We need to select 7 musicians with at least 2 pianists, at least 1 guitarist, and more violinists than guitarists. Let's consider the possible scenarios:

1. 2 Pianists, 3 Violinists, 2 Guitarists:

   \(\binom{8}{2} \times \binom{4}{2} \times \binom{6}{3} = 28 \times 6 \times 20 = 3360\)

2. 2 Pianists, 4 Violinists, 1 Guitarist:

   \(\binom{8}{2} \times \binom{4}{1} \times \binom{6}{4} = 28 \times 4 \times 15 = 1680\)

3. 3 Pianists, 3 Violinists, 1 Guitarist:

   \(\binom{8}{3} \times \binom{4}{1} \times \binom{6}{3} = 56 \times 4 \times 20 = 4480\)

4. 4 Pianists, 2 Violinists, 1 Guitarist:

   \(\binom{8}{4} \times \binom{4}{1} \times \binom{6}{2} = 70 \times 4 \times 15 = 4200\)

Summing the number of ways for all scenarios gives:

\(3360 + 1680 + 4480 + 4200 = 13,720\)