Recently got a new single speed bike which I'm enjoying riding fixed, but still very much learning.
Is it normal for there to be a degree of eneven chain tension when turning the pedals? There are spots on each chainring revolution where the tension is tight making it difficult to adequate tension across the whole chainring. I think it is excessive, perhaps around 2-3cm difference in vertical chain movement between the low/high spots if I wiggle the chain up and down with my fingers.
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Howdee, as you surmised this is typically caused by either an un-round chainring or a chainring that isn't aligned on the spider. Can you identify where in the pedal stroke the chain is tightest? If so it may be as simple as slackening the bolts on the spider to allow you to center the chainring on the spider.
As matthew says, put the bike on a work stand (or upside down) and slacken the chainring bolts just a small amount, so they're still holding but not fully gripping. Tighten the chain a little and turn the pedals a few revolutions. Then repeat. You need to do it gradually or you'll just be pulling the chainring from one side to the other. When you're done tighten up the bolts fully. Then loosen the chain off a bit, you don't want it too tight, a few mm movement at the pedals when the wheel is held still is fine and won't be noticeable as slop when riding.