Why 13 months?
In short, it’s the only option if we want
- a fixed calendar
- equal months
- equal quarters
- seven-day weeks
To create a calendar layout, you first have to start with a cycle length. Because nature is messy, it’s never an exact integer. For Earth solar calendars, this is slightly less 365.2424 days (see leap years for details). So we need years with fewer days than our target values and years with more days than our target value, and some rules on how often and when we switch between those years.
Because we typically want to subdivide our year into smaller, roughly equal parts like quarters, months, and weeks, it helps to list prime factorizations for some days around the 365-day mark.
Columns:
- Day count for that year
- Difference to a year of 365 days
- Prime factors
- Divisible by four (equal quarters)
- Divisible by seven (equal weeks)
- Possible factors that could be considered a “month” (equal months)
- Equal quarter length
| Days | Δ | Prime factors | 4 | 7 | “Month” | Quarter |
|---|---|---|---|---|---|---|
| 356 | 9 | $ 2^2 \times 89 $ | ✅ | $ 89 $ | ||
| 357 | 8 | $ 3 \times 7 \times 17 $ | ✅ | $ 21 $ | ||
| 358 | 7 | $ 2 \times 179 $ | ||||
| 359 | 6 | $ 359 $ | ||||
| 360 | 5 | $ 2^3 \times 3^2 \times 5 $ | ✅ | $ 20, 24, 30, 36, 40 $ | $ 90 $ | |
| 361 | 4 | $ 19^2 $ | ||||
| 362 | 3 | $ 2 \times 181 $ | ||||
| 363 | 2 | $ 3 \times 11^2 $ | $ 33 $ | |||
| 364 | 1 | $ 2^2 \times 7 \times 13 $ | ✅ | ✅ | $ 26, 28 $ | $ 91 $ |
| 365 | 0 | $ 5 \times 73 $ | ||||
| 365.2… | 🎯 | |||||
| 366 | -1 | $ 2 \times 3 \times 61 $ | ||||
| 367 | -2 | $ 367 $ | ||||
| 368 | -3 | $ 2^4 \times 23 $ | ✅ | $ 23 $ | $ 92 $ | |
| 369 | -4 | $ 3^2 \times 41 $ | ||||
| 370 | -5 | $ 2 \times 5 \times 37 $ | $ 37 $ | |||
| 371 | -6 | $ 7 \times 53 $ | ✅ | |||
| 372 | -7 | $ 2^2 \times 3 \times 31 $ | ✅ | $ 31 $ | $ 93 $ | |
| 373 | -8 | $ 373 $ | ||||
| 374 | -9 | $ 2 \times 11 \times 17 $ | $ 22, 34 $ | |||
| 375 | 10 | $ 3 \times 5^3 $ | $ 25 $ |
Now we only need to use this messy table to build our calendar. To create a calendar that even works at all, without drift, we need to pick a year with a length below our target 🎯 and another year above our target length. Later we’ll find a set of rules that determine how often and when we switch between those two years we picked so the average year length hits close to the target length.
Technically, you only need the first column to do that. The closest years would be 365 days and 366 days. It’s actually a sensible choice because this results in a calendar that has the lowest possible year variability, with only one day difference between the year below and the year above the target. This is also what the Gregorian Calendar does. But looking at the table we can note the following things about both lengths:
- Not divisible by four; a requirement for equal quarters
- Not divisible by seven; a requirement for fixed calendars with seven-day weeks
- They don’t have any factors between 20 and 40, which would allow for equal “month” lengths
But luckily there is a great choice very close to our target; 364. It’s divisible by four and seven. It can support equal months (13 months with 28 days each, or 14 months with 26 days each). The month length of 28 days is even divisible by seven, which is perfect because that allows for months with exactly four weeks.
But 364 is below the target and we also need a year above that. Looking at the table the next length that’s divisible by seven is 371. This is what calendars with leap weeks are doing. But note that 371 supports neither equal months nor equal quarters, and the year variability is now seven days. So the challenge is now to somehow squeeze in the extra week without completely ruining the benefits that 364 has.
Another approach would be to base the year layout solely on 364 days and then add one or two extra days to reach 365 and 366. This is what 13-month calendars (including SAC13) do. However, this approach has the downside that we either won’t have a fixed calendar, or we will have to break the week cycle once or twice a year.
I believe it’s important to generally stick to the seven-day week cycle (see why seven-day weeks). However, it’s also very important to keep the year variability as low as possible (see leap weeks), and thus it’s acceptable to allow some exceptions to maintain a highly symmetrical calendar.