Benade was wrong about there being no negative effects on intonation as players have long suspected.
Thank you for that citation of a new study by Nederveen. It is great to find someone else with an interest in musical instrument acoustics.
Going just from memory, I mistakenly applied Benade's comments about the effects of closed toneholes making the bore wider and longer to the effects of a "bend". I apologize for giving the wrong information. In reviewing "Fundamentals of Musical Acoustics", Benade mentions the acoustical effects of bends only with regard to brass instruments. On page 409 he writes
The more or less sharp bends that are made in the tubing of brass instruments to fold them enough for easy portability have and acoustical effect of their own. . . .For all these reasons instruments with many sharp bends act quite differently from their straighter cousins. The resonances can be shifted quite enough to be noticed in playing steady tone, and the beginnings of notes can be affected even more."
In reviewing "Acoustical Aspects of Woodwind Instruments" it is Nederveen who describes the effect of a sharp bend in a conical bore is to produce a bore that is both "shortened and widened". He writes, "This will cause unevenness in the tuning
when the bend is part of the oscillating air column in more than one register. It would be better to reduce a little both diameter and conicity in bends".[emphasis added] I think it is important to note that the sharp bend in the bow of a saxophone is normally part of the air column in only the low register.
In a specific reference to the saxophone on page 87 he writes:
In two places, tenor and alto saxophones have bends in the bore. The crook in the upper part of the bore shows very low curvature and does not exercise any appreciable influence. The reverse is the case for the sharp bend in the lower register. The length of the bend is about 12% of the acoustical length of the lowest fundamental. In Section 37 it was shown that such sharp bends correspond to a shortening and widening of the equivalent straight tube, where the differences may be of the order of 5% for the curved parts. This means that a perceptible increase of the resonance frequency is not unlikely.
He goes on to state, "no correction was applied, however, since . . . no accurate numerical values are available yet". Perhaps the new study that was cited provides this additional information.
Going back to the original statement:
"Curved saxophones are more difficult to play in tune due to the shortness in length as well as the fact that the sound needs to bend around the curve." I believe the science indicates that a) the "bend around the curve" only affects the intonation of the notes below the curve, and b) these effects can be compensated for by engineering the bell bow to have a slightly reduced diameter and conicity.
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