He studied theology at Uppsala University and became interested in natural history, accompanying Carl von Linné (1707-1778) on field trips.
Encouraged by Linné and with the prospects of travel and natural history collection, he joined the Swedish East India Company as a chaplain.
The ship was held up by the monsoon and Tärnström fell ill and died on the island of Pulo Condore (Con-Dao).
Linnaeus described the domestic fowl as Phasianus gallus and gave the location as Pouli candor, based on the notes of Tärnström.
The reaction of his widow made Linnaeus subsequently choose only unmarried apostles.