A · Canonical textChapter 6

Sacred Calendars, Adhika Māsa & 100-Year Birthdays

The mathematical harmony between the Solar Year (365.25636 days) and Luni-Solar synodic months (354.36707 days), intercalary month (*Adhika Māsa*) calculus, and lifetime *Janma Tithi* birthday synchronization.

Sūrya SiddhāntaCh. 1 (Madhyamādhikāra), v. 38

A · Canonical text

असंक्रान्तिमासोऽधिमासः स्फुटः स्याद् द्विसंक्रान्तिमासः क्षयाख्यः कदाचित्।

asaṃkrānti-māso 'dhimāsaḥ sphuṭaḥ syād dvi-saṃkrānti-māsaḥ kṣayākhyaḥ kadācit |

Translation:A lunar month within which NO solar ingress (Sankranti) occurs is declared an Adhika Masa (intercalary month). A lunar month within which TWO solar ingresses occur is known as a Kshaya Masa (omitted month), occurring only rarely.

1. The Solar-Lunar Metonic Synchronization

A purely lunar year of 12 synodic months contains $12 \times 29.530588 \approx 354.367$ days, falling short of the tropical solar year ($365.2422$ days) by approximately 10.875 days per year. Without adjustment, lunar festivals would drift across all four seasons.

Adhika Māsa (Intercalary Month) Condition

Solar_Ingress(m) = ⌊ λ_Sun(t_end) / 30° ⌋ - ⌊ λ_Sun(t_start) / 30° ⌋

if Solar_Ingress(m) == 0: Adhika_Māsa (Intercalary)

if Solar_Ingress(m) == 2: Kṣaya_Māsa (Omitted)

Every 32.5 to 33 solar months (approx 2.7 years), the accumulated deficit reaches 30 days. An extra lunar month (Adhika / Mala Masa) is inserted carrying the same name as the subsequent month (prefixed as Adhika vs Nija), restoring perfect seasonal alignment.

Parameters & variables

t_start: New Moon Conjunction: Beginning instant of synodic lunar month(Julian Day)
t_end: Subsequent New Moon: Ending instant of synodic lunar month(Julian Day)
λ_Sun / 30°: Zodiac Sign (Rāśi): Integer zodiac sign index from 0 to 11(0 to 11)

2. Amānta vs. Pūrṇimānta Regional Systems

Amānta System (New Moon to New Moon)

The lunar month begins on Śukla Pratipadā and concludes on Amāvāsyā (New Moon). Prevalent across South India (Karnataka, Andhra Pradesh, Telangana, Tamil Nadu, Kerala) and Western India (Maharashtra, Gujarat). The month begins with the waxing moon.

Pūrṇimānta System (Full Moon to Full Moon)

The lunar month begins on Kṛṣṇa Pratipadā and concludes on Pūrṇimā (Full Moon). Prevalent across North India (Uttar Pradesh, Bihar, Rajasthan, Madhya Pradesh). The waning fortnight (Kṛṣṇa Pakṣa) leads the month.

Crucial Epistemic Principle: Complete Festival Identity

Although Amānta and Pūrṇimānta calendars assign different month names to Kṛṣṇa Pakṣa,the physical astronomical moment, Tithi, Nakshatra, and festival days (e.g. Dīpāvalī, Śivarātrī, Janmāṣṭamī) occur on the exact same solar day nationwide. For example, Dīpāvalī occurs on Āśvina Amāvāsyā in the Amānta system, which is simultaneously Kārttika Amāvāsyā in the Pūrṇimānta system.

3. 100-Year Janma Tithi Birthday Algorithm

In Vedic tradition, an individual’s true spiritual birthday (Janma Tithi) is observed when the identical Māsa (Lunar Month), Pakṣa (Fortnight), and Tithi (Lunar Phase) recur at local sunrise:

Janma Tithi Matching Criterion

Match(Year_y) = { Date d | Tithi(Sunrise_d) == Janma_Tithi AND Masa(Sunrise_d) == Janma_Masa }

Pūjāvāni's birthday engine scans all 100 years of an individual's lifetime, resolving Adhika Masa births (differentiating between Adhika and Nija occurrences) and Kshaya/Vriddhi Tithi edge cases.

Parameters & variables

Janma_Masa: Birth Lunar Month: Chaitra through Phalguna(1 to 12)
Janma_Tithi: Birth Lunar Phase: Pratipada through Purnima/Amavasya(1 to 30)