Step 1: Concept
Musicological treatises serve as the foundational textual basis for historical developments, raga classifications, and technical terms in Carnatic music.
Step 2: Analysis of Options
- Swaramelakalanidhi was authored by Ramamatya in the 16th century ($1550$ AD), introducing systematic mela classifications.
- Brihaddesi was written by Matanga Muni around the 8th or 9th century AD, which is legendary for defining the word ‘Raga' for the first time.
- Raga Vibhodha was composed by Somanatha in $1609$ AD, focusing on raga structures and veena playing techniques.
- Sangeeta Saramrita is an authoritative 18th-century work written by King Tulaja I (Tulajaji), the Maratha ruler of Thanjavur (reigned $1728$–$1736$ AD), capturing the contemporary musical practices of his era.
Step 3: Conclusion
King Tulajaji authored the comprehensive treatise Sangeeta Saramrita, matching option (C).
Final Answer: (C)