The sidereal period of the moon is about 27 days. There are
27x27 or 729 sidereal days every 2 years. With some
adjustment we could have a trinary sidereal calendar.

For every 2 revolutions of the earth around the sun there
would be 3 "years" (243 days each) of 3 "seasons" (81 days
each)
of 3 "months" (27 days each) of 3 "weeks" (9 days each) of 3
"clusters" (3 days each) of "days" (1 day each). A sidereal
day
is about .997 regular days so a drawback would be that the
dawn and dusk of the day would drift as the revolutions
went
by.

This has the advantage that it would be easy to keep track of the Moon's phases. I never know when the full moon is coming, and sometimes howl at the moon on the wrong night.

However, I don't fancy paying taxes every 243 days, or paying bills every 27.

The number three is an important symbol in some religions, so your invention could be popular with them. But for people who are struggling just to "Go metric," "Going Sidereal" is too much.

To make the shorter day, would you have an occaisional leap-hour with 59 minutes?

(The lunar calendar aspect of this was baked by the Hebrews 5000 years ago)