This can be achieved with a single 555, provided that the 260us pulse is allowed to be a negative-going pulse.
The discharge time is controlled only by one resistor (the one generally called Rb) and the capacitor.
The charge time can be controlled by a potentiometer to give a repetition rate of 1-16Hz.
Starting with the potentiometer, 2.2Meg is a reasonable value - that should give the 1Hz.
It needs a series resistor of 1/15th of its value (150k) to give the 16Hz. The pot and the 150k resistor make up Ra (the top resistor in the usual 555 astable circuit)
Then the 260us pulse needs Rb to be 620 ohms.
0.7 RC = 260us gives C = 600nF.

 

The first 555-timer circuit controls the repetition rate, hence the frequency.
The second 555-timer circuit generates the desired pulse width, i.e. 260μs.

 

but I don't really know what the one on the left does.

If you mean the one on the right, that changes the pulse width (as I previously noted).
To view that pulse width, use a faster sweep speed (say 500µs for a full sweep).

 

This can be achieved with a single 555

Yes, but its difficult to control the frequency and the pulse-width independently, which the TS apparently wants.

 

Yes, but its difficult to control the frequency and the pulse-width independently, which the TS apparently wants.

No SPICE on the ipad, so fountain-pen-CAD will have to do, although the output device isn’t quite so neat.
Pulse width depends ONLY on Rb and C, so the frequency can be varied without changing the pulse width.
If a positive-going pulse is required the circuit on the right can be used, with a slight loss of accuracy due to the volt-drop of the schottky diode, which produces a positive-going version of the DISCHARGE pin.

why did my iPad think I meant “schoolyard” when I typed “schottky” ? - bizarre.