How is length of a month determined?

Answer 1

Months originated as lunar cycles, but the the amount of time that it takes the Earth to orbit its sun is not an integral multiple of any number of lunar cycles.

Cultures that use the time of the Earth's orbit as more fundamental in marking time that the lunar cycle typical redefine the month so that there is an equal, integer number of them in each year.

There are between twelve and thirteen lunar cycles in a years, so these cultures have twelve or thirteen months. The number of cycles is closer to twelve, and 12 is easily divided by 2, 3, 4, and 6, whereas 13 is a Prime number.

If we had 7 months that were 30 days long, and 5 months that were 31 days long, then this would work pretty well. But the ancient Romans thought that February was unlucky, so Gaius Julius Caesar, when he was designing their calendar, got cute and made February a short month. (March proved to be an unlucky month for him!) They ended-up with four months that were 30 days, 7 months that were 31 days, and February which has 28 or 29 days. (We'll get back to February in a bit.) We inherited that silly business of a short February.

But, also, the number of times that the Earth spins in a year also isn't a nice even number. Trying to live with days that are plainly out-of-synch with the spinning of the orbit is just too hard, but we also don't want to have the year get noticeably out-of-synch with the time that it takes the sun to orbit the earth. So we have rules that let the year be just a little too short, and then play catch-up about every four years by adding a day to February, that short month. I say "about every four years" because doing it absolutely every four years would cause the average year to be a little too long, and these things can add-up. So there are rules about not having an extra day in the last year of each century, except every 400 years. (Even that's not perfect, but it'll be a long time before the difference is noticeable!)

Modern science doesn't try to mess around directly with the lengths of months, but they like to define the "second" so that physical laws are as regular as possible. This creates a problem in that, by such a standard, the Earth does not spin at a constant rate! It certainly doesn't take exactly 86400 seconds for each turn! So there's a group who monitor the number of seconds that have passed and the number of times that the Earth spins, and every now-and-then they declare that some day is going to get an extra second or lose an extra second just before the next day. So, measured in terms of these seconds, this group indirectly changes the lengths of months, but not ever by very much.

Flippen heck that's a complicated answer!!!