The fine-tuned universe

Points of interest to violin and bow makers

The harmonic properties of a strings afterlength have been examined in the past - but what about the before-length, from the peg to the top nut? Andre Theunisand Gunnar Gidionfind some surprising results in their investigations.

IMAGES COURTESY J.&A. BEARE LTD

In the process of making a violin, luthiers generally refer back to old models and methods when it comes to positioning the tuning pegs. How their exact placement was originally calculated, however, is rarely questioned. The most important consideration, obviously, is the practical use of the pegs: the player should have enough room to tune the instrument comfortably. But is this the only criterion on which we should base their positioning?

The violin maker usually calculates the afterlength (the length of the string between the bridge and the tailpiece) to 1/6 of the vibrating string length, in order to optimise the overall timbre. When I started measuring the distances from the top of the pegs to the edge of the top nut, I realised that that of the D peg string was exactly the same as the afterlength, i.e.also 1/6 of the vibrating string. Harmonic relationships between the vibrating string section and the pegbox section can also be found for the other strings: see figure 1, in which the relationships for Stradivari’s ‘Messiah’ violin are shown. The G-string pegbox section shows a 1/20 relationship; the A string 1/5; and the E string 1/10.

WHEN THE STRING TOUCHES THE D PEG, IT ACTS AS A NUT

The length from the A peg to the top nut is often slightly longer than 1/5 of the vibrating string, such as on the c.1570 Andrea Amati shown in figure 2. It is likely that the measurement has been calaculated 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 1/6 of the vibrating string length.

Even if the Baroque string length is taken as a reference, the proportions still work out approximately the same because the length of the pegbox string section should vary from 0.5mm to 1mm compared with the modern string length.

FIGURE 1Pegbox string length proportions of Stradivari’s 1716 ‘Messiah’ violin, in comparison with the main vibrating string length

LINE DRAWINGS COURTESY ANDRÉ THEUNIS

FIGURE 2Measuring the pegbox string lengths of a c.1570 Andrea Amati violin gives similar proportions

The idea to build a stringed instrument in such a way that the unused string sections are in a harmonic relation to the main section is also realised in the Steinway grand pianos. In a 2012 study this was referred to as ‘duplex stringing’.

After examining the Andrea Amati violin, I concluded that violin makers were using a very similar approach to stringing even 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.

THE PEGBOX SECTION CAN BE REGARDED AS A RESONATOR COUPLED TO THE VIBRATING STRING

Wile 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 are harmonically related to the vibrating open string. In many cases, specific intervals could be distinguished by ear.

To understand these observations, we need to examine the harmonic relationships between the open string and the pegbox section. 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, 196Hz, this corresponds to a fundamental frequency of 3,920Hz 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 (or 20:10), the pegbox section has an octave relationship to the tenth harmonic of the open G. This would be a perfect 4th (4:3 = 20:15) or major 3rd (5:4 = 20:16) when the fifteenth or sixteenth 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 660Hz, the pegbox fundamental has a theoretical frequency of 6,600Hz. This again could be regarded as an octave in relation to the fifth harmonic of the E string, for example, or as a major 3rd in relation to the eighth harmonic of the E string.

The length from the D peg to the top nut is equal to 1/6 of the vibrating string, which generates a perfect 5th, for example, when compared to the fourth harmonic of the open string (the ratio is 6:4 = 3:2). From the A peg to the nut, the length is equal to 1/5 of the vibrating string. Here, a major 3rd can be conceived owing to the fourth harmonic of the open string.

In conclusion, as the pegbox section can be regarded as a resonator coupled to the vibrating string, it is possible that the described placement of the pegs might influence the vibrating string, especially the open strings, which could enhance the overall tone. But these are speculations that will have to be confirmed. We hope that this initial study of peg placement might provide an incentive for future research on the function of the pegbox.