Sobriety is the state of being sober, which can mean either not intoxicated or being solemn. If he takes a drink, an alcoholic ends a stretch of sobriety. You might note sobriety in the stands when your team is losing.
When thinking about sobriety, think about the Puritans, and the way they kept the Sabbath. For them, even laughing on a Sunday was seen as an excessive display. Known for sobriety, they preferred a day marked by prayer and contemplation. It goes without saying that they frowned upon the consumption of any form of alcohol––favoring sobriety of both kinds.