Please be very careful if you try this. To do instantly what the vinegar takes a long time to do we use "The Works" toilet bowl cleaner for the worst sax necks and brass crooks. You must wear rubber gloves when doing this and must not get any of the cleaner on the lacquered surface of the instrument. If you do rinse it off immediately.
First run a flexible brush through to remove any "groceries" then plug the small end with a rubber stopper. Remove the octave key and cover the hole with blue tac or electrical tape. Carefully pour enough "The Works" to fill the neck, wait about five seconds and then pour it out (into the toilet). Rinse immediately with lots of water and then refill the neck with a mixture of 1/3 baking soda to 2/3 water and let it work about 30 seconds. Then rinse again, brush vigorously with soapy water and then pull a clarinet "hanky" swab through.
The brass will look like new again.