Post History

## Precise alignment for a limited time
### Alignment of one ratio
In your world, how long do you require the alignment to last?

As [Olin Lathrops's answer](https://scientific-speculation.codidact.com/posts/288191/288193#answer-288193) explains, stable resonances are very unlikely to arise for large number ratios. A whole number of lunar months in a year, or a whole number of days in a year or in a lunar month, is not going to be stable long term.

However, the drift in the orbital periods of the planet and its moon, and in the period of rotation of the planet, will tend to be slow compared to the lifespan of an individual, which may be sufficient for your purposes depending on how precise you need the alignment to be.

On Earth, the change in the length of a day is so slow that even across millenia, the length of a day varies by only a few seconds. When the number of days in a year eventually reaches a whole number, it will stay a whole number to within a second for centuries. Even if the people in your world can measure time to sub-second accuracy, there may be a period when the length of a year is so close to a whole number of days that the difference is indistinguishable from noise in the measurement.

So a story can be set during a period of history when the number of days in a year just happened to be passing through a whole number. Alternatively the number of lunar months in a year could be temporarily a whole number. Neither of these are unrealistic. During history both these ratios pass through whole numbers several times, very slowly, staying near to a whole number for millenia. On our particular planet, our species is too young to have lived through such a time yet, but the species in your world could have developed writing just as the alignment was being reached.

### Alignment of two ratios
What is much less likely to happen is for both ratios to pass near a whole number at the same time. So an Earth-like planet where both the number of days in a year and the number of lunar months in a year are whole numbers may be unrealistic.

If one of the ratios is changing much more slowly than the other, there may be time for the other to catch up and become a whole number while the first has barely moved. This would allow both to be a whole number for a long period of time without being unrealistically improbable.

This is also helped by the fact that the lunar month and the year are both getting shorter when measured in contemporary days (the day length at the time of measurement, rather than our current roughly 24 hour day). In seconds, both the lunar month and the year are getting longer, but days are getting longer at a faster rate, meaning there are fewer days in a lunar month and fewer days in a year than previously. The fact that the ratios are both moving in the same direction means that when they do both align with a whole number, they stay aligned for longer than if they were moving in opposite directions.

### Deviation from the average ratio
Bear in mind that even if there is a long period of history when the average lunar month and the average year are both a whole number of days, there will still be variation from the average, particularly in the lunar month. On Earth the length of a lunar month varies by several hours from the average during a year. You could reduce this variation in your story by having the planet and its moon both have orbits that are much closer to circular than for our Earth and Moon.

### A 1 day month
In the long term, the length of a day is likely to continue increasing until it matches the length of a lunar month. At this point the Earth will always face the same side to the Moon, in the same way that the Moon already always faces the same side to the Earth.

The drifting of the lunar month length (at least its length in days) will then stop, so that every time the year length passes through a period of being a whole number of days, both ratios will be whole numbers. So if you wait long enough, you can set your story on Earth. There are two problems with this:
1. You may not want a 1 day month and a much shorter than 365 day year for your story
2. Earth will be uninhabitable by that point, due to the expansion of the Sun.

### Alignment when the planet is part of a binary
With a binary planet, the reasoning will be similar if the secondary planet replaces the moon. However, if there is a secondary planet and a separate moon, the dynamics may get more complex. There will now be a year, a lunar month, and what the inhabitants might call a "planetary month" (based on the other planet in the pair). I'd expect it to be even less likely for all 3 of these to have their ratios line up with whole numbers at the same time, and even more variation from the average.

## Culture of precision 
If the fussiness is part of the story, another approach might be to have ratios that are farther from whole numbers than on Earth, so a calendar based on whole numbers would drift so quickly that the inhabitants would never have tried. Being forced to include fractions in their dating system from early in their history might be part of the reason for them having a culture of precision. Their mathematics may have advanced far more rapidly than ours due to this necessity.

If you decide to go with such an approach, you could try having a much slower rotating planet with much longer days, so the lunar month and year are only a small number of days. Alternatively you could have the moon and sun much closer to the planet, so the lunar month and year are a small number of days without having to make the days longer. You could also use a combination of these two methods to avoid either being too extreme. Either way, the result is that a whole number of days will be such a poor approximation that it might never occur to them to describe a lunar month or a year that way.