What specifically are you asking? What the belt titles are, what colors they are, what material is learned at each rank, at what level someone is considered to be a "master", or something else entirely?
the question is as stated what (IS) 4th thru 10th master belt ? What does fourth degree mean,fifth degree mean,so forth and so on
there are 8 Dans, I believe there are only Cho Dan, E Dan, Sum Dan, Sa Dan, etc. Cho Dan is initiate E Dan is like Apprentice or adept Sum Dan is Disciple Sa Dan is Master as for the next 4 I am not sure, but 8th Dan is Grandmaster, I believe. There may be 10 dans but I have never met someone higher than 8th, nor do I know of anyone off hand. Perhaps a founder of the art is, that would make sense. Usually once you reach 7th the only way to advance is to do certain things, start a school, federation, then art, I suppose.
The only 9th dan i have ever met was grandmaster lee, and i think he's reached 9th dan for founding the UK Tang Soo Do federation.
There is no Dan higher than 9th in the International Tang Soo Do Federation. Wether Grandmaster Kang Uk Lee (9th Dan) reserves 10th Dan for our founder, I'm not sure. Anything above 5th Dan is generally achieved through merit, you reach your Master's level at 4th Dan. Signified by a black belt with a red stripe.
To a certain degree, my master, Master J. M Tang (6th Dan) achieved his 6th through the constant dedication to his/our federation. This was over a period of about thirteen years to the best of my knowledge. I think the merit comes in different forms, you aren't as likely to be 'promoted' just because you started a new club. The usual like expanding the federation into a different country, I assume would have some high yield.