Harmonic Structure

Representation of the harmonic structure of radiated sound through computation of the levels at the frequencies of the harmonics. Comparison between different quality violins (Stradivarius vs. a student instrument).

Some preliminary tests on different instruments have shown that it is possible to make conjectures about the subjectively perceived hearing impression based on the resonance profile. It is clear that the sense of hearing will be more accurately modeled the more closely the chosen form of representation corresponds to the musical situation. This is why the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING has developed a program which presents the resonance profile of the sound radiation in a "musical" way: The following three figures show the "musical" contour diagrams for the three violins presented above in terms of their "acoustic maps":

•    Antonio Stradivari, 1712 ("Schreiber")
•    Joseph Guarneri del Gesu, 1733
•    A basic student violin (ca. 1900)

A "musical" contour diagram showing the sound radiation for a violin by Antonio Stradivari (1712):
The abscissa has a chromatic scale while the ordinate has the harmonics (fundamental and associated overtones). The levels (ratio of sound pressure p to excitation force F) can be seen based on the color scaling (the ranges with strong radiation have red colors while the weaker regions have blue colors). Depending on the force spectrum produced by the musician's bow stroke, the amplitude of the ith harmonic is divided by i ("sawtooth" function). This is why the higher harmonics have decreasing amplitudes.
Note: Since the notes in the scale are displayed at equal distances on the x axis, it has logarithmic frequency scaling. On the y axis, the harmonics are plotted at equal distances so it has linear frequency scaling. Due to the scaling types, lines representing the same frequency on the contour diagram are curved.

A "musical" contour diagram showing the sound radiation for a violin by Guarneri del Gesu (1733):
To better understand this diagram, please read the text for the previous figure.
The "musical contour diagram" helps to reveal the musical relevance of the physical resonance profile. For example, the resonance peak at the frequency of the fundamental note a1 (440 Hz) reappears simultaneously (follow the curved line!) as a resonance peak at the second harmonic of the fundamental note a (220 Hz). The reason is that the 2nd harmonic of the fundamental note a at 220 Hz as well as the fundamental note a1 at 440 Hz both coincide with the resonance peak of the instrument's T1 corpus resonance.
There is a similar situation in the "musical contour diagram" of the Guarneri del Gesu for the resonance peaks around 2000 Hz and 2300 Hz. On the one hand, they make a strong contribution to the sound radiation in the range of the very high fundamentals h3 to #d4, while on the other hand these resonance peaks are responsible for the strong sound components ("mountain peaks") of the 2nd harmonic of the notes which are an octave lower and again the sound components of the 3rd harmonics of the notes which are another fifth lower (and so on). This clarifies the great importance of the high-frequency resonances of the instrument in terms of the higher harmonics of the low-frequency notes.
When listening to a chromatic scale, the figure should correspond to what is heard. For example, the Helmholtz range on the g-string around c1 sounds much more sonorous than, say, the open g-string. Likewise, the weakly resonant range around f1 is clearly audible and is recognizable in the figure in the form of the green incursion.

A "musical" contour diagram showing the sound radiation for a simple student violin (ca. 1900).

When compared to the "musical" contour diagram for the "Guarneri del Gesu", the student instrument has some clear musical deficiencies:

The Helmholtz resonance range (responsible for the sonorous color of the g-string) is much less pronounced.
The range in the 1st position on the d-string is characterized by a strong, large-scale incursion around f1. The d-string has a sound which lacks power and color due to the weak resonance in the fundamental and also in the overtone range for this region.  
The overtone range is highly fractured. This causes unevenness in the instrument.
There are strong peaks in the sound radiation in the nasal range. This tends to produce a tight, nasally dominated tonal color.
The "musical contour diagram of the sound radiation" thus represents the harmonic structure of the radiated sound, i.e. it shows the levels of the fundamentals and associated harmonics for each of the musical notes. As we saw above, this harmonic structure is represented in the form of a map. An alternative representation involves presenting each harmonic (fundamental and overtones) as a separate "mountain profile" using cross-sections through the "musical contour diagram". Corresponding to the force spectrum produced by the musician's bow stroke, the amplitude of the ith harmonic is again divided by i. The "harmonic cross-sections" of the sound radiation can be seen below for two violins (above: Antonio Stradivari 1712; below: a student violin ca. 1900).

"Harmonic cross-sections" of the sound radiation. The fundamental (gray) and overtones (2nd through 10th harmonic in red to violet) are shown as individual "mountain profiles". Here too, the musical note is plotted on the x axis and the associated level (ratio of sound pressure to excitation force) is plotted on the y axis. Above: Antonio Stradivari 1712; Below: A student violin from ca. 1900.
A reliable interpretation of these cross-sections would require more work and a number of further analyses. These examples are useful primarily in terms of the notable similarities between violins which are said to "sound good" or the differences when compared to instruments which sound unexciting (or worse!). For example, in the Stradivarius instrument there is a uniformly decreasing spiked pattern that is evident in the harmonic structure. Such a uniform pattern is missing in the average student instrument. In addition, the levels of the harmonics in the Stradivarius instrument tend to exhibit a clear, decreasing trend, while in the average instrument the levels of higher-order harmonics often exceed those of the lower-order harmonics. (Note in particular the range on the d-string with the strong incursion of the 1st and 2nd harmonics and a dominant peak of the 3rd harmonic which juts out on one side. The absence of the low-order harmonics results in a "thin", "rough" sound that is "without substance". The sound will also be lacking in volume and warmth.

Our experience has shown that if one listens frequently to instruments whose sound radiation has been measured and plotted in this "musical presentation format", an intuitive understanding of these colored maps will quickly follow. This is a good way to learn how to hear the harmonic structure of instruments. This is true particularly if one is able to listen to A-B comparison recordings for two different violins while simultaneously viewing the colored maps of the radiated sound.