Introduction
Here I compare some experimental measurements of pitch vs bore perturbation with the predictions of two well-known algorithms by Benade and Nederveen. The experimental setup is described here.
In implementing code for Benade's W Curve calculation (described here), I wondered how accurate it was. Regarding accuracy, Benade says only (in a footnote) "This expression is accurate for a Boehm type cylindrical bore, and reasonably dependable when applied to a conical-bore instrument of normal proportions." I had also seen a description by Prairie of an impedance-based resonance calculation by Nederveen so I have implemented that as well. Both of these methods analyze bore resonance only, and do not consider the effect of tone holes, unlike more complete models such as that of Dickens or WIDesigner.
The Embouchure Correction
In these calculations, the effect of a blown embouchure (or fipple) hole is lumped into an equivalent added length (the embouchure correction). This is found by measuring the blown frequency, then calculating the length of an ideal tube that would resonate at the same frequency. Subtracting physical length and end correction from the acoustic length then gives the embouchure correction. One such calculation on my experimental setup gave a value of 1.4". For comparison, the embouchure correction for a flute has been given as 1.65" (Benade) and 2" (Boehm). This does vary with frequency, so it will be re-calculated when a different harmonic is being measured.
Results
One set of results is shown below.
The outer vertical dashed lines show the acoustic extent of the bore--i.e. with end corrections added--and the inner lines show the physical extent from the blowing edge to the end of the bore.
I made a change to the software having to do with the end correction. The end correction adds an additional length of tube whose diam is the bore diam and whose length is .3 * bore diam. But when the ptb sleeve is right up against the open end, the diam is less. The software change takes this into account, and uses the smaller ptb diam for the last point. This gives a result that is much closer to the actual measurement, and departs from the classic cosine-bell shape (shown as the "Cos" line). The measurements depart increasingly from the cosine shape starting at the zero crossing, which is interesting. I wonder if there an end effect which is extending into the bore...
Another aspect of the graph is that we can judge how accurate the .3 * diam formula is for the end correction by seeing if the peak of the measurements coincides with the peak of the predictions. They are pretty close; it looks to me like the measurement curve is very slightly offset to the right, which I think indicates the actual end correction is slightly larger...
Here are the measurements for the second harmonic:
The agreement between theory and experiment is fairly good here, except at the midpoint the Benade formula underestimates the change, where the Nedeveen estimate is quite accurate.
Large Perturbations
The next chart shows how the Benade and Nederveen methods compare when predicting the effect of an increasingly large perturbation. We are modeling a bore perturbation at a fixed location, then looking at the predicted pitch change vs. perturbation diameter.
The Benade formula is linear with respect to area, so we see a relatively straight line (since we are using diameter rather than area, there is a bit of a curve). The predictions agree for small perturbations, but diverge for large perturbations.
Below is a chart showing a 1st harmonic measurement for a very large perturbation where the area is reduced to 40% of the original.
Because the bore change is so large and abrupt, I would not expect either formula to be particularly accurate here, but the Nederveen formula has much better agreement with experiment.
Conclusions
Both Benade and Nederveen formulas produced accurate predictions for small-to-moderate perturbations, up to 15%. It was necessary to recalculate the end correction when the perturbation was right at the open end of the tube. The two methods give increasingly divergent results for larger perturbations; in this case the Nederveen method seems to be more accurate.
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