Conversion from physical quantities to psychoacoustic quantities. Representation of the sound radiation of bowed stringed instruments using colored "maps" showing the excitation patterns of the inner ear: Specific loudness (after MOORE) as a function of the frequency group (ERB) for chromatic excitation of the instrument. Presentations using the example of a violin by Guarneri del Gesu (1733) and a cello by Domenico Montagnana (1740).
The final analysis step used in sound analysis of bowed stringed instruments by the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING is intended to display the measured sound radiation of the instrument for visualization of the excitation pattern of the inner ear. In other words, the radiated sound is depicted in a further step using psychoacoustic perception quantities (instead of just physical stimulus quantities). The MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING has developed its own analysis software for this application which computes and displays the specific loudness pattern for all of the playable notes based on the measured sound. All of the masking effects of the human ear are taken into account and not just the specific filtering characteristics. The basis for this work is the LOUDAES program by the British researcher Brian Moore et.al. (see references).
The following figure shows the result based on the the example of various individual notes of a violin (Guarneri del Gesu 1733) which were computed from the sound radiation transfer function and then subjected to psychoacoustic evaluation.
The red curve represents what is known as the "specific loudness" which was computed from these discrete sound components. It shows the extent to which the fine hair cells are excited on the different regions of the basilar membrane in the inner ear. The excitation of the basilar membrane controls the neuronal activity which makes up our hearing. As we can see by comparing the "physical information" (black dots) with the "psychoacoustic processing" (red curve), excitation of the hair cells of the inner ear does not occur only at the "physically present" discrete individual frequencies but instead over a significantly more broadband "soft" range (in the form of the specific loudness curve).
The blue dot with the dashed line designates the "spectral center" of the specific loudness. The spectral center is defined as the point where the loudness components on each side of it are equally large. Its position on the frequency axis (ERB) is thus a measure of the tonal color equilibrium of the musical note that is represented. The further to the left it is, the darker the tonal color will be and vice versa.
By evaluating the specific loudness curve (shown in red), we can recognize essential characteristics of the musical notes of the instrument. For example, there is a clear dominance of the 2nd harmonic (= first overtone above the fundamental) in the area of the 1st position up to h1. Why? The frequency of the 2nd harmonic is in the range of the corpus resonances. Starting with c1 (3rd finger, 1st position on the g-string), the tonal color begins to undergo a significant change. The influence of the "heavy breathing" Helmholtz resonance begins to appear. Strong sound components are now added to the fundamental so the sound becomes "sonorous" and "full".
Our program for psychoacoustic evaluation of sound radiation by bowed stringed instruments finally displays the specific loudness of all of the individual notes in the form of a contour plot (next figure). The colors represent the specific loudness as a function of the frequency groups (ERB) of the basilar membrane (horizontal axis) and all of the notes in the chromatic scale (60 semitones on the vertical axis). To the left next to the colored contour diagram of the specific loudness, the overall loudness of these 60 chromatic notes is computed (which involves adding up the specific loudness values for all of the frequency groups).
Performed on a number of different violins, this type of psychoacoustic evaluation of sound radiation has yielded some remarkable results. We find it particularly illuminating that the sound differences between different violins are hardly apparent when the absolute loudnesses are formed. Clearly seeing sound differences requires a display of the difference in the loudness values between two violins.
The following figure represents a complete display of the loudness for all of the musical notes of an instrument based on the example of the violin by Guarneri del Gesu (1733) mentioned above:
Overall psychoacoustic evaluation of the sound radiation of a violin.
Line graph (above): Physical transfer function (FRF). The FRF is the input quantity for the algorithms used to compute the specific loudness. Color diagram: Specific loudness (in sone) for 60 semitones (chromatic scale). Line graph (left): Overall loudness levels (in phon) for the 60 semitones.
About the line graph (above): On the horizontal axis, the frequency is plotted in ERB (frequency groups 2 to 37) which corresponds to the local excitation on the basilar membrane of the inner ear. On the vertical axis, the level of the ratio of the sound pressure to the excitation force is plotted. The individual resonances are recognizable in the form of resonance peaks.
About the color diagram: The color scale Ls represents the value of the specific loudness S (in sone) such that an increase by a value of 1 corresponds to a doubling of the volume impression: S=2^Ls–k; where k = diagram offset (large loudness values are red, small loudness values are gray).
Here again, the horizontal axis (identical to the horizontal axis of the transfer function, FRF) represents the ERB frequency axis with frequency groups 2 to 37 (gray vertical grid lines); the vertical axis represents the 60 musical semitones in the chromatic scale (note#); black horizontal grid lines are arranged at intervals of a fifth: Note#1 = Open g-string; Note#8 = Open d-string; etc.
Dashed diagonal grid lines = Position of the harmonics: On the first (slightly curved) line to the left are all of the 1st harmonics (fundamentals). On the 2nd line, all of the 2nd harmonics, etc. (Note: The 1st harmonic is the fundamental, the 2nd harmonic the 1st overtone, the 3rd harmonic the 2nd overtone, etc. of the played note).
The white dots in the color diagram indicate the position of the spectral center of each musical note. The position in the horizontal direction is an indicator of the tonal color of the corresponding musical note.
About the line graph (left): Overall loudness levels (in phon) for the 60 semitones. The overall loudness results from adding up all of the partial loudnesses of the individual frequency groups (horizontal addition of the color values). In this manner, we can display the "loudness evenness" and the absolute loudness of an instrument over its entire playing range.
Another example of psychoacoustic evaluation of sound radiation is found below for the previously discussed cello by Domenico Montagnana (1740).
- Montagnana top plate
- Montagnana back plate
- Montagnana scroll
- Montagnana f-hole
- Corner of top plate
- Varnish Craquele
- Scroll
- Label
If we compare the different members of the family of bowed stringed instruments, then we will note the fundamental lack of power of any cello on its lower two strings compared to the violin. Due to the increasing "hardness of hearing" that humans have towards lower frequencies as well as the basic lack of resonant power for the cello in this frequency range, significantly lower excitation occurs in the inner ear at lower notes than does at higher notes. These effects are entirely incorporated into the described psychoacoustic evaluation of the sound radiation. (The "fireworks" begin further up.) Here, we have found a basic problem of the cello which can make it difficult for listeners to hear the cello during, say, a cello concerto with a full orchestra. Even with the best instruments (such as the renowned "Isserlis" Montagnana which we have discussed here), it is still not possible to produce enough sound in this low-frequency range on the bottom two strings.
Overall psychoacoustic evaluation of the sound radiation by a cello (Domenico Montagnana 1740). For more information, see also the text for the figure above.
- The loudness level Ls is represented by a color value. The relationship between the loudness level Ls, loudness S and offset k is given by the following equation: S=2^Ls-k. This means the distance between two colors in the diagram always corresponds to the same subjective loudness difference and a color difference of Delta-Ls=1 corresponds to a doubling (or a halving) of the perceived loudness. These computations take into account the specific perception characteristics of the human ear such as the frequency dependency of the loudness perception and the spectral masking.
- About the instrument: The low-frequency Helmholtz resonance (first resonance peak of the transfer function FRF at 3.4 ERB or 100 Hz) produces an excitation only when playing the 8th semitone (open g-string); see the bright blue arrow and circle. On the other hand, the influence of higher-frequency resonances is more diverse: For example, the distinct plate resonance in the range at 10 ERB (or 444 Hz) is responsible for the strong loudnesses in the fundamental range of Note# 31 to 35 as well as in the range on the open a-string (Note# 22) for the first overtone with its strong presence in each case. The same applies to the second overtone in the range on the g-string (Note# 15); see the three bright-blue circled loudness regions.
- In other words, the resonance strength of the cello in the higher frequency groups benefits a number of musical notes, namely the fundamental range of high notes as well as the overtone range of low notes. We can do the same for each of the instrument's resonances by going from the resonance peak of the transfer function (FRF) vertically down into the colored loudness diagram to check which of the 60 semitones are benefited by the selected resonance (whether in the fundamental range, i.e. on the first diagonal dashed grid line to the left which is labeled "Fundamental" or in the overtone range, i.e. on diagonal curved grid lines lying further to the right).
Note: Due to the lower specific loudness S of the cello (compared to the violin), the offset k in the caption to the figure was increased from k = 2 to k = 2.5.
The next figure shows the corresponding individual display of the specific loudness (colored contour diagram) for a cello by F. Gofriller. Here again, the basic problem of the cello is that the low-frequency range of the basilar membrane does not undergo sufficient stimulation due on the one hand to the lack of effective resonances in this frequency range and on the other hand due to our increasing "hardness of hearing" at lower frequencies.
As can be seen from the color variations for the specific loudness in this diagram, the first two fifths (notes #1 ... #15) exhibit a low loudness value. Not only is this playing range lacking in fundamental resonance (see the drop in the loudness on the first dashed grid line of the first harmonic) but also the overall loudness (see the diagram to the left) is about 10 phon lower than the overall loudness that is present starting with the d-string.
- Gofriller Decke
- Gofriller Boden
- Gofriller Schnecke
The next figure shows the corresponding individual display of the specific loudness (colored contour diagram) for a cello by F. Gofriller. Here again, the basic problem of the cello is that the low-frequency range of the basilar membrane does not undergo sufficient stimulation due on the one hand to the lack of effective resonances in this frequency range and on the other hand due to our increasing "hardness of hearing" at lower frequencies.
As can be seen from the color variations for the specific loudness in this diagram, the first two fifths (notes #1 ... #15) exhibit a low loudness value. Not only is this playing range lacking in fundamental resonance (see the drop in the loudness on the first dashed grid line of the first harmonic) but also the overall loudness (see the diagram to the left) is about 10 phon lower than the overall loudness that is present starting with the d-string.
The next figure shows the corresponding individual display of the specific loudness (colored contour diagram) for a cello by F. Gofriller. Here again, the basic problem of the cello is that the low-frequency range of the basilar membrane does not undergo sufficient stimulation due on the one hand to the lack of effective resonances in this frequency range and on the other hand due to our increasing "hardness of hearing" at lower frequencies.
As can be seen from the color variations for the specific loudness in this diagram, the first two fifths (notes #1 ... #15) exhibit a low loudness value. Not only is this playing range lacking in fundamental resonance (see the drop in the loudness on the first dashed grid line of the first harmonic) but also the overall loudness (see the diagram to the left) is about 10 phon lower than the overall loudness that is present starting with the d-string.
Overall psychoacoustic evaluation of the sound radiation of a cello (Francesco Gofriller) makes clear the fundamental problem of this type of instrument (compared, say, to a violin; see the text for details).
It has been used to carry out many studies of construction parameters which can significantly influence the sound of the violin. These parameters are as follows:
- Bridge
- Strings
- Soundpost position
- Wood types
- Fingerboard thickness graduation
- Design differences
- Variations in the plate thickness graduation
- Cutting a Violin Bridge
Sound analysis has proven to be an effective tool for monitoring the progress of an instrument under construction and as an optimization tool during tonal adjustment of valuable instruments and also during creation of tonal copies of reference instruments.
