Method

The techniques developed in the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING for assessing sound radiated by violins are based on the following steps:

1. Spatial measurement of the radiated sound
2. Averaging of the RMS power levels: Resonance profile of the instrument
3. Determination of levels of the harmonics for each playable note
4. Representation of the excitation of the basilar membrane (inner ear)

Measurement of the radiated sound is based on a dual-channel Fourier transform. The ratio of the radiated sound pressure (p) to the excitation force (F) is determined. The analyzed quantities are displayed as absolute values. Both quantities (p and F) vary with frequency. This frequency dependency is normally represented as a function (meaning a diagram curve). Based on the shape of the curve, it is possible to read off the transfer factor for p/F ("power of the radiated sound") (y axis) for each frequency that is shown (x axis). This curve is known as a transfer function.

The transfer function which is measured using this setup and computed using a Fourier transform is not sufficient alone for comprehensive representation of the sound radiated by the instrument. This is due to the highly directional radiation characteristics of the instrument, i.e. the instrument does not radiate sound uniformly in all spatial directions. The curve shape of the transfer function is thus highly dependent on the direction of the measurement and where in the room it is made. To obtain a representative result for the sound which is actually radiated by the instrument, it is necessary to measure a large number of transfer functions in different spatial directions. The best approach involves representing these transfer functions as a "colored map" instead of as individual curves.

The following figures show such "sound radiation maps" in the form of contour diagrams. The magnitude of the level p/F (dB) is plotted using a color coding system (red = strong, blue = weak). The x axis represents the frequency. Now, however, the y axis is used to show the radiation angle around the instrument. This is based on an imaginary circle around the instrument with the axis of the circle in the direction of the violin's neck. The height of the circle is at the position of the stop. 0° and 360° correspond to the direction perpendicular to the top plate, and 180° is perpendicular to the back plate.

If we examine the color in the horizontal direction on these contour diagrams, the changes in color will reflect the frequency dependency of the sound radiation (for a single radiation angle). In the vertical direction, the changes in color reflect the directional dependency (for a single frequency).

In other words, if a vertical band does not change color, this means that the sound radiation is non-directional at this frequency. As a basic rule, this is true only in the low-frequency range (left). On the other hand, a "fracturing" of the sound radiation clearly occurs in the high-frequency range: The resonances "fire off" the sound with greater power in certain spatial directions. This is the main source of the "liveliness" and "spatiality" of the sound. Thanks to the fracturing and resonance strength in this frequency range (with many small "islands of coloration"), the musician can use vibrato and bowing variations to play with the directional, spatial "fire" in the resonances.

These "sound radiation maps" make it possible to clearly visualize small sound differences between different instruments. In addition, different states of the same instrument can be compared during restoration work or tonal adjustments, for example. This enables conclusions about the effects of changes in the thickness graduation, soundpost displacement, bridge, etc. on the sound of the instrument.

The following figure shows how it is possible to derive many characteristic traits from a "map" of the sound radiation. The instrument we analyzed in this case was the "Isserlis" Montagnana cello from the year 1740. The resonance profile of the sound radiation is shown under the colored contour diagram. This diagram was produced by averaging the RMS power levels over all of the spatial directions. The heights of the individual resonance peaks in the resonance profile are thus a measure of the magnitude of the sound radiation. The colored bar which is associated vertically with each resonance peak is in turn a measure of the spatial directivity characteristic of that peak. The magnitude of the sound radiation and the directivity characteristic both result from the associated mode shape for any given resonance. (For more information about measuring mode shapes and how they are relevant, see modal analysis.)

Below the gray resonance profile, seven mode shapes of the instrument are shown for illustration purposes. The view towards the top plate (left) and the bottom plate (right) is from the outside. The gray rings are the contour lines of the amplitudes. The black-gray regions are vibrating with a 180° phase shift with respect to the white-gray regions. The blue dashed lines refer to the resonance peaks they are associated with. The two low main corpus resonances T1 and B1 both produce similarly strong sound radiation. The fact that for the T1 resonance the large-scale top plate amplitudes in the region of the bass bar are greater than the back plate amplitudes is also reflected quantitatively in the color variation of the associated vertical color bar for the sound radiation. This is particularly clear in the case of the plate resonance at 440 Hz. Here, the maximum sound radiation in the direction of the strongly vibrating top plate is about 10 dB above that which is occurring in the direction of the back plate. Of course, the cello has a significantly greater number of mode shapes than the seven shown here: In the range up to 1500 Hz, we measured 61 individual modes. Each of these modes has its own character in terms of the frequency, mode shape, damping, radiation level and directional characteristic. It is the interplay between the many modes which determines the acoustic character of the instrument.

Above: "Sound radiation map" for a cello by Domenico Montagnana (1740) "Isserlis".
The frequency (Hz) is plotted on the x axis and the y axis represents the directional angle: 0° = perpendicular to the plane of the top plate (normal direction); 90° = parallel to the plane of the top plate and perpendicular to the neck in the direction of the bass bar side; 180° = perpendicular to the back plate plane (normal direction); 270° = parallel to the top plate plane and perpendicular to the neck in the direction of the soundpost side. The color scale reflects the sound level divided by the excitation force. Note: Absolute value (re 1 Pa/N) in dB = Caption color (dB) + 76 dB.
If we follow the color in the horizontal direction on these contour diagrams, the changes in color will reflect the frequency dependency of the sound radiation (for a single radiation angle). In the vertical direction, the changes in color reflect the directional dependency (for a single frequency).

Middle: Resonance profile of the sound radiation using RMS power averaging.

Below: Mode shapes for a few modes. They reflect the vibrations of the instrument corpus and are the source of the sound radiation that is produced.
 

Our sound analysis of the Montagnana instrument helped to reveal the following acoustic characteristics:

1. The Helmholtz resonance (air resonance: The air in the f-holes vibrates on the air buffer of the air enclosed in the corpus): Perpendicular (0°) to the top plate plane (max. level), the level of this lowest main resonance of the cello (just below 100 Hz) is almost 10 dB above the minimum level perpendicular (180°) to the back plate plane.
2. The main corpus resonance radiated at 217 Hz is relatively non-directional whereas the corpus resonance at 194 Hz is highly directional. Here, the maximum at 105 dB is in the angular range 0 to 90°, i.e. in the direction of the top plate on the bass bar half whereas the minimum at 77 dB lies at an angle of 150° in the direction of the back plate (also on the bass bar side).
3. In the octave range between 200 Hz and 400 Hz, the cello exhibits many very effective and largely non-directional resonances. This is seen in the form of the strong, i.e. red/orange-colored and constant vertical bands.
4. In the resonance range from 400 Hz to 1000 Hz, directional beams with directional level differences of over 20 dB in some cases begin to appear.
5. The resonance range between 1000 Hz and 3000 Hz is characterized by a huge number of small "level islands" within the sound radiation map. Due to the large slope of the resonance peaks (low resonance damping), the levels of the overtones which fall in this resonance region are highly vibrato-dependent. This causes the played notes to be very lively and easy to shape.
6. Above 3 kHz, a strong lowpass filtering effect begins to take over. This steep level dropoff in the range around 3000 Hz is very desirable in order to achieve a "velvety" sound (some musicians speak of "Italian sand") that is still not too harsh.
7. After averaging the RMS power levels, the total radiation (averaged over all of the frequencies) in the direction of the top plate (angles of 270° to 90°) is almost 2 dB higher than the radiation in the direction of the back plate (angles of 90° to 270°).

It is also possible to display the spatial sound radiation of the instrument for any arbitrary frequency. Here, a network diagram turns out to be very useful. The information shown above in the form of colored "sound radiation maps" is now reproduced here in the form of directional diagrams (level as a function of angle) for many of the resonance peaks. Here again, the described characteristics are plainly evident in the form of directional maxima, directional beams, etc.

Here again, the directional characteristics of individual resonance peaks in the sound radiation are presented using the network diagrams that were explained above. The following figure shows the sound radiation network diagram corresponding to the "acoustic map" for the Guarneri del Gesu described above: