As with a lot of things concerning flutes there is always some grit between theory and practice. The organ builders of Europe ran into this problem centuries ago. On the face of it, it would appear that the scaling principle for pipe organs should result in a doubling of the pipe radius every octave, 'doubling on the 12th pipe' as builders often call it. This makes all pipes geometrically similar. But it doesn't work, as the timbre across a rank of pipes begins to shift and the organ sounds funny. For a satisfactory scale the bass pipes must be narrower and the treble pipes wider.

The problem of finding a scaling rule that gives tonal coherence and balance across a rank of pipes is of central importance in organ building and one to which the great builders have found satisfactory empirical solutions. A scaling with doubling at the fifteenth to eighteenth pipe is generally satisfactory for pipe organs.

Now back to our shak. To make a replica pitched an octave lower and having the same timbre, how much should the bore be increased? Not 2, but 1.78 times. This is about the same as doubling between the fourteenth and fifteenth pipe. When scaling shakuhachi down one note change the length by 1.0595 and the bore by 1.0493. To scale to higher pitches use the inverse of these values. The length changes by about 6% per note and the bore about 5%.

For the mathematically curious: 1.0595 is the twelfth root of 2 and 1.0493 is the twelfth root of 2 raised to the 5/6ths power. 1.78 is 1.0493 raised to the twelfth power.

Anyway, the point is that shakuhachi scale using different factors for length and bore.

The need for a separate bore factor arises because a flute loses energy in two ways: out the end of the pipe and into the walls. The loss out the end of the pipe is affected by frequency, while loss to the walls isn't. So what we're really talking about is the fact that frequency effects one loss more than the other. And of course length is what determines frequency, so doubling the length cuts the frequency in half, effecting the end loss more than the wall loss.

Let's bring this discussion closer to home. It may not have occurred to you or perhaps you've never really listened but the timbre of the first and second octaves isn't the same. Same flute, same length, same bore diameter, so what's different? The frequency. The second octave is the same as building a replica with measurements divided by 2. From above we already know that it would play with a brighter timbre, so we know that the timbre of the second octave is brighter than the first. How much brighter? We can measure it indirectly. If the aspect ratio of our flute is 30, the second octave would sound as if it had an AR of 33.7. For the second octave to have the same timbre as the first, our flute's bore diameter would have to magically expand by 12% whenever we shifted to second octave.

The following graph plots the theoretical relationship between length and bore diameter in red and the practical in green. It extends an octave below and above D. The green line indicates proper bore diameters which will result in a timbre matching that of our D shakuhachi.