You need a programmable interval timer.
Intermatic and Tork make digital timers with interval settings that can
be set for specific short-duration with 1 minute minimum.
(Or set to pulse bell ringing with 1-127 seconds.
10 minutes is 600 seconds)
For example ET70115 single-channel timer.
The timer would be set for 24-hour operation, and then programmed for
Pulse, and then set timer with 24 programs, with 1 program for each
Mechanical cycle timers from Intermatic:
1) Talento 1 hour cycle timer/ each tripper represents 1.25 minutes
2) Model C8835/ each tripper represents 15 seconds.
To operate these timers certain hours of the day, add another timer in-line.
The in-line timer could be digital, but remember digital timers are not protected from power surge and tend to have more problems, and do not have replaceable clock motors and trippers.