Precision broken clock

It's a cliché that 'even' a stopped clock is right twice a day, but one which is fast or slow is never right.

This is intuitively true, but does it have to be the case? There are some common changes made to timepieces such that when stopped they show the correct time less than twice a day on average - a 24-hour clock (either with the hour hand going round once a day, or a separate am/pm indicator) would be right once a day, while a face showing the day of the month is right only twice a month at most. I consider these degenerate cases and won't consider them further.

But it occurs to me that a very fast clock could be right three -or more- times a day.
The precision broken clock is a post-manufacture modification, made solely by adjusting its existing parts. For the "three times/day" model this causes the clock to run exactly two and a half times too fast. One can choose one of the three times a day that the clock is correct simply by setting it at that time (the other two times are at one- third day intervals).
Suppose one sets the correct time at midnight. The clock will then increment by 2œ hours per hour, so that by 8 am it will have proceeded through 20 hours; the face will show 8 o'clock. At 4 pm it will overtake the true time again, and the cycle will repeat at midnight.