The White Star Line
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The White Star Line archive was largely lost when the Cunard Steamship 
Company took them over. They would most certainly have had a band, yes. 
You are almost certainly not going to discover this information through 
company records. Your best chance of success lies with the passenger list 
for the Adriatic.  Musicians are very frequently listed. They could be 
listed as either passengers or crew, and I cannot speak for what you might 
find on the arrival lists into New York. However, the British departure 
and arrival passenger lists are very likely to show the names of musicians. 
They are sometimes listed separately (often with a crayon line through them 
to indicate that they were not to be included in the Board of Trade passenger 
statistics) but often there is nothing to identify their special status.  
In that case you need to look at the given occupations of each passenger. 
They frequently show up in the lists and I guess you are searching for a 
specific name so should be able to spot them fairly easily. Incidentally 
the same applies to other persons such as postal workers, nurses, chaplains, 
gymnasium workers, and various others working for the shipping lines but not 
officially 'crew.'