The month that gets skipped
India's lunisolar calendars: the month that repeats, the month that vanishes, and the two rule-books that have disagreed for nine centuries about which name dies. ← Calendar Contraptions
The previous piece was about rules that insert a month — China's machinery, which cannot break because it only stretches. India's calendars do the opposite thing too. The lunar month runs 29.5 days and the sun needs 30 to 31 days to cross one of the twelve zodiac signs, so now and then a lunar month comes around in which the sun never crosses a boundary at all. That month has to go somewhere. The Hindu calendars give it the name of the month that follows and call it adhika — the extra one, repeated. And once in a long while the balance tips the other way: a single lunar month catches two sign-crossings, twelve months of names have to fit into eleven slots, and one name is simply never used. The year skips. That dropped month is a kṣaya — literally a decay.
Both events are rare enough to be remarkable and regular enough to predict. An adhika month comes roughly every 2.7 years — 2026 just had one, an adhika Jyeṣṭha that ran from May 18 to June 15, so this year prints the name Jyeṣṭha twice. A kṣaya comes roughly every 141 years. The last one took the month of Mārgaśīrṣa away in 1963–64, a second took Māgha in 1982–83, and the machine on this page says the next is due in the winter of 2123–24 — a Pauṣa that will never be printed.
And there is a quarrel underneath it all: two classical rule-books agree the sky deletes a month, and disagree about which name it deletes. The split is nine hundred years old, both sides are quotable, and one published description of the 1982 event contradicts another because of it. This page runs the machine and shows both books.
The reading
The sun's uneven clock
A saṃkrānti is the instant the sun crosses from one sidereal sign into the next. The twelve solar months between saṃkrāntis are not equal — the sun runs fastest in December (perihelion) and slowest in July. That unevenness is the whole gear train: long solar months leave lunar months with no saṃkrānti (adhika), short ones can push two into one lunar month (kṣaya).
The month walker
The two rule-books
The census, 1100–2131
Find a month
Why the months drift: a sun that changes speed
The Earth's orbit is an ellipse, and the sun's apparent march through the zodiac speeds up and slows down accordingly: fastest at perihelion in early January, slowest at aphelion in early July. A solar month — one saṃkrānti to the next — therefore runs from about 29¼ days (December) to about 31½ days (July), while the lunar month stays near 29.5. The clock panel above shows the exact lengths for any year.
Everything on this page falls out of that inequality. When the solar months are long (roughly April to October) a lunar month can fail to catch a saṃkrānti — adhika. When they are short (roughly November to February) a lunar month can catch two — kṣaya. The census bears it out: 342 of the 393 adhika months sit between Vaiśākha and Āśvina, and every one of the thirteen kṣaya months is named between Kārtika and Māgha.
The sidereal frame
A saṃkrānti is not an equinox: it is the sun crossing a boundary of the sidereal zodiac — the stars, not the seasons. The Hindu almanacs measure longitude from a point fixed in the sky, conventionally the star Spica (Chitrā) at exactly 180°, and the gap between that frame and the tropical one — the ayanāṃśa — is currently about 24°13′. It grows by 50.3 arc-seconds a year, which is precession: the same slow wobble that, in the Saka piece, moves the national calendar's new year against the equinox.
One more convention hides in the printed dates: the almanac's day runs sunrise to sunrise, not midnight to midnight. The machine puts the 2026 Kumbha saṃkrānti at 03:51 IST on February 13 — nearly three hours before Ujjain's sunrise — so panchangs print February 12. The Mīna saṃkrānti does the same trick three weeks later: 00:42 IST on March 15, printed as March 14. The sky is indifferent; the printing is local.
The month that repeats
The naming rule for the extra month is stated in the Kalatatva-vivechana (attributed to Vyāsa, followed by Bhāskarāchārya in 1150, and by most of India today): a lunar month in which no saṃkrānti falls is named for the saṃkrānti that comes next, and wears the label adhika. So an empty month whose next crossing is into Mithuna is adhika Jyeṣṭha, and it walks immediately in front of the nija Jyeṣṭha. In older documents the pair is called prathama and dvitīya — first and second — which is exactly what they are; the scholar F. Kielhorn noticed that "adhika"/"nija" is modern usage.
The folk calendar is fonder of the repeated month than its name suggests: adhika māsa is Puruṣottama Māsa, the month of the Supreme Person, inserted — so the Purāṇas tell it — because the lunar calendar felt insulted at being the odd one out among the twelve. Extra months cluster where the sun is slow (the census counts 69 adhika Āṣāḍhas and 68 Śrāvaṇas in ten centuries), and almost never happen where the sun is fast: no adhika Pauṣa exists at all in 1031 years, and an adhika Kārtika happened just six times — 1963 was the only one between 1901 and 2150. An adhika month, like a kṣaya, is usually not alone: in this census every one of the thirteen kṣaya months has an adhika partner within seventy-five days, and every adhika Kārtika from 1125 onward pairs with a kṣaya. The saṃkrānti count has to balance; the sky borrows and repays.
The month that vanishes
When one lunar month catches two saṃkrāntis, eleven names must cover twelve slots: one name is never printed. The month itself takes the first crossing's name, the second crossing's name dies. Sewell and Dikshit's The Indian Calendar (1896) — still the standard treatment — tabulates the suppressed months for sixteen centuries and notices that the gaps between them fall into three classes only: multiples of 19 years, the constant 141, and 141 plus or minus a multiple of 19. The census panel's chips are colored by exactly those classes, and every gap this machine finds — 76, 65, 19, 19, 57, 141, 20, 140, 160, 141, 20, 140 — lands in one.
The classes are strong enough to predict on. Sewell's tables end with the 1822 event and 78 unfinished years; he wrote that the next should fall "at earliest" 122 years on — A.D. 1944 — "and possibly not till (141 years =) A.D. 1963". It fell in 1963. And the court astronomer Ganeśa Daivajña, writing in 1520, had already listed the suppressed months his Sūrya- Siddhānta arithmetic would produce for the next six centuries: Śaka years 1462, 1603, 1744 and 1885 — A.D. 1540, 1681, 1822 and 1963. Sewell confirmed the first four against his own tables. This page's modern machinery agrees exactly on 1822 and 1963, and puts the earlier stretch at 1502, 1522 and 1662 instead of 1540 and 1681 — the same 141-and-19 rhythm, shifted by six centuries of ephemeris refinement. Sewell himself warned that different Siddhāntas disagree about suppressed months near the knife-edge; the rhythm survives the disagreement.
Two rule-books
Here is the quarrel. The naming verse Sewell quotes ("the twelve lunar months at whose first moment the sun stands in Mīna and the following signs are called Chaitra…") names a month by where the sun stands when it begins. A second rule, which Sewell traces to a Brahma-Siddhānta tradition, names a month by where the sun stands when it ends. For ordinary months the two rules are the same rule. For a kṣaya month — which begins with the sun in one sign and ends in the next — they disagree, and the disagreement decides which name dies. Bhāskarāchārya followed the first rule; under it the 1982–83 event deletes Māgha. The old school, following the second, deletes Pauṣa.
Both are live in the sources. The English article on the Hindu calendar says "in the Hindu year spanning 1982–83, Maagha did not occur" — the present rule. The Japanese study of the 2033 problem, in its note on India, says that in 1982 the month after the ninth was the eleventh, the tenth slot having been skipped — the old rule. Same moon, same sky, different dead name. Bhāskarāchārya's own claim that "Kārttika, Mārgaśīrṣa and Pauṣa only are liable to be suppressed" is true of his school and false of the other: run this page's census under each book and the expunged sets are exactly {Mārgaśīrṣa, Pauṣa, Kārtika} and {Pauṣa, Mārgaśīrṣa, Māgha}. Sewell's dry footnote about almanac lists that show "the same month both added and suppressed" — impossible, he protests — is this same confusion, nine centuries deep: the list-makers had used one book for the adhika and the other for the kṣaya.
The other book: pūrṇimānta
Everything above is the amānta book — months ending at the new moon, the reckoning of the south and west and of the national calendar. The north counts full moon to full moon. Sewell states the conversion in one line: the bright fortnight keeps its name in both books, and the dark fortnight takes the name of the following month — so the pūrṇimānta book runs a fortnight ahead. That is why Diwali, the new moon that ends amānta Āśvina, is Kārtika amāvasyā in the north; why Holi, the full moon inside amānta Phālguna, is Phālguna Pūrṇimā in both books; and why Shivaratri's dark fortnight is Māgha to the south and Phālguna to the north — the 2026 almanac prints it as "Phalguna, Krishna Chaturdashi, February 15". The Makara saṃkrānti falls, as Sewell puts it, invariably in amānta Pauṣa — but in the kṛṣṇa fortnight, which the north files under Māgha.
Under a kṣaya the two books drop the same name: the expunged month is a missing month in both. For 1982–83 both books lose Māgha, which is what the English article records. The walker panel shows both readings side by side for any year.
What the machine can't tell you
Knife-edges. Sewell's warning, quoted in the rule-books panel: authorities diverge when a saṃkrānti falls within six ghaṭīs — 2.4 hours — of the amāvasyā. Nine of the thirteen census events sit inside that zone — the tightest, in 1304, has a saṃkrānti 23 minutes from the amāvasyā — which is exactly why the old almanacs disagree about these years while agreeing on the rhythm. The published anchors hold: Sewell's 1822 pairing, the 1963 and 1983 records, and the 2124 forecast all classify the same way here. 2026 offers a tamer look at the same machinery: the Mithuna saṃkrānti lands at 12:35 IST on June 15, four hours after the new moon that births Jyeṣṭha — comfortably clear of the warning zone, but still on the month's own birth day, so the walker shows the month named by a crossing that happened before its first sunrise.
The frame. The ayanāṃśa is a convention with schools; the spread between Lahiri variants is about an arc-minute, some 25 minutes of saṃkrānti time. This page uses a linear Lahiri frame (Spica at 180°, 23°51′11.5″ at J2000). Delta-T before 1600 comes from the Morrison–Stephenson table, good to about a minute — immaterial here, disclosed anyway.
Dates before October 1582 are proleptic Gregorian, ten days ahead of the Julian civil calendar the period actually used.
Days, not tithis. The panchang labels each sunrise with a tithi, and festivals ride those labels — which is why Nirjala Ekādaśī can print a day either side of this page's day-count, and why a single tithi can happen at two consecutive sunrises or none. That is a whole machine of its own: the adhika and kṣaya tithis, the day-scale version of this page. A later piece. And the oldest kṣaya on record — Underhill's example of a dropped Pauṣa in 1 BCE, when no new moon fell between Makara and Kumbha saṃkrānti — is quoted, not computed: Delta-T at two millennia's distance is honest to hours, not to the minutes that case would need.