Not sure the 16 bell sequence has any significance Rosie tbh, other than tradition dictates that 8 bells shall be rung just prior to midnight on New Year's Eve with a further 8 immediately afterward.
A ship's "watch" is four hours. The "first watch" would be 8pm to midnight.
The bell was rung every half hour of the 4 hour watch. A 24 hour day was divided into six 4 hour watches, except the dog watch (16:00 - 20:00 hours) which could be divided into two 2 hour watches to allow for the evening meal.
00:30 1 bell
01:00 2 bells
01:30 2 bells, pause, 1 bell
02:00 2 bells, pause, 2 bells
02:30 2 bells, pause, 2 bells, pause, 1 bell
03:00 2 bells, pause, 2 bells, pause, 2 bells
03:30 2 bells, pause, 2 bells, pause, 2 bells, pause, 1 bell
04:00 2 bells, pause, 2 bells, pause, 2 bells, pause, 2 bells
Middle Watch: Midnight to 4 AM (0000 - 0400)
Morning Watch: 4 AM to 8 AM (0400 - 0800)
Forenoon Watch: 8 AM to Noon (0800 - 1200)
Afternoon Watch: Noon to 4 PM (1200 - 1600)
First Dog Watch: 4 PM to 6 PM (1600 - 1800)
Second Dog Watch: 6 PM to 8 PM (1800 - 2000)
First Watch: 8 PM to Midnight (2000 - 0000)
So, midnight would be 8 bells.
That is why the New Years tradition would be 16 bells, the only time a ship's bell would be struck more than 8 times.