|We've finally derived formulas which incorporate much of the foregoing pages within simple mathematical expressions. When the energy losses in a flute (open end rate, wall losses, boundary layer, etc.) are balanced, a single formula appears. It's unique, in that, it strictly defines the length/bore/hole parameters over a huge range and at the same time contains an inherent equivalent timbre. Use of the formula guarantees your shakuhachi will operate with fundamental acoustical balance--a subtle, yet critical characteristic. Further, it's sound will be of the timbre distinctive to optimal bore geometry--at any length and in any key. For metric measurements only.
General, all purpose metric (mm) equation:
Length = (186500/ Hz) - 5 Bore
You can do these calculations at Google. Just put it into the search box and click search.
Example: Enter (545/17)^(5/6) into the search box, click and you'll get the optimal bore diameter for a 1.8 shakuhachi in millimeters. (545 / 17)^(5 / 6) = 17.9868999
For more Google calculator instructions.
How many octaves should/could your flute play? Aspect Ratio (bore length divided by bore diameter) is the greatest determinate of note range.
AR / 12 = Range (in octaves)
So, a bore with an AR of 24 should play two octaves, an AR of 30 will support 2.5 octaves and so on. Consider this formula an approximation.
Here's another set of formulas--from the Dark Side.
These are approximations which will produce flutes a few percent darker (timbre-wise) than the equations above.
Make all measurements in millimeters.
Comparison of the 'Light' and 'Dark' formulas
See The Synthesis for a final flute design.