1. Frequencies in organ pipes

For a pipe of length $L$:

• Open organ pipe: $f_n = \frac{nv}{2L}, \qquad n=1,2,3,\dots$

• Closed organ pipe: only odd harmonics are present: $f_n = \frac{nv}{4L}, \qquad n=1,3,5,\dots$

2. Seventh overtone of each pipe

(a) Open organ pipe

In an open pipe, all harmonics are present.

• 1st overtone = 2nd harmonic
• 2nd overtone = 3rd harmonic
• ...
• 7th overtone = 8th harmonic

So, $f_{\text{open}} = \frac{8v}{2L} = \frac{4v}{L}$

(b) Closed organ pipe

In a closed pipe, only odd harmonics occur. The overtones are:

• 1st overtone = 3rd harmonic
• 2nd overtone = 5th harmonic
• 3rd overtone = 7th harmonic
• ...
• 7th overtone = 15th harmonic

So, $f_{\text{closed}} = \frac{15v}{4L}$

3. Ratio of frequencies

Taking the ratio of the seventh overtone frequencies of closed pipe to open pipe: $\frac{f_{\text{closed}}}{f_{\text{open}}} = \frac{\frac{15v}{4L}}{\frac{4v}{L}} = \frac{15}{16}$

Given, $\frac{a-1}{a} = \frac{15}{16}$

4. Solve for $a$

Comparing, $a=16$

So the required value is: $\boxed{16}$