Tad Lincoln's music teacher, William Withers jr?

[NHjohn]

Greetings folks,
I play a Civil War vintage bass saxhorn in the Federal City Brass Band out of Baltimore and recently authored the book, "I Was Detailed to the Regimental Band," with Volume One, Massachusetts. Currently, I am researching New York state, including William Withers Jr., Band Leader 12th New York State Militia in 1861. That band played numerous concerts on the White House lawn in May-June of 1861, Withers' Orchestra entertained at Lincoln's second Inaugural Ball, and I have found several references to Withers "teaching" the drum to Tad Lincoln in 1864, at the White House, mainly based on Withers' off-hand statement that he did so. Can anyone confirm this fact, does anybody know how/why Withers was chosen?