzu tianwen lisuan 藏族天文历算).

literature

Nachum Dershowitz, Edward M. Reingold: Calendrical Calculations . Third edition. Cambridge University Press, Cambridge u. a. 2008, ISBN 0-521-70238-0 , pp. 315-322.
Winfried Petri: Indo-Tibetan Astronomy . Habilitation thesis to obtain the venia legendi for the subject history of natural sciences at the high natural sciences faculty of the Ludwig-Maximilians-University in Munich. Munich 1966
Dieter Schuh : Studies on the history of the Tibetan calendar calculation . Steiner, Wiesbaden 1973 ( Directory of Oriental Manuscripts in Germany Supplement 16, ZDB -ID 538341-9 ).
Dieter Schuh: Basics of the development of the Tibetan calendar calculation . In: Wolfgang Voigt (Ed.): XVIII. German Orientalist Day. From 1st to 5th October 1972 in Lübeck. Lectures . Steiner, Wiesbaden 1974, ISBN 3-515-01860-3 , pp. 554-566 ( Journal of the German Oriental Society . Supplement 2), uni-halle.de
Dieter Schuh (Editor): Contributions to the History of Tibetan Mathematics, Tibetan Astronomy, Tibetan Time Calculation (Calendar) and Sino-Tibetan Divination. Four volumes. Archive for Central Asian Historical Research. Edited by Dr. Karl-Heinz Everding Issue 17-20. Publishing information
Zuiho Yamaguchi: Chronological Studies in Tibet . In: Chibetto no rekigaku. Annual Report of the Zuzuki Academic foundation . X, 1973, pp. 77-94.
Zuiho Yamaguchi: The Significance of Intercalary Constants in the Tibetan Calender and Historical Tables of Intercalary Month . In: Ihara Shōren, Yamaguchi Zuihō (Ed.): Tibetan Studies . Proceedings of the 5th Seminar of the International Association for Tibetan Studies. Volume 2: Language, history and culture . Naritasan Shinshoji, Narita-shi u. a. 1992, pp. 873-895 ( Monograph series of Naritasan Institute for Buddhist Studies . Occasional papers 2, 2, ZDB -ID 1225128-8 ).

Web links

Svante Janson , Tibetan Calendar Mathematics . (PDF; 268 kB) math.uu.se
Tibetan astronomical almanac calculates solar eclipse on July 22
Yin Ba and its relation to calculating the Tibetan calendar
Article "Tibetan calendar calculation" in: Tibet-Encyclopaedia, edited by Dieter Schuh