Need an equation to measure frequency of a certain period over days, months, years.

Question

Example) Curious as to how many times a certain song of 3:46 will repeat over a given amount of time. How many times over a day, month, year, and the remaining growing fraction of time left over.

Answer 1

Just express a (standard) day, month, and year in seconds. Then divide those by $226$ seconds.

A decent expression for a day is $86400$ seconds. A month is $30$ days (more or less) and a year is $365$ days (more or less).

Leap seconds aside, the song will start playing again at the same time every $113$ days (since the least common multiple of $86400$ and $226$ is $113 \times 86400$). Every year that is divisible by $4$, except those years that are divisible by $100$ but not $400$ are leap years. This lets you predict the exact times that the song will start to play again for hundreds of years if you play it continuously.

The song will play completely $381$ or $382$ times a day. There is an extra $68$ seconds each day: $$86400 \text{ mod } 226 \equiv 68 \text{ mod }226,$$ so depending on when the song starts you may lose a full play of the song.

Also interesting is that $86400/226 = 282.30088...$ -- that is, it's almost $282.3$ but not quite, so the times almost repeat every $10$ days. But not quite. That could be, um, annoying to your character.