More About Summer

I've given it a lot of thought and I have an idea for a new calendar. The ancient Romans only had 10 months. The period after what is now December, but before March was not part of any month. Similarly, I'm thinking that Summer should begin on Memorial day, with the middle of summer on the solstice (of course), and it being essentially over once Bastille day has passed. Yet it should remain technically summer throughout July. After July comes the month of August, which is not part of any season, but simply August. Fall begins on the first day of September.