Fun question. How often does the liturgical cycle – with its myriad variables that are specific to each hour, day, week and year – how often does this cycle, as it is currently practiced, (or rather, as it is currently set forth according to the Typikon) how often does it perfectly repeat (i.e. for one day’s full cycle of services to be the exact same as another day)?
Though it came from a reliable source, my memory is not very reliable, I was told once in a casual conversation that the cycle repeats something like every 140 years or something, but again that’s me trying to remember a casual conversation.
That’s a really good question and it would require a good deal of mathematics. The date of Pascha sets the cycle of modes, so it would need to be based on that. As it stands currently, Pascha can fall between April 4th and May 8th (on the New Calendar), so that certainly gives us a lot of variants. Of course, as Pascha changes dates from year to year, the beginning of the Triodion is also variable, so it would be very difficult for one year to be exactly the same as another. I can outsource this question to my more math-minded typikon friends and see if they can give us an answer!