Karl,
>It occurs to me that the rules you give
> here suggest that the there's a cross sectional area or volume effect at
> play here (i.e. that scaling for consistent timbre must be somewhat
> dependent on one or the other of those quantities rather than just bore
> diameter) i.e. that the scaling is more than a simple linear scaling.
The reason to scale with a bore factor other than 2 when doubling has to do
with energy loss in the pipe. The loss at the end of the pipe involves
cross-sectional area. And the loss to the walls involves total wall area.
Both calculations are computed using bore radius. Both are second degree
functions--that is functions of area. The bore scaling factor comes from the
ratio of these two functions. Thus no function survives which is second
degree (or greater) which would make it non-linear.
If the two loss amounts (end and wall) were equal when doubling that'd be
the end of the story--but they aren't, they're slightly different.
Because of this difference we need a bore scaling factor slightly smaller
than 2 (in this case 1.78) when doubling to keep the energy losses
equalized.
Nelson
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