|
Benade and French's 1965 paper “Analysis of the Flute Head Joint” [Benade1] is one of the classics of flute acoustics. I was curious about what their calculations might say about octave stretch. I digitized their head joint measurements (except the Quantz) from their Fig. 9. Their "Modern" curve is an average of 12 flutes "including instruments made by Powell and both the Haynes brothers". From personal interest I have also added (as series "D") a recently made (ca 2018) student flute of good quality that I own.
In the original diagram, there are dual range axes; one is bore diameter and the other is bore perturbation. The cylindrical bore diameter is shown as 19mm for all heads; unless they were all really 19mm they were probably normalized.
The Rudall Carte profile looks quite different from the others. Benade mentions only that it is an ebonite flute and that "The modern Artley flute has a head very similar to the Rudall Carte design...".
Their Fig. 10 is a calculation of length correction for the measured head joints (i.e. the effect of the head joint taper vs. frequency). Using my implementation of Benade's equation I have recalculated those values from the digitized measurements. This serves as a useful check on my code. A visual comparison of this chart and the original Fig. 10 shows that they indeed agree closely.
These units (length change and frequency) are good for physicists but not for musicians. Here is the same information, but converted into more musically useful units and limited to the flute's range. I've also added the "Modern" head which is not included in the original Fig. 10:
And now we get to my original goal, which was to calculate the octave stretches for these headjoints:
The Boehm and Modern tapers have very similar effects. The "D" taper gives a less stretched octave by about 5 cents. They state the Rudall Carte "...certainly gives a rather stretched octave when played in the French manner" and this graph certainly supports that.
...and here are the third octave stretches:
These stretches are about twice as large as the second octave ones. As mentioned in the introduction, these stretches are calculated relative to the first octave.
| 
