After a little more searching, I found that the tyler-setting 5,5,30 does produce a series of three pentagons that touch an edge. The other series, the 6 pentagons that span the figure, also is exact.
Andrew Weimholt sent me a pair that he discovered with tyler-settings of 5,5,5,15 and 4,4,4,12. I have redone these so that the enclosure is fairly circular. The assorted connections that these present are exact. For example, the dodecagon in 4,4,4,12 is as high as three squares.
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