Mathematical Exploration of Sections of Tantraratna and Tāladīpikā: Unraveling Tippa Gopendra Bhūpāla’s Contribution in Astronomy and Rhythm

Abstract

The Vijayanagara Empire stands as a significant epoch in Indian history. Exerting a profound and lasting influence on the cultural, educational, and economic development of southern India, particularly in the regions corresponding to present-day Karṇāṭaka, Andhra Pradesh, and Tamil Nāḍu. This period was marked by extraordinary advancements in intellectual and artistic pursuits, fostering an environment that enabled the proliferation of knowledge across multiple disciplines. Beyond the celebrated contributions of eminent scholars and the distinguished Aṣṭadiggajas, who played a crucial role in shaping the educational and literary landscape, the rulers of the Vijayanagara Empire were equally remarkable. These kings were not only adept administrators but also patrons of the arts, literature, and scientific inquiry, significantly enriching the intellectual and cultural heritage of this region.

Among the illustrious figures of this era, Bhūpāla Tippa Gopendra of the Sāluva dynasty remains largely overshadowed despite his significant intellectual contributions. His scholarship spans multiple domains, particularly in musicology and astronomy, where his works exhibit a remarkable synthesis of mathematical precision and artistic insight. This interdisciplinary approach reflects the intellectual traditions of the Vijayanagara period, where scientific inquiry and artistic expressions were not differentiated but deeply interconnected. Among his known works, three stand out: Tantraratna, an astronomical treatise; Tāladīpikā, a comprehensive study on music systems; and Kāmadhenu, a commentary on Vāmana’s Kāvyālaṅkāra Sūtra Vṛtti. This thesis delves into two of his works, Tantraratna and Tāladīpikā, through an analysis of primary manuscripts.

The first section of this thesis examines the Candravākyas presented in Tantraratna. These vākyas are decoded, mathematically analyzed, and their underlying principles are explored alongside a comparative study with earlier works by Mādhava and Vararuci, assessing their methodological continuity and advancements. The second section focuses on the concepts of tāla, particularly the Prastāra methodology as discussed in Tāladīpikā. Tāla Deśa Prāṇas, Māgadeśī and Prastāra systems are studied with references to earlier texts such as the Nāṭyaśāstra and Saṅgīta Ratnākara. Through this study, the enduring relevance of Tippa Gopendra’s intellectual legacy is brought to light, reaffirming his place in the scholarly traditions of medieval India.