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Pipe Scaling II

Sometimtes you're given the diameter of two C pipes and need to calculate the diameters of the pipes in between. Here's the equation:

DIA(n) = C1 / [2^((n-1)*ln(C1/C2)/8.3178)]

  • DIA(n) is the diameter of pipe number n
  • C1 is the diameter of the lower C, or pipe n=1
  • C2 is the diameter of the upper C, or pipe n=13

Here's an example of how to calculate the diamter of tenor G given the diameter of tenor C and middle C. Tenor G is the 8th note above tenor C, therefore n=8. Let's say the diameter of tenor C is 85.0 (C1 = 85.0) and middle C is 48.0mm (C2 = 48.0).

DIA(8) = 85 / [2^((8-1)*ln(85/48)/8.3178)]

Which gives you a diameter of 60.9mm for tenor G.

The above equation is the way I figured it out first. I've since realized there is a simpler equation:

DIA(n) = C1 * [(C2/C1)^(1/12)]^(n-1)

They give the same result so somehow one must simplify to the other.