The acoustics of violins

There are four strings in a violin. The musical notes of the four open strings are

G3, D4, A4 and E5 respectively. The range of the violin pitch is from G3 to E7. When

we play the violin, the string produces the fundamental frequency and many related

harmonic frequencies. The sting transmits the vibrations to the bridge, then to the

chamber and finally to the air which we hear of. The bridge and the chamber would

enhance the intensity of several specific harmonics if these harmonics match the

vibration modes on them. On the other hand, the intensity of several specific harmonics

would be weakened if these harmonics are correspondent to the forbidden mode of them.

Bissinger associated the enhanced frequency bands to the violin structure (Figure 1-

6).¹⁸ This figure is a combination of several different experimental methods, because

the influence of different violin structures can only be observed in different experiments.

Figure 1- 6. Summary schema for “building” violin radiative profile.

A0: the air mode, produced by the resonance of the violin plates. B1^- and B1⁺: the baseball modes, produced by the resonance of the violin plates. bridge-island: the resonance produced by the bridge and

the island. f rock: the rocking mode, produce by resonance of the bridge.¹⁸

There are two bands related to the chamber in the frequency ranging from 200 to

800 Hz. Bissinger used the modal analysis to observe these three modes, A0, B1^- and

B1⁺, which are called the signature modes. In this method, a hammer is used to hit the

bridge corner, and the mechanical response of the violin plates could be measured.¹⁹

Each violin has a unique set of signature modes. Many researchers also proved it and

explained it well. A0 is called the air mode and is produced by the resonance of violin

plates and the air in the violin body. If we suppose the string is pulled to the right. The

left foot of the bridge goes up, and the bassbar transmits the force to the top plate. The

right foot of the bridge goes down, and the soundpost transmits the force to the back

plate. At this time, the top moves up, and the back moves down. It expands the chamber

left, and the air flows out. In this kind of motion, the f-holes acts as the nose of the

violin, and this is why the motion is called the breathing mode. B1^- and B1⁺ are called

the baseball modes, because of the shape of the nodal lines. These modes are produced

by the resonance of the violin plates. In the previous research, there have two characters,

the bending and the breathing mode, with opposite phases.²⁰ The sum and difference

combination cause the B1^- and B1⁺ modes respectively. In Figure 1- 7, the left pair

shows the top and the back plate measurements while the right pair shows simplified

finite element computations.²¹ The two patterns are close to each other, proving the

correct explanation of signature modes. In brief, the signature modes were mainly

influenced by the vibration of the violin chamber.

In Figure 1- 6, the resonance bands from 800-5000 Hz are mainly influenced by

the bridge and the islands in the frequency. Although the full explanation of this

frequency range is complicated and nuanced, that the resonance of the bridge leads to

this feature is not controversial.²² As noted earlier, the soundpost is not directly linked

to the right foot of the bridge but slightly displaced from it. Besides, in this frequency

range, the inertia of the soundpost and the back plate become significant enough to

reduce the vibration. Therefore, the soundpost can be seen to be stationary and acts as

a fulcrum to allow a teeter-totter motion of the bridge and the top plate. This motion is

called the rocking mode and assigned around 3500 Hz in Figure 1- 6. Several possible

modes of bridge rocking have been proposed in Figure 1- 8.²³

Figure 1- 8. Sketches of the resonances of the bridge.

This figure is adopted from Bissinger et al., 2006.²³

Besides, Bissinger used zero-mass-loading laser scans to analyze the mechanical

response of the islands, finding that there was a vibration peak around 1200 Hz in the

Y-direction (perpendicular to the violin plate) and a vibration peak around 2500 Hz in

the X-direction (across the violin) (Figure 1- 9).²⁴ Compared with the bridge vibration,

when the energy is transferred from the bridge to the island, Bissinger found stronger

vibration on the island than on the bridge. It suggested some excitation due to the X

motion of the island happened (Figure 1- 10).²⁴ Hence, he assigned the band in 2500

Hz which was traditionally called bridge-hill (BH) as the bridge-island peak.

Finally, Bissinger used the far-field radiativity scans in an anechoic chamber to measure

the pressure response. Compared with the total mechanical response of the violin, he

obtained a parameter, the fraction of the vibrational energy radiated (FRAD), finding

that the most effective frequency was related to B1 mode.¹⁸

Figure 1- 9. The vibration of the islands.

Cross: X-direction. Triangle: Y-direction (perpendicular to the violin plate). Circle: Z-direction.²⁴

Figure 1- 10. Impedance ratio of bridge to island.

Cross square: X-direction. Solid square: Y-direction. BH is the abbreviation of bridge hill.²⁴

Based on the works of Bissinger, the knowledge of the violin acoustics is extended

far, Then, people want to know how the acoustics evolved and what is the relation

between the acoustics and the construction design or the material chemistry.

To get the answers, we go back to the time when the violin was created. After the

invention of the modern violin by Andrea Amati, the shape and form of violins has been

almost fixed. As time went on, the shape had evolved little. Chitwood used linear

discriminant analysis to study more than 7,000 violins. He separated luthiers by the

shape attributes and made hierarchical clustering (Figure 1- 11).²⁵ The four major

groups have been visualized (Figure 1- 12).²⁵ Nevertheless, as the saying of Chitwood,

the acoustics of violins are little influenced by the body shape, compared to other traits,

such as arching patterns, thickness distribution, and wood properties. The body shape

may be more influenced by the personal styles and the customer demands.

Figure 1- 11. Hierarchical clustering of violin shape.

Clustering based on averaged harmonic coefficients by prolific luthier (.45 violins). Four main clusters, named by prominent luthiers they contain, are indicated by color. Blue, Maggini cluster; red, Stradivari

cluster; purple, Amati cluster; green, Stainer cluster.²⁵

Figure 1- 12. Thin plate splines of major violin clusters.

Thin plate splines, deforming grids to transform violins from members of reference clusters (vertical) with those of targets (horizontal), are provided.²⁵

On the other hand, Alam tried to connect the structure evolution to the acoustic

evolution. After measuring the f-holes of 470 Cremonese violins, he found that the

elongation of the f-hole length was correspondent to enhanced radiated power for the

air resonance (Figure 1- 13).²⁶ Interestingly, the violins made by Stradivari have both

longer f-holes and stronger air resonance than the violins made by Amati.

Figure 1- 13. Time series of changes in radiated power and f-hole length.

The upper:The solid line is the total radiated acoustic air-resonance power. Colored shaded patches represents the standard deviations. The lower: The colored markers are the f-holes measured

from 470 Cremonese violins. The black line represents 10-instrument running average.²⁶

However, the acoustic evolution is still not clear and there are still many

unanswered questions. Some of the important questions are why the sound of some

violins are better than the others, and why many soloists prefer to play Stradivari violins.

The answers are hard to be found because the frequency response of the different violins

often look similar to one another. It made us start to think of how to use other methods

to analyze the frequency response and resonance properties, how to see all the peaks

Bissinger suggested in a single experiment, and how to simulate the perception in our