Examples of psychoacoustic sound analysis of bowed stringed instruments: Comparison of violins by Antonio Stradivari (1712) and Joseph Guarneri del Gesu (1733); Comparison of cellos by David Tecchler (1719) and Stradivari-Lott; Comparison of cellos by Francesco Gofriller and Domenico Montagnana (1740); Viola by Sannino (1906); Results of tonal optimization of a violin based on the soundpost and bridge; Comprehensive tonal adjustment of a cello (Stradivari-Lott) including a new bass bar, soundpost and bridge.
The method of psychoacoustic evaluation of sound radiation developed by the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING is a diagnostic tool which allows visualization of specific sound differences and peculiarities of bowed stringed instruments. The following examples show that with bowed stringed instruments, we need to talk about the specific character of the instrument since a simple "good" or "bad" is never adequate.
Example 1: Difference in excitation of the inner ear. The violins are a Stradivarius (1712) and a Joseph Guarneri del Gesu (1733).
These violins are first-class representatives of the old Italian school. We can see that the Stradivarius produces much stronger excitation in the middle range of the basilar membrane while the Guarneri produces stronger excitation in the low-frequency and high-frequency ranges. It is probably reasonable to talk about typological differences here. Further studies of other works by these two masters are planned.
Typological differences between violins by Stradivari (red coloring) and Guarneri del Gesu (blue coloring) based on a display of the different excitation patterns of the inner ear. The colors clearly show which instrument produces stronger excitation of each region of the basilar membrane of the inner ear.
Example 2: Comparative sound analysis of a cello by David Tecchler (Rome 1719) owned by the Bavarian State Orchestra and a cello by Stradivari-Lott (top plate by Antonio Stradivari, back plate and ribs by J.F. Lott).
Resonance profiles of two celli: (D. Tecchler: red; Stradivari-Lott: blue). What is plotted is the average RMS power level (for various spatial directions) which is the ratio of the radiated sound pressure to the excitation force.
The above figure shows the resonance profiles (with the RMS power levels averaged over the different spatial directions) of the sound radiation for the two celli. It is clear that the Tecchler cello has two strong corpus resonances in the low-frequency range at 156 Hz and 173 Hz. In the Stradivarius cello, the comparable corpus resonances are at significantly higher frequencies of 204 Hz and 223 Hz. These acoustic differences are evident when listening to these instruments: Due to its "bass power", the Tecchler cello is considered as one of the best sounding celli at the Bavarian State Orchestra. The high levels in the fundamental range (100 Hz to 200 Hz) are noteworthy as they are considerably higher than the comparable levels in the Stradivarius cello. Due to its different design, the Stradivarius cello is a more "tenory" instrument with a middle range that stands out. We can also see this in quantitative terms in the resonance profile: In the frequency range between 2 kHz and 3 kHz (which is responsible for the brilliance of the tone), the Stradivarius cello exhibits resonance levels that are up to 10 dB higher than the Tecchler cello. We can also see the differences in tonal color if we compute what is known as the spectral center of gravity: For the Tecchler instrument, it is at 252 Hz while for the Stradivarius-Lott instrument, it is at 355 Hz.
The spectral center of gravity is defined as follows: The sums of the power components of the resonance profile above and below the frequency of the spectral center of gravity should be equal. For a higher spectral center of gravity, the instrument will radiate more high-frequency (as opposed to low-frequency) components. A high spectral center of gravity will thus be associated with a brighter sound and a low spectral center of gravity with a darker sound. The spectral center of gravity has units of Hz (vibrations per second).
The average RMS power level of the entire resonance profile has a value of 95.2 dB for the Tecchler, which is a good 2 dB greater than that of the Stradivarius-Lott cello at 93.0 dB.
The differences in design are obvious just from looking at a few of the dimensions: The Tecchler cello with a C-bout width of 240 mm has a significantly wider contour than the Stradivarius cello (211 mm).
Click here for photos of these two instruments
Psychoacoustic comparison of two celli: Tecchler (red); Stradivarius-Lott (blue). The colors clearly show which instrument produces stronger excitation of each region of the basilar membrane of the inner ear. A color difference of 1.0 (caption value) represents a doubling of the specific loudness. On the vertical axis, a five-octave chromatic scale is plotted and on the horizontal axis, the "frequency scale" of the inner ear (in ERB).
Here, we can see the psychoacoustic effect of the instrument resonances measured using the transfer function: The Stradivarius-Lott cello produces significantly stronger excitation of the entire middle region of the basilar membrane of the inner ear (frequency groups 6 to 18 ERB all exhibit blue coloration) whereas the Tecchler cello produces strong excitation in the low-frequency part of the basilar membrane (2 to 6 ERB exhibit strong red colors). The very high-frequency range undergoes stronger excitation by the Tecchler cello (18 to 25 ERB).
We can see the differences in the effect on human hearing in the color contour diagram shown above (difference in specific loudness level Ls). Those ranges in which the Tecchler cello produces stronger excitation of the inner ear are shown in red and those where the Stradivarius cello is stronger are shown in blue.
In the frequency range of the "a-formant" (600 Hz to 900 Hz) which is responsible for an "open" sound, the sound radiation of the Stradivarius resonances is up to 6 dB greater (4x sound power) in certain cases compared to the Tecchler cello. Due to this strong "resonance potential", a specific loudness that is almost twice as strong is produced to some extent in the related frequency groups of the inner ear. Vice versa, the specific loudness of the Tecchler cellos in the low-frequency fundamental range on the c-, g- and d-strings is almost twice as large as in the Stradivarius cello. (We also noted that instruments by Stradivari have a greater emphasis on the middle spectral range when comparing his violins, e.g. with violins by Guarneri del Gesu. We can possibly conclude that a very strong emphasis on the middle register is a typological trait of Stradivarius instruments.)
By adding up the specific loudnesses for all of the frequency groups, we obtain the overall loudness of the computed, bowed note. This loudness level in phon (which, like the specific loudness, takes into account the frequency-dependent perception of the human ear) is shown in the lower line graph in the figure. We can see that, despite the differences in tonal color between the celli as described above, the overall loudnesses of the two instruments are relatively similar. The computations show that when playing chromatically on one and the same instrument, maximum differences in loudness level of about 12 phons will occur. The two instruments have in common the substantially lower loudness levels in their low notes (c- and g-strings) in comparison to notes starting on the open d-string. This sort of "low-frequency weakness" is generally more problematic with the cello than the violin. This is due to the fact that the cello is significantly smaller than the violin in relation to the wavelength of the notes that are emitted. This problem is made worse by the fact that the sensitivity of the human ear drops off greatly at lower frequencies.
As we have seen from this example, we need to talk about the specific character of the instrument since a simple "good" or "bad" is never adequate.
Example 3: Tonal differences between two celli: F. Gofriller and D. Montagnana (1740). About the instruments: The "Gofriller" is a solo instrument played by Daniel Müller-Schott (Munich). The "Montagnana" is one of the most renowned and best sounding instruments by this master. It is played by Steven Isserlis (London). You can see pictures of both of these instruments in our photo gallery.
The following diagram shows the resonance profiles (based on the RMS power level averaged over different spatial directions) of the sound radiation for the two celli. The most significant differences are in the two main corpus resonances (T1 and B1). The eigenfrequencies as well as the sound radiation levels are higher in the Montagnana we studied than the Gofriller. Also apparent are the significantly higher sound radiation levels of the plate modes between 1 kHz and 2 kHz.
Comparison of the sound radiation resonance profiles. Domenico Montagnana (red) vs. Francesco Gofriller (black).
The excellent insight provided by the described psychoacoustic sound radiation evaluation technique is particularly clear in this case. The measured resonance profiles underwent further processing using the psychoacoustic evaluation software developed by the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING. The results are shown in the following figure:
Comparison of the specific loudness for two celli: F. Gofriller (blue) and Montagnana (red). The colors clearly show which instrument produces stronger excitation of each region of the basilar membrane of the inner ear. A color difference of 1.0 (caption value) represents a doubling of the specific loudness. We can see the differences in the effect on human hearing in the color contour diagram (difference in specific loudness level Ls). Those ranges in which the Montagnana cello produces stronger excitation of the inner ear are shown in red and those where the Gofriller cello is stronger are shown in blue. The acoustic superiority (greater sound volume, wider dynamic range) of the Montagnana cello is clear: Almost of the excitation area is colored in red. The Gofriller cello produces a greater specific loudness only around the 26th frequency group and in the range of the fundamentals of a few of the low-frequency notes.
Example 4: Sound analysis of a viola (Sannino, Naples anno 1906)
Method: Measurement of the sound radiation using impulse excitation and subsequent computation of the excitation patterns of the inner ear for each of the playable notes.
There are the following variable parameters:
- How large is the frequency shift of the vibrato?
- How far apart are the instrument and the listeners?
- What are the dynamics?.
Viola (Sannino, Bavarian State Orchestra). What we see are eight individual notes: The four open strings (on the left from top to bottom: c-, g-, d- and a-string) as well as the notes a half tone above the open strings (each to the right). Here, we can see a fundamental weakness of the viola: The ratio of the wavelength of the radiated sound to the size of the corpus is more of a problem with violas compared to violins. This explains the lack of effective resonances in the low-frequency range on the bottom string. Due to this lack of resonance in the low-frequency range, the fundamentals in the entire 1st position on the c-string exhibit almost no excitation. On the other hand, the strong resonances in the range of the d- and a-strings produce very effective radiation of the fundamental and first overtone. This produces very "sonorous" broadband excitation of the basilar membrane: This instrument sounds very "full" and "big" in this range.
The following contour diagram shows all 60 notes in the viola's playing range as specific loudness patterns (excitation of the inner ear). The x axis represents the 60 semitones in the chromatic scale and the y axis the "ear's frequency axis" (in frequency groups, ERB). The color caption represents the loudness values. Here again, we can clearly see the "fundamental hole" on the c-string. Not until we reach the g-string is there sufficient resonance in the instrument's resonance profile for the fundamentals. The dominance of the fundamentals can be seen in the lower, colored region which curves upwards.
Specific loudness pattern for a viola. Specific loudness perception based on excitation of the basilar membrane of the inner ear (frequency scaling of the horizontal axis in ERB); chromatic scale with 60 notes (vertical axis).
Example 5: Tonal optimization of a violin based on the soundpost and bridge (violin: Sannino)
In the "Tonal adjustment" section, there is a detailed description of how the sound was altered using the soundpost and bridge (Example 2: Vincenzo Sannino). We will now subject the same results to psychoacoustic evaluation for an additional check:
Sound differences in a violin based on visualization of the different excitation patterns of the inner ear: Example - Influence of the soundpost and bridge (setup).
Before (old setup): Red notes; After (new setup): Blue notes. A color difference of 1.0 (caption value) represents a doubling of the specific loudness.
Through tonal adjustment, we were able to achieve a considerable gain (particularly in frequency groups 22 to 32 ERB). This range is very important for the brilliance of the sound. We were able to boost the specific loudness by almost half. In the low-frequency range as well, we were able to achieve a clear increase in the specific loudness (by increasing the level of the two main corpus resonances).
As we can see from the following comparison curves for the overall loudness of all bowed notes as computed from the specific loudness (before vs. after), it was possible to achieve a sizeable gain, particularly on the e-string.
The +1 dB to +2 dB gain in the overall loudness is significant starting in the 2nd position on the e-string (fundamental frequencies starting at 1 kHz). The result was a noticeable improvement in the soloistic qualities of the instrument as well.
Example 6: A process check while making a "tonal copy" of a Stradivarius.
For a description of the procedure we use when making instruments that are oriented towards the resonance properties of a given reference instrument, see the article about tonal copies in our "Violin acoustics" handbook. Sound radiation analysis is an important acoustic monitoring tool for use during the tonal creation process in the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING. The results are subjected to psychoacoustic evaluation as part of a subsequent check and then documented with additional recordings.
Fig.: An acoustic check while making "tonal copies"
The red curve represents the frequency response of the sound radiation of the reference instrument (a violin by Antonio Stradivari from the year 1712). The black curve shows the frequency response of a tonal copy of this reference instrument made by MARTIN SCHLESKE.
The objective when making a tonal copy is to reproduce the main properties of the individual resonance profile in the newly constructed instrument. If we compare the two frequency response curves, it is clear that this was a success since the characteristic resonance peaks and valleys were replicated. Important characteristics include the frequency range and level of the Helmholtz resonance, the shape of the corpus resonances, the envelope (overall curve) of the plate resonances and the frequencies of the recesses between the various resonant regions.
At 94.0 dB, the average RMS power level of the tonal copy is slightly above that of the Stradivarius instrument (at 93.1 dB). At 1276 Hz, the spectral center of gravity is also slightly above that of the Stradivarius (at 1241 Hz).
For comparisons (resonance profiles, psychoacoustic excitation patterns, audio examples, etc.) between new instruments from the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING and the related (mostly old Italian) reference instruments, please read the article entitled "Tonal copies in the field of violinmaking" in our "Violin acoustics" handbook.
Example 7: Tonal optimization of a cello (Stradivarius-Lott) based on the bass bar, soundpost and bridge
Analysis work performed for this example:
- Modal analysis of the instrument in its playing state
- Modal analysis of the free top plate (Stradivarius) with the old (not original) bass bar
- Modal analysis of the free top plate without the bass bar
- Modal analysis of the free top plate with the new bass bar
- Analysis of the sound radiation before and after the tonal adjustment
Besides minor restoration work, the following was carried out:
- Fitting of a new bass bar
- Preparation of a new fingerboard
- Correction of the neck angle
- Preparation of a new soundpost
- Preparation of a new bridge
- Usage of other string types
Pictures of this cello by Stradivari and J.F. Lott can be found in our photo gallery. It has undergone extensive tonal work in the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING. We did nothing that would change the original character of the instrument. We did fit it with a new bass bar, soundpost and bridge. During this work, we performed comprehensive analysis of the instrument's acoustics and construction. The following figure shows an example of the many measurements that were required:
Example of the measurements performed during analysis of the sound radiation. Frequency response (bottom): Level (ratio of radiated sound pressure to excitation force) in the direction perpendicular to the top plate (0 to 12.8 kHz). Frequency response (top): Zoom (0 to 1 kHz).
The work we performed was successful in liberating up some of the considerable sound potential of this cello. The following figure shows the frequency response of the sound radiation before (red resonance profile) and after (black resonance profile) our extensive tonal adjustment work.
Frequency response of the sound radiation for a cello before (red) and after (black) our tonal adjustment. What is plotted is the average RMS power level (for various spatial directions) which is the ratio of the radiated sound pressure to the excitation force.
We achieved a sizable boost in the radiated sound (the average RMS power level increased by +3 dB). Particularly the increase in the low-frequency resonance range from 100 Hz to 300 Hz had a very favorable effect on the fundamental projection in the lowest register. Due to the increase in radiated sound in the resonance range between 1 kHz and 3 kHz, the brilliance of the sound when playing fortissimo down close to the bridge was improved along with the focus. We did not change the overall balance of the tonal color of this instrument, i.e. the overall sound did not become any brighter or darker. This can be heard subjectively and is also made clear by the fact that the position of the spectral center of gravity was more or less unchanged (before: 365 Hz; after: 355 Hz).
Modal analysis of the top plate of the Stradivarius cello. <o:p></o:p>