Since several of us have independently thought of it, and other than the dates it's a good fit, I'm going to propose The Praxis, first book of the Dread Empire's Fall trilogy by Walter Jon Williams. The only problem with this identification is that it was not published until 2002.
The last of the Shaa, Anticipation of Victory, has just decided that it is time for him to die. The Shaa are a race of extremely long-lived (Anticipation of Victory is over 12 000 Shaa years old) and very large aliens (twice the size of a Terran).
The viewpoint character, Lieutenant Gareth Martinez' commanding officer, Commander Enderby, announces that he will join the ritual mass suicide to honour the Shaa's death:
"Of course, following the Great Master's death, I will kill myself."
Lieutenant Gareth Martinez, keeping pace alongside the longer-legged Fleet Commander Enderby, felt himself stumble as he heard the words.
"My lord?" He drove his legs through the stumble, to stride once more off Enderby's left shoulder. Their heels rang again in unison on the shaved, glittering asteroid material that floored the Commandery.
"I've volunteered," Enderby said in his prosaic, literal tone. "My family needs a representative on the pyre, and I’m the most suitable candidate. I'm at the apex of my career, my children are well-established, and my wife has given me a divorce." He looked at Martinez from beneath his level white brows. "My death will assure that my name, and my family's name, will be honored forever."
Enderby has arranged for Martinez to transfer to another position aboard a ship:
"I've spoken to Captain Tarafah about you," Enderby went on. "He's agreed to take you aboard as communications officer on the Corona."