Simon Polak: Early Flutes
The position of the cork of a traverso is essential for the intonation and the tone quality. It is not straightforward where this should be and often problems simply come from a mistake in this position.
Let me first give briefly what the rules are and roughly why. Then I will further give some basic reasoning behind those simple rules.
The cork should be positioned such that d2 and d3 are exactly an octave, not necessarily d1 and d2.
Then the intonation and tone quality should be optimal. The lowest d on most (original also) traverso then is a bit too low. It should not be tried to correct this with the cork position. But it is easily corrected by embouchre.
Simplified the reason is the following. If the cork is moved 1mm. d1 changes in the order of 1.5 cent, d2 changes in the order of 3 cents and d3 in the order of 6 cents. I will be more precise in the following. If we change the cork 3 mm. d1 goes in the order of 4.5 cents, d2 in the order of 9 cents, d3 18 cents.
Let us consider an example (in reality the cents depend non linearly on the cork position but because the changes are small with respect to the wave length this is also practically linear).
Suppose that the cork is in the position where d2 and d3 are an exact octave and d1 is e.g. -5 cents without compensating. The cork would have to move 3 mm. to make d1 5 cents higher. But then d2 is 10 cents higher and d3 20 cents higher. This gives a d2-d3 that is 10 cents off. And on top of that d1 is still 5 cents low with respect to d2!
Experience shows that with many originals the rest of the intonation and the tone quality are optimal with d2-d3 an exact octave. This further illustrates that this is the right choice.
We could easily change the intonation of the traverso to make d1 higher but it effects the character of the whole flute negatively.
The above rule is correct but the numbers (1.5 cent is 1 mm.) need some further consideration.
In the following I will try to be a bit more precise although an exact calculation would not be simple as will be seen. First I will look at it as if the cork position change has exactly the same effect on the sounding length.
The above more precise.
In the following we take the a1 to be 415Hz.
Then the frequency of d1 is 415*3/4=311.25 Hz because a-d is a fifth. Let us take the speed of sound 343 m/sec. Then the wave length of d1 is 1.102 m. So d2 is 622.5 Hz and has a wave length of 0.551 m. and d3 is 1245 Hz and is 0.275.5 m. A shortening of the wave length of 1 mm by moving the cork position therefore is a freqeuncy of 311.535 Hz for d1. Translating this in equal temperament cents gives 1.6 cents (for the experts: 1200*2log(1102/1101), yes the 343 cancels)
For d2 the same story gives 3.15 cents and for d3 it is 6.3 cents (again for the experts Taylor development shows that because 1 mm. is small it is almost linear).
For a shift of 3 mm. this gives for d1 4.7 cents, for d2 9.5 cents for d3 19 cents.
This is not correct because the mouth hole stays in the same position and the cork end is not a node. The graph on p 494 in [1] shows this. I am simply not able to adapt the calculation for this.
Everybody can do the practical experiment by pushing in the cork for instance 3mm. The d1 will hardly change but d2 will and d3 about double.
[1]Arthur H. Benade Fundamentals of Musical Acoustics. Dover 0-486-26484-X
