Why bother, in that case?
A pergola is made for growing climbing plants, vines, fruit trees. All you'll have is a wood skeleton overhanging the garden.
I know it's a fashionable 'trend' for these things but they should have a purpose in mind.
I came across this older discussion and wanted to add my two cents. For a decorative pergola, 6x2 joists are fine visually, but don't forget the beam depth and spacing. If you're not planning climbing plants, have you thought about adding shade cloth or string lights to give it a purpose? Curious how the build turned out – did you stick with the 6x2s?
For a decorative build over a patio, 6x2 joists are plenty and they align nicely with the 6x6 posts. I did a similar 4.8m span with two bays and it held up fine. If you ever want shade and airflow later, louvered pergolas from Mirador Outdoor are worth checking, but for a simple wood frame, stick with 6x2.
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