VEDANGA JYOTISHA

PANCHANGA 1

 Types of Panchanga

The Southern Amaanta Calendar

The Southern Amaanta Lunisolar Calendar is predominantly followed in the South and South-West Indian states of Andhra Pradesh, Karnataka and Maharashtra.

It is essentially a lunisolar one; i.e., its days and months are calculated based on the

motions of the moon. Like the Chinese calendar, the month is calculated from new moon

to new moon. It however, differs from the Chinese calendar in that the lunar day

(“thithi”) of the new moon is considered the last day of the previous month. Again, as in

the Chinese calendar, a leap month, an adhika maasa, is added to the calendar every 2.7 years on an average to offset the disparity in lengths between the lunar year and the sidereal year. In addition, a month, the kshaya maasa, is occasionally subtracted. This is discussed in a later segment.

The Southern Amaanta Calendar differs from the Western Amaanta Calendar in its treatment of kshaya maasas, the New Year Day and the Era followed. We believe that the Southern Amaanta Calendar follows the Southern School for treating kshaya maasas.  Saha and Lahiri suggest that it follows the Salivahana Saka Era starting with Chaitra Sukla Pratipada5, the lunar day after the last new moon before Mesha Sankranti. The years are also named according to the names of the Jovian years (Southern School6). The Eras and handling of kshaya maasas will be discussed in detail in their respective segments.

Western Amaanta Calendar

As already mentioned, we believe it’s important to distinguish between the Amaanta calendar practised in South and West India. In West India, specifically, in the state of Gujarat, the Amaanta calendar is of two forms7, one that starts with Aashaadha (followed in the Kathiawar region) and one that starts with Kartika (followed all throughout Gujarat). Both calendars follow the Vikrama Era and both also possibly follow the North Western School for kshaya months.

 

Purnimaanta Calendar

The Purnimaanta Calendar is followed in most of North India, i.e., in the states of Bihar, Himachal Pradesh, Uttar Pradesh, Haryana, Punjab, Jammu and Kashmir and Rajasthan8. (Earlier literature fails to mention Uttaranchal, Chattisgarh, Jharkhand and Delhi, but they are off-shots of bigger states and would necessarily follow the same calendar). The last of the three Indian lunisolar calendars, this one differs from the Amaanta calendar in that the months are reckoned from full moon to full moon.  Therefore, the Purnimaanta calendar starts two weeks before the Amaanta calendar does; that is, it starts with the lunar day after the last full-moon before Mesha Sankranti. The Vikrama Era is followed9, along with the Northern School of Jovian-year names10.

The Malayali Calendar

We now come to the list of Solar Calendars. The Malayali Calendar is predominantly followed in the South Indian state of Kerala. It is essentially a solar calendar; as we shall see later, the months are defined according to the raasis. The year starts with the Simha Sankranti and follows the Kollam Era.11 The month begins on the same day as a Sankranti if it occurs before aparahna, i.e., three-fifths of a day.  Otherwise, it begins on the next day.

 

Tamil Calendar

The Tamil calendar is followed in Tamil Nadu. This calendar is also solar; the month begins on the same day as a Sankranti if it occurs before sunset12. The Kali Era is followed along with the Southern Jovian cycle. One peculiarity about the Tamil calendar is that its month names start with Chittirai13 (Chaitra).

Bengali Calendar

The Bengali calendar is predominantly followed in West Bengal, Assam and Tripura. The Era is the Bengali San. The rule for the beginning of the month is again different; the month begins on the day after a Sankranti, if it occurs before midnight.Otherwise, it begins on the third day.

Oriya Calendar

The Oriya calendar is followed in the Eastern state of Orissa. In addition to the

Bengali San, the Saka, Vilayati and Amli eras are followed.15 The month begins on the

same day as that of the respective Sankranti.16

The Nanakshahi Calendar

Promulgated in 1998 CE, the Nanakshahi Calendar is followed in Punjab. It’s

intrinsically linked to the Gregorian calendar, except in its usage of the Nanakshahi Era.

 

 

 

National Calendar of 1957

Proposed by the Calendar Reform Committee of 1952 and promulgated in 1957 CE, the National Calendar is a tropical calendar with fixed lengths of days and months.

 

 Maasa Naama

 

There are two types of month names: -

1) Months named after Nakshatras

The set of month names named after nakshatras is used by both solar and lunisolar calendars, adding to seeming complexity of the Indian calendar system. Indeed, as we shall see, this type should actually called as ‘Months initially named after Nakshatras’; there has been an infusion of solar rules into an essentially lunar convention.

Let us then, first consider the original rule. Saha and Lahiri mention that pakshas

or fortnights were differentiated based on the nakshatra “where the moon is full”.20 That

is to say, if a particular full moon occurs near, say, the lunar asterism, Visakha, the full

moon would be called as Vaisakha Purnimaasi, and the month would be Vaisakha. The

earliest lunisolar months, then, were purnimaanta, that is, the name of the full moon

corresponded to the name of the month. Of course, the full moon occurs at all nakshatras.

Fifteen were taken into account for naming of the month, spaced more or less equally.

We thus have the following set of names along with their respective nakshatras21:

-

Nakshatra on Purnima Month Name

Chitra                                      Chaitra

Visakha                                  Vaisakha

Jyestha                                   Jyaistha

(Purva & Uttara) Aashaadha          Aashaadha

Sravana                                  Sraavana

(Uttara & Purva) Bhaadrapada       Bhaadrapada

Asvini                                      Asvayuja (Aasvina)

Krittika                                    Kaarthika

Mrugasira                               Maarghasira

Pushyami                               Pausa (Pushyam)

Maghaa                                  Maagha

(Uttara and Purva) Phalguni         Phalguna

It may be noted that the months of Aashaadha, Bhadrapada and Phalguna are linked to two nakshatras respectively. Chatterjee and Chakravarthy give the following criteria for choosing nakshatras for month names –

1) The yogataaras or the identifying stars of the nakshatras are prominent or have traditional significance.

2) They are spaced more or less equidistant from one another.

It must be mentioned that this rule is now an approximation largely due to Earth’s

precession; for instance, this year’s Chitra Purnimaasi had Swati as its nakshatra. Also,

possibly for historical reasons, and allowing for regional variation in pronunciation, the

Oriya, Bengali, Assamese, Punjabi and Tamil solar calendars also use the same set of

month names. To reconcile all this, we might frame a new rule; that, the amaanta lunar

 month takes its number from the solar month that starts in it, but its name from the solar month in which it starts, while following the purnimaanta months in chronological order.  That is to say, since Chitra occurred during the purnima of this year’s first purnimaanta month, we call this month as ‘Chaitra’. Consequently, the first amaanta month would also be ‘Chaitra’, which also would be the name of the solar month during which the amaanta ‘Chaitra’ started. However, the ‘number’ of the solar month (‘1’ in the case of amaanta and purnimaanta Chaitra) is not quite the same; the solar Chaitra is the last (i.e., 12^th) month of the year. The lunisolar Chaitra’s number is taken by the solar month that begins in it, namely the solar Vaisakha. All this can be seen in the graphic in the next page.

 

 2) Months named after Rasis

Only solar months share their names with rasis. SK Chatterjee and Apurba Kumar Chakravarthy give the following names along with the associated raasis25.

Raasi             Sanskritised Version             Malayalam Version

Mesha                  Mesha                               Medam

Vrshava          Vrshava                Edavam

Mithuna       Mithuna                      Midhunam

Karkata       Karkata                      Karitaka

Simha                   Simha                               Chingam

Kanya                   Kanya                               Kanni

Tula           Tula                             Thulam

Vrischika       Vrischika                    Vrischikam

Dhanus           Dhanus                 Dhanu

Makara           Makara                       Makaram

Kumbha       Kumbha                    Kumbham

Mina          Mina                            Minam

That is to say, the month shares its name with that of its corresponding Sankranti.

For instance, if Mesha Sankranti occurs on a certain day, then the period until the next

Sankranti would be Mesha maasa (Medham maasam).

This naming rule is followed primarily in the Malayalam calendar. 

 

 

Lunisolar

Ugadi / Gudi Padwa Chaitra S 1 None Lunisolar

Rama Navami Chaitra S 9 Must cover Madyahna

Tamil New Year,

Vishu, Bengali New

Year

Mesha Sankranti Respective Sankranti

 

 

 

 

 

Era Zero Year Beginning of Era with respect to

individual year

Saka 78 CE Mesha Sankranti, Chaitra S 1

Vikrama 57 CE Mesha Sankranti, Chaitra S 1,

Kartika S 1, Ashadha S 1

Kali 3101 BCE Mesha Sankranti, Chaitra S 1

Kollam 824 CE Kanya Sankranti, Simha Sankranti

Bengali San 963 + solar years since 1556 CE Mesha Sankranti

In addition, some regions also name their years according to the names of the Jovian years. Saha and Lahiri point out that there are two schools for this; the Southern school names its years in continuous succession, while the Northern school names its years corresponding to the present Jovian year30.

 

Kshaya Sutra –

How certain months are dropped.

Observing the reaction, she enquired,

To calendars you seem to be an active saakshya31,

But have you studied the ephemerally confounding kshaya?

To which, Apara Ganita looked at some fallen leaves and replied thus:-

One of the most interesting aspects of the Indian lunisolar calendar is its kshaya maasas, literally “decayed months”. Occasionally, certain months are dropped from the lunisolar calendar. We now try to understand the modalities behind this omission; we try to answer how, why, when and where it happens.

First, let’s try to define a kshaya month. Chatterjee, in his work on Indian calendars, says that a certain lunar month “may completely overlap any of the short three nirayana solar months of Margasira, Pausha and Magha”, with the result that there will be no new moon in the respective solar month. Consequently, there will be no lunar month named “after …this solar month”.32 A graphic describing this interaction is given in Appendix C.

We learn the following from this statement: - a) that the solar months of Margasira, Pausa and Magha are small, b) that at a certain time, there might be no new moon in these months, and c) the corresponding lunar month is dropped from the calendar. Note that Chatterjee is silent on whether the dropped lunar month is amaanta or purnimaanta; a naïve assumption would be that since he talks about new moons, the month would be amaanta. But, a study of the (Chaitradi) amaanta and purnimaanta calendars for the present year reveals that the difference between these two calendars is still two weeks. Therefore, it’s safe to conclude that kshaya months were dropped from the purnimaanta calendar as well.

 

Moreover, the statement about “corresponding lunar month” is unclear; are we talking about the lunar month with the same number as the new-moon-lacking solar month? Or are we talking about the lunar month with the same name of the solar month?  Running the calendrica code provided by Dershowitz and Reingold with their book Calendrical Calculations – The Millenium Edition (see table for values), we see that it’s the lunar month with the same name that gets dropped.  To account for a purnimaanta kshaya, and to further clarify which month to drop, we re-phrase the definition of a kshaya month to be thus: - “in any given lunar year, if two consecutive Sankrantis occur between two consecutive new moons, then the lunar month, whether amaanta or purnimaanta, with the same name as the solar month in which this occurs, is dropped.” As we shall see, such a re-phrasing is useful for computational purposes.

Indeed, as we mentioned earlier, we ran the Dershowitz and Reingold’s

calendrica package to get values for the occurrence of a kshaya month. Since searching

for a kshaya month is computationally very heavy33, we used a table prepared by Saha

and Lahiri (table 22 in the book)34 as a starting point. We also tabulated results for nonkshaya

months, specifically years with gaps of 19, 46, 65, 76, 122 and 141 years respectively. The results and the graphs from these results are tabulated in the appendix.

It must be noted that all cases tabulated previously have been calculated according

to Surya Siddhantic rules and that we may get a different set of results if calculated

according to ephemeris calculations. Indeed, as Chatterjee has pointed out, there was a

difference in 1964 CE; ephemeris calculations showed Margasira to be kshaya (and

33 Dershowitz, Nachum and Reingold. Calendar Tabulations – 1900 to 2200. (2002: Cambridge) Cambridge

 

Karthika, Chaitra to be adhika), while as we’ve seen, Surya Siddhantic computation showed Pausa to be kshaya (and Asvina and Chaitra to be adhika).35 Chatterjee, however, seems to be in agreement with Dershowitz and Reingold in saying that there was a kshaya in Magha in 1983 CE36, despite his use of ephemeris calculations.

What do we get from all this? We see that a kshaya month can occur every 19, 46,

65, 76, 122 or 141 years. Indeed, Saha and Lahiri’s tabulation provide us with the

following frequencies of occurrences for gaps between kshaya months: -

Interval Number of times occuring

19 11

46 3

65 1

76 1

122 1

141 6

Table – Number of times a particular interval gap occurred

We therefore see that between 525 CE and 1985 CE, kshaya occurred 11 times with a gap of 19 years, thrice with a gap of 46 years, six times with a gap of 141 years, and once each with gaps of 65, 76 and 122 years. The obvious question one would like to ask would be why. Why does kshaya occur only in these gaps?

To answer this better, we re-iterate what causes kshaya in the first place. We

already said that a kshaya would occur when two consecutive Sankrantis occur between

two Amavasyas. That is to say, when a solar month is shorter in length than, and is

completely enclosed by, a (an Amaanta) lunar month. Saha and Lahiri go on to say that

the “maximum duration of a lunar month exceeds the lengths of the solar months only in

35 Chatterjee, SK. p. 38

36 Dershowitz, Nachum and Edward M. Reingold. Calendrical Calculations – The Millennium Edition.

(2001: Cambridge) Cambridge University Press. p. 269

Panchanga- Tantra: The Magic of the Indian Calendar System / 22

the case of Margasira, Pausa and Magha”37 and that, therefore, kshaya is possible only in these months.

This would explain the solar month part, but what of lunar? How can the lunar month be bigger than the solar month? Ala’a Juwad has some answers; in his article, he suggests that the canonical synodic month, a lunar month between two consecutive phases of the moon, is not constant in length. Indeed, he goes on to say that between 1600 and 2400 CE, the synodic month extends in length from 29 days 6 hours and 31 minutes to 29 days 19 hours and 59 minutes.38 Moreover, he says that the “longest lunar months … occur when the date of the new Moon coincides with apogee”.39 A brute-force search for the longest synodic month definitely won’t give us a kshaya; for kshaya to occur, the lunar month needs to be only bigger than its solar counterpart and more importantly, completely encompass it. Indeed, Jawad says that the longest synodic month occurred in 1610 CE, a year which occurs within the 141 year long kshaya hiatus between 1540-1541 CE and 1680 – 81 CE.

We therefore search for other clues to unscramble kshaya. On a purely arithmetic

perspective, we observe the following: -

19 = 19 * 1

46 = 19 * 2 + 8

65 = 19 * 3 + 8

76 = 19 * 4

122 = 19 * 6 + 8

141 = 19 * 7 + 8

That is to say, the year-gaps are in the form 0, 8 mod 19.

37 Saha et al. p. 250

38 Jawad, Ala’a. “How Long Is a Lunar Month?” in Sky & Telescope, November 1993. p. 76 39 Ibid.

Panchanga- Tantra: The Magic of the Indian Calendar System / 23

Is it possible then, that the kshaya month has something to do with the Metonic cycle? The Metonic Cycle is a fairly well documented phenomenon; first observed by the Greek astronomer Metos, every 19 years, the lunar dates overlap with the tropical ones.  The underlying mathematical reason is simple: - 19 sidereal years contain 19*365.242189 = 6939.6 solar days, while 235 synodic months (with a mean of 29.53 solar days) contain 235*29.530588853 = 6939.68 solar days. The lengths overlap. But this obviously is neither necessary nor sufficient; it might be useful for the dates to repeat, but it definitely doesn’t fulfil the requirement for kshaya.

One suggestion therefore, might be that the kshaya occurs when the number of solar days of a sidereal year is equal to that of a synodic month, which in turn is equal to that from an anomalistic month. An anomalistic month is defined to be the time – period between two consecutive perigee passages and has a mean value of 27.55455 days.

Taking these average values, we calculate the average values of solar days in whole

numbers of synodic and anomalistic months (canonical kshaya years shaded for

reference): -

Interval Occurrence Modulo Solar Year Synodic Months Anomalistic Months

19 11 1*19 6939.601591 6939.68838 6943.7466

27 0 1*19+8 9861.539103 9863.216677 9864.5289

38 0 2*19 13879.20318 13879.37676 13887.4932

46 3 2*19+8 16801.14069 16802.90506 16808.2755

57 0 3*19 20818.80477 20819.06514 20831.2398

65 1 3*19+8 23740.74229 23742.59344 23752.0221

76 1 4*19 27758.40636 27758.75352 27774.9864

84 0 4*19+8 30680.34388 30682.28182 30695.7687

95 0 5*19 34698.00796 34698.4419 34718.733

103 0 5*19+8 37619.94547 37621.9702 37639.5153

114 0 6*19 41637.60955 41638.13028 41634.92505

122 1 6*19+8 44559.54706 44561.65858 44555.70735

133 0 7*19 48577.21114 48577.81866 48578.67165

141 6 7*19+8 51499.14865 51501.34696 51499.45395

Broadly speaking, we might summarize the above table as thus: - for the most

part, the number of solar days in solar years, synodic and anomalistic months overlap in

Panchanga- Tantra: The Magic of the Indian Calendar System / 24

kshaya years. However, this overlap doesn’t occur only in kshaya years; as the table shows, there’s an overlap for 133 years as well. Does this, then, explain the kshaya phenomenon? We might summarize it as being strongly suggestive, but definitely not conclusive.

Treatment of Kshaya Months40

We may complete our discussion of kshaya months by describing the three Kshaya Schools of thought.

The North Western School is followed in the north-western part of the country, presumably in Gujarat and/ or Rajasthan, where the lunisolar calendar is used.  Essentially, the North Western School treats the adhika month before kshaya as a normal month and the one after the kshaya month to be intercalary. This contrasts with the Eastern School where the reverse is followed; the adhika month before the kshaya is deemed intercalary, while the one after it is deemed normal. The Eastern School is followed in the eastern parts of the country, where the lunisolar calendar is followed. The final of the trio, the Southern School, treats both adhika maasas as intercalary, instead reckoning the kshaya month as a “jugma”, i.e., the first half of the thithi of this month is deemed to be that of the first month, and the second half as that of the second month.  This is presumably followed in the Southern parts of the country where the lunisolar calendar is followed.

MATERIAL FOR BA PART I PAPER 3