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I'm also tending to associate YATS with this...http://www.bennygrunch.com/song_lyrics.html

Unless I've missed something, I think we're overcomplicating the calculations. If the spars and supporting tie-outs are to form a regular tetrahedron (equivalent 3-sided pyramid), then the tie-outs...

Corresponding colors match up vertical height on the left side to spar length on the right side.

I was having trouble envisioning the relationship between the spar length vs. vertical height for the desired 60deg angle so I did a graphic....

10-foot span at what height?

Right. If the top triangle was reduced to say 12' sides, you could move it downward on the boards keeping the same angles. This would also shorten up the outer triangles.

Nice.
Do you think it could be shortened up a bit?