Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. What is $q-p$?

$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 $
To solve the problem, we need to find the smallest positive integer \( q \) such that there exists a positive integer \( p \) satisfying the inequality:
\[
\frac{5}{9} < \frac{p}{q} < \frac{4}{7}
\]

1. **Express the inequalities in terms of \( p \) and \( q \):**
   \[
   \frac{5}{9} < \frac{p}{q} < \frac{4}{7}
   \]
   This can be rewritten as:
   \[
   5q < 9p \quad \text{and} \quad 7p < 4q
   \]

2. **Manipulate the inequalities to find bounds for \( q \):**
   - From \( 5q < 9p \):
     \[
     5q < 9p \implies p > \frac{5q}{9}
     \]
   - From \( 7p < 4q \):
     \[
     7p < 4q \implies p < \frac{4q}{7}
     \]

3. **Combine the inequalities:**
   \[
   \frac{5q}{9} < p < \frac{4q}{7}
   \]

4. **Find the smallest \( q \) such that there exists an integer \( p \) satisfying the combined inequality:**
   - We need to find the smallest \( q \) such that:
     \[
     \left\lceil \frac{5q}{9} \right\rceil < \left\lfloor \frac{4q}{7} \right\rfloor
     \]

5. **Test values of \( q \) starting from the smallest possible integer:**
   - For \( q = 1 \):
     \[
     \left\lceil \frac{5 \cdot 1}{9} \right\rceil = \left\lceil \frac{5}{9} \right\rceil = 1, \quad \left\lfloor \frac{4 \cdot 1}{7} \right\rfloor = \left\lfloor \frac{4}{7} \right\rfloor = 0 \quad \text{(not valid)}
     \]
   - For \( q = 2 \):
     \[
     \left\lceil \frac{5 \cdot 2}{9} \right\rceil = \left\lceil \frac{10}{9} \right\rceil = 2, \quad \left\lfloor \frac{4 \cdot 2}{7} \right\rfloor = \left\lfloor \frac{8}{7} \right\rfloor = 1 \quad \text{(not valid)}
     \]
   - Continue this process until \( q = 16 \):
     \[
     \left\lceil \frac{5 \cdot 16}{9} \right\rceil = \left\lceil \frac{80}{9} \right\rceil = 9, \quad \left\lfloor \frac{4 \cdot 16}{7} \right\rfloor = \left\lfloor \frac{64}{7} \right\rfloor = 9 \quad \text{(valid)}
     \]

6. **Verify the values of \( p \) and \( q \):**
   - For \( q = 16 \) and \( p = 9 \):
     \[
     \frac{5}{9} < \frac{9}{16} < \frac{4}{7}
     \]
     - Check:
       \[
       \frac{5}{9} \approx 0.5556, \quad \frac{9}{16} = 0.5625, \quad \frac{4}{7} \approx 0.5714
       \]
     - The inequality holds true.

7. **Calculate \( q - p \):**
   \[
   q - p = 16 - 9 = 7
   \]

The final answer is \(\boxed{7}\)