If I understand the question correctly, you’re asking how you as a worldbuilder might go about developing a flavorful, plausible, not-insanely-unwieldy timekeeping system. In addition, it appears that the various societies in your world more or less agree about the system. What follows are some practical suggestions for something of a step-by-step, noting a number of the more significant choices you might make along the way. There are other ways of doing this, of course—this isn’t a question that can be answered definitively—but I hope this may be helpful.
As several have noted already, sexagesimal numbers are exceedingly convenient for quick calculation, because 60 is divisible by 2, 3, 4, 5, and 6. This naturally lends itself to divisions of 12 & 5, thus the division of our clock face into 12 blocks of 5 minutes, our 12 months of the year, and so forth.
I suggest that you keep to something like this. The obvious alternative is a lunar calendar, as traditionally used in China and its broad area of influence (among other places), but you have to understand it clearly enough that you can use it fluently and fluidly in passing conversations among characters.
A further point of pragmatism: since your readers will almost certainly assume a sexagesimal solar calendar and timekeeping system, the only reason to draw a great deal of attention to your system is if it is much different. And yet, if you already had a great idea for how the weird calendar system was going to be a big deal in your world, you wouldn’t be asking this question. So I think what you want is window-dressing on a sexagesimal system. It’s easier for you and for the reader.
Start by coming up with an arbitrary list of images or objects to fit the 12 Zodiac signs. They don’t have to be animals: they could be plants, for instance. I think it will be a bit easier for the reader to follow if you are consistent (all animals, all plants, etc.), but there’s no reason you have to do it this way if you prefer not.
It may be convenient and elegant to come up with a sort of story in which each of the 12 signs leads naturally to the next. This can also provide extensive flavor-material for mythology and whatnot.
For the sake of high precision, cut each item in three parts: head, body, tail; blossom, stem, root; etc. It needs to be clear to you which end comes first.
You now have a complete system of decans. These label every ten-degree arc of a complete circle: lion-head, daisy-stem, etc. Under normal circumstances, however, everything will be labeled by the main image (hour of the lion, month of the daisy, etc.).
Cutting the Year
The year is traditionally cut at the four “corners”: the vernal and autumnal equinoxes and the summer and winter solstices. These are very readily observable with a low level of precision necessary, and have been known around the world since Neolithic times. The only reason to remove them from your calendar is because you have some clever idea about that—in which case, again, you’d not be asking this question. So we can assume that the year has four corners.
Now you get more choices:
Does each corner begin or end something? (Begin and end are of course chronologically the same, but they are conceptually quite different.)
At what point does the Zodiac begin? (Since the Mesopotamians, it begins at the vernal equinox, but you can pick any corner.)
Line up your 12 zodiac signs with the proposed rough calendar. You may wish to tweak this or that sign to fit well with the relevant season, but since that’s not the case on earth, it may be best to leave the list apparently incoherent to the seasons.
One more choice:
- What happens to intercalary dates?
This system presumes that there are exactly 360 days in a year, and there just ain’t. And your medieval society knows that. So what do they do about it? There are many options, such as:
Add one festival (or penitence) day right before or right after each corner, plus one more at the very end of the year cycle, plus another every four years.
Add five days at New Year’s, plus one every four years. (This is probably the most popular option, historically, and lends itself to really exciting and complex New Year rituals—great stuff for more flavor!)
Make a dramatic adjustment every six or twelve years, or some other convenient unit, by adding a whole month somewhere.
Add a special counting number to each year’s name, and make a dramatic adjustment—skip a year, most likely—when this counting number reaches X (probably 6, 7, or 12, but possibly something else). Thus this year is 3-Lion. This will tend to make your months get way out of whack with the seasons, which is less of a problem the closer to the equator you are.
Note that every intercalary system is a nod to practicality, making the system more usable by normal people. People whose job it is to study the stars—astrologers, for one, but also navigators and possibly many others—will need to use the celestial year that always runs equinox to equinox.
Cutting the Day
This is just a matter of deciding what the major unit is. Two-hour watches have been convenient and useful around the world for much of human history, so you might want to just stick to that. These can and should be named for your zodiac signs, in the same order, but you do need to decide what “zero” is. That is, does the day begin at midnight, sunup, sundown, or noon (to list the most obvious choices)? Midnight and noon have the advantage of being consistent, but are a pain in the tail to use practically. Sunup and sundown are easy to use, but they have an infuriating habit of drifting. As with seasonal calendars, you may want to think how close these folks live to the equator, because the closer they are, the easier it is to use the sun.
Counting these is easy enough: water clocks, sundials, hourglasses, marked candles.
This may seem like an unnecessary refinement, but a House system is a great way to add flavor and depth without making things more difficult for you. If you’re not trying to do precise astrological calculations (and I assume you’re not!), the House system really amounts to nothing more than labeling the sky with hours. You know how you might say “enemies coming in at 9:00,” and that basically means “from your right”? Same deal, only vertically. Due east is 0:00, straight up or down is 6:00, straight west is 12:00, and straight down or up is 18:00.
In the western system developed by the Mesopotamians, down is 6:00, and up is 18:00. This is because whatever is currently at 6:00 will rise on the eastern horizon in six hours, and so on. But for purposes of flavoring your world system, it makes no difference: you could reverse up and down, and while you’re at it you could focus on the western (setting) horizon rather than the eastern (rising) one. Take your pick—four options.
Since your hours/watches are named, this means that people used to paying close attention to the sky may remark that they see the moon “in the house of the Lion” or the like.
Don’t. It’s not necessary, and it’s very hard to justify. How often have you heard someone have a conversation about the nature and structure of the calendar and time system we use? That’s about how often it should happen in your world too.
The thing is, if you’re absolutely consistent about it, the whole thing will start to hang together by itself. And here and there, there are some opportunities for a little fleshing out:
An ordinary person and a sailor try to set a time for the attack. The sailor says he’ll start the attack when the moon enters the Lion’s house, and the ordinary person will say he’s got no idea what the sailor is talking about. The sailor now has to explain very simply, without justification: east is Zebra, straight up is Bullock, west is Crab, so Lion is from just east of straight up, so entering the Lion’s house is when the moon gets to 20 degrees east of straight up (you can come up with another term for “degree” if you choose, remembering that they’re using the things anyway).
An astrologer gets all mystic-weird about crossings and paths and houses and decans and whatnot. He won’t really explain, but in the process of his making it sound like he’s being coherent (if perhaps pretentious), he’s going to use a lot of terminology that the reader has seen in passing before.
Remember that the fine details are irrelevant, but their consistent application will give a little flesh to the bare bones. And because the system is actually, behind all the window dressing, essentially identical to what we’ve been using for the last 3000+ years, it will all seem eerily comprehensible even without explanation.