I looked and hope this comment isn't redundant. I took a silver polish cloth - the two-part cloth, one for initial cleaning and the other for polishing - and cut it into 1 inch strips. I wet the first cloth strip with water and used it to clean the horn. Ran it into all the nooks and crannies like a piece of dental floss. Then I took another strip from the second cloth and used it to polish the horn running it into the same areas the same as with the first cloth. Result was well worth it and took about an hour to really get in there and clean it. Horn should be much easier to keep clean with minimal maintenance on the exterior.
The horn was Arthur Lehman's old horn which he bought 60 years ago in 1948. There are places on the horn where the silver plate has worn away and brass is visible but over all the horn is in good condition and polished up nicely. I guess many of you have newer horns and in better condition, but they would sparkle and shine using the method I mentioned. Probably be less work for you since they are "newer" and "cleaner"?