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@Biggles, I'm wondering if you're looking at the manual for the correct model.
Note that Yamaha makes a YPT-360 keyboard, which is another name for the PSR-E363, and there is a 76-key version called the PSR-EW300. Those three models share the same manual-- the
PSR-E363, PSR-EW300, YPT-360 Owner's Manual-- and they do indeed have a button that is labeled "HARMONY/ARPEGGIO" for accessing the Arpeggio feature.
However, the PSR-E360 is a different model than the PSR-E363 and YPT-360, despite the similarities in the model names. If the PSR-E360 is the model you have, it does not include the Arpeggio feature per se, hence there is no "HARMONY/ARPEGGIO" button. It does have a style called "Arpeggio," which is Style Number 125; but it doesn't have a collection of 176 different arpeggios like the PSR-E363/YPT-360 does.