Arrangement of musicians into distinct bands - combinatorics solution? Real life problem. 
I have 8 different musical instruments and 14 players of each instrument. There will be 9 sessions where the players are placed into 14 bands (each band has 1 of each instrument). So if each instrumentalist was labeled 1-14 for their respective instrument, the first session could be band 1: all instrument 1s, band 2 all instrument 2s etc. 
Is it possible for each band in each session to be unique? I.e. Trumpet 1 doesn't play with clarinet 1 more than once (and doesn't play with any other instrumental person from other instruments more than once). I am able to brute force this with 3 or 4 sessions but past that it gets difficult. 
While I have a math degree and took some combinatorics classes, I am definitely not an expert. I think this might fall under Combinatorial Design but  I am not really sure. 

Yes, you can set this up and solve as an integer program. If I'm understanding your requirements correctly, it could be something like the following. (Then this could be solved using a variety of optimization software. E.g., SCIP in python)
Sets:
$S$ Set of sessions
$B$ Set of bands in each session
$P$ Set of people
$I$ Set of instruments
Parameters:
$c_{pi} \in \{0,1\}$ if person $p$ plays instrument $i$, $0$ otherwise
Variables:
$X_{psb} \in \{0,1\}$ if person $p$ plays in band $b$ in session $s$
$Y_{pp'sb} \in \{0,1\}$ if person $p$ plays with person $p'$ in band $b$ in session $s$
Formulation:
min $0$
subject to:
$ \sum_{p \in P} c_{pi}X_{psb} = 1, \forall s \in S, b\in B, i \in I$
$ Y_{pp'sb} \geq X_{psb} + X_{p'sb} - 1, \forall p \in P, p' \in P \setminus \{p\}, s \in S, b\in B$
$ \sum_{s\in S}\sum_{b \in B} Y_{pp'sb} \leq 1, \forall p \in P, p' \in P \setminus \{p\},$
I believe you're looking for any feasible solution that satisfies your constraints - so the objective doesn't matter. The first constraint set requires there to be exactly 1 instrument of each type in each band. The second constraint set assigns an indicator variable if a person has been assigned to the same band as another person in the same session. The third constraint set only allows you to assign 2 people to the same band once.
