An Unsuspected Function of the Violin's Pegbox.
In the process of making a violin, when the time comes to position the pegs, the luthier refers to old models and ways. However, how exactly was their placement originally calculated?
Of course, one must take into consideration the practical use of the tuning pegs: they should allow the player to have enough room to tune the instrument comfortably. But is this the only criteria on which we should base their positioning?
The violin maker usually calculates the afterlength (the distance between the bridge and the tailpiece) to 1/6 of the vibrating string length in order to optimize the overall timbre.
I realised that the distance from the D peg to the top nut is the same as that of the afterlength, i.e., also equal to 1/6 of the vibrating string. Harmonic relations between the vibrating string section and the peg-box section can also be found for the other strings, see Figure 1, in which the relations for the Stradivari “Messiah” are shown. The G string peg-box section shows a 1/20 relation, the A string slightly longer than 1/5, and the E string 1/10.
The length from the A peg to the top nut is often slightly longer than 1/5 of the vibrating string. It is likely that the usual measurement has been modified in order to accommodate more space for the player to manipulate both the A and the D pegs.
But here is another observation: when the placement of the A peg is low and the string touches the D peg, the latter acts as a nut. The distance between the A string touching on the D peg and the top nut again equals to 1/6 of the vibrating string length.
Even if the Baroque string length is taken as reference, the proportions are still approximately the same, as the length of the pegbox string section should vary from 0,5 to 1 mm compared with the modern string length .
The idea to build a string instrument in such a way that the unused string sections are in a harmonic relation to the main section is also realized in the Steinway grand pianos, for example, see the study of Öberg and Askenfelt*. In Steinways, this is called “duplex stringing”.
* Öberg, F. und Askenfelt, A.: Acoustical and perceptual influence of duplex stringing
in grand pianos. Journal of the Acoustical Society of America, 131:856–871, 2010.
From the present discoveries, it seems that the violin used a very similar approach to stringing already at the beginning of its conception.
The positioning of the pegs, however, varies slightly depending on the individual instrument. Moreover, when this concept is applied, the G peg might look abnormally high in the pegbox. Some have been bushed and placed lower.
In any case, on the well-preserved Amati and Stradivari instruments, it is quite clear that the positioning of the pegs follows the described principle of proportions.
Peg box string length proportions in comparison with the main vibrating string length.
While studying the sounds produced by the strings in the pegbox, I unexpectedly found that the frequencies of the string sections from the top of the peg to the back edge of the top nut is harmonically related to the vibrating open string. In many cases, specific intervals could be distinguished by ear.
To understand these observations, the harmonic relations between the open string and the pegbox section are discussed in the following. The lengths of the pegbox sections are measured from the top of the peg to the edge of the nut.
The pegbox length of the G string is equal to 1/20 of the vibrating string. Based on the fundamental frequency of the G string, 196 Hz, this corresponds to a fundamental frequency of 3920 Hz of the pegbox section of the G string. Moreover, the relation to certain harmonics of the G string may be of interest: with a ratio of 2/1 (=20/10), the pegbox section is in an octave relation to the tenth harmonic of the open G or in a fourth (4/3=20/15) or major third (5/4=20/16) when the 15th or 16th harmonics are regarded respectively.
For the E string, the pegbox section has a length 1/10 of the vibrating string. As the open E string has a fundamental frequency of 660 Hz, the pegbox fundamental has a theoretical frequency of 6600 Hz. This again could be regarded as an octave in relation to the 5th harmonic of the E string, for example, or as a major third in relation to the 8th harmonic of the E string.
The length from the D peg to the top nut is equal to 1/6 of the vibrating string.
1/6 of a string generates a fifth, for example, when compared to the 4th harmonic of the open string (the ratio is 6/4=3/2).
As the pegbox section can be regarded as a resonator coupled to the vibrating string, the described placement of the pegs may influence the vibrating string, especially the open strings, which could enhance the overall tone. But these are speculations that will have to be confirmed.
In conclusion, this research on the placement of the pegs may be the incentive for additional understanding of the function of the pegbox.
André Theunis / Violin Maker Brussels, Belgium www.violons.eu
Dr. Gunnar Gidion / Physicist-Acoustician Albert-Ludwigs-Universität Freiburg