Sali guitars

I also produce flamenco guitars and acoustic bass guitars (intonation is one octave lower than for classical one).

The procedure of building is based on sound optimization of the guitar. A high quality guitar tone cannot be ensured merely by copying the dimensions, design features, and wood species of a high quality guitar. The main reason for this lies in non-homogeneity and anisotropy of wood. The task of a luthier is to achieve quality sound, even if wooden parts of an instrument have to be modified. These modifications have to be made because the properties of wood vary from one sample to another. For example, additional cutting of the braces on the sound board of a violin may lead to a better sound, but the procedure depends on the luthier's ear. In guitars the luthier has to add the braces with optimal shape and dimensions on optimal places on boards to achieve optimal sound. The additional cutting of these braces is also possible to improve the sound. In order to define the quality of guitar sound I worked several years to define the differences between “bad” and “good” guitar sound, and the result of this work is so called rule of consonance-dissonance (SALI, Samo, KOPAC, Janez. Measuring the quality of guitar tone. Exp. mech., 2000, vol. 40, no. 3, p. 242-247).

When the sides, neck, sound board, fingerboard and bridge are assembled together, the procedure of sound optimization can begin. The first phase is a measurement of the Frequency Response Function (i.e., FRF) of the system (an unfinished guitar without braces on a sound board, without backboard and without bindings). More about FRF measurements can be found in the paper:

SALI, Samo, KOPAC, Janez. Measuring a frequency response of a guitar. V: WICKS, Alfred L. (ed.), DEMICHELE, Dominick J. (ed.). Proceedings of IMAC-XVIII: A Conference on Structural Dynamics, San Antonio, Texas, February 7-10, 2000, (Proceedings of the International Modal Analysis Conference & Exhibit, 18). Bethel, Connecticut: SEM, 2000, vol. II, p. 1375-1379.

Briefly, FRF is a measurement where the object (unfinished guitar without braces on a sound board, without backboard and without bindings) is excited with a mechanical impulse and its response is measured by a microphone at a certain distance. The place of excitation in our case is on a saddle. The obtained FRF is a basis for the two following actions:

-approximately 5,000 individual measurements of FRF on a sound board (this approximately corresponds to the mesh of co-ordinates with density 5 millimeters).

-use of Artificial Neural Networks (artificial intelligence) to calculate the FRF for practically each point on a sound board.

When the dynamic behavior of a mounted sound board without braces is well known (described by thousands of FRF's), the calculation of brace position and their shape follows. This results in well balanced registers, high sound loudness, long duration, and a pleasant sound (see rule of consonance-dissonance) for all tones of Sali guitars.

An example of Frequency Response Function of a “bad” (black) and “good” (green) guitar is shown below. The ratio in price of these two guitars is about 1:100. Horizontal axis represents frequency in Hz and vertical axis represents the ratio between the sound output and mechanical impulse (input) in dimensionless units. In a very simplified way one can say that “good” guitar has many distinctive and sharp (relatively low damped) resonant peaks in comparison to a “bad” guitar. Damping of key resonances is important; also, the damping characteristics of the body should not be too high and not too low. High damping results in a high decay of the played tones and low damping in an unacceptably slow response time of the instrument. Excessive damping of the played tones is a characteristic of “bad” instruments. The FRF, or , a ratio between the output signal (acoustic response of a guitar) and input signal (mechanical impulse), also tells us much about the impedance of a guitar. The acoustical impedance of a guitar depends directly on mechanical impedance of guitar parts and on the interaction between these parts. The impedance of a guitar has to be optimal. Too low impedance can result in a Wolf tone, which is pulsation of tone amplitudes. Too high impedance means that energy drain from strings into a guitar body will be insufficient - the guitar cannot accept it and tone quality will be poor. More about this topic can be found in:

SALI, Samo, KOPAC, Janez. Positioning of braces on a guitar soundboard. V: WICKS, Alfred L. (ed.), SINGHAL, Raj (ed.). Proceedings of IMAC-XX: a Conference on Structural Dynamics, Los Angeles, California, February 4-7, 2002, (Proceedings of the International Modal Analysis Conference & Exhibit, 20). Bethel, Connecticut: SEM, 2002, vol. 1, p. 709-715.

Figure: FREQUENCY RESPONSE FUNCTIONS OF TWO GUITARS (black - bad, green - good).

If positioned and shaped in a proper way, the braces can improve the guitar's impedance, or more precisely, its FRF (Frequency Response Function). Sound board with braces acts like "reservoir for energy". As commonly understood, the energy from the guitar strings is transformed into sound in the guitar body. The majority of this transformation occurs in the sound board and the remainder occurs in the back board and other guitar components. Therefore, the sound board with braces should accept vibrational energy of strings and transform it in an optimal way into the radiated sound. Due to an interaction between the sound board, back board, and internal air the situation is more complex, however the role of the braces is clear. During the vibrations at several frequencies (see distinctive peaks in FRF above) kinetic energy is transformed into potential energy and vice versa. Due to internal losses (friction) and sound radiation this transformation from one form of energy into another stops after a relatively short period of time. It is clear that in “good” instruments the sound board with braces can accept a lot of kinetic energy and it can also transform it into potential energy with minimal losses. We can say that here lies "the secret" of high-quality guitars (and other wooden string instruments). The frequency of resonant peaks is as important as their amplitude and damping (damping is defined by sharpness of peaks). Namely, the frequency of a resonant peak depends on the ratio of modal stiffness to modal mass (a mode can be seen as a vibration at certain frequency with certain intensity and damping). Modal mass and modal stiffness are proportional to the ability to store kinetic and potential energy, respectively. However, it is logical that if the modal masses (each mode has its own modal mass) of a sound board are too high, then this can result in an inefficient excitation of a sound board by the strings. Another parameter which can be easily seen from a FRF is the damping factor which is evident from a shape of a resonant peak (sharp peak means low damping, round peak means high damping). Once the damping factors and resonant frequencies for the most important modes (resonant peaks) are extracted from a FRF plot, then the values for modal mass, stiffness, and damping coefficient (a measure for internal losses) of these modes can be calculated, at least in a relative way. Finally, these parameters are used in the algorithm for sound optimization of Sali guitars.

The following figure shows an example how a certain resonant peak in the FRF of a guitar can be defined as a consequence of damped oscillation of a one-mass mechanical-acoustic system. This system consists of (i) mass (discrete mass m) which can accept kinetic energy, (ii) spring (stiffness k) which can accept potential energy, (iii) damper (coefficient of viscous damping b) which represents energy losses, and (iv) massless membrane with area A (surface A) which radiates sound energy.

Thus, for a certain mode, due to an impact at a certain place on a sound board (see impulse I) the guitar responds with damped oscillations (see sound pressure p' in dependence on time t). Irrespective of the place of excitation, surface A is a constant for each mode, or in other words, for each resonant frequency. For a constant place of excitation, calculation of parameters m, b and k is a relatively simple task (see D.J. EWINS: Modal Testing; Theory and Practice. Letchworth: Research Studies Press Ltd., 1984). This calculation is based also on a following relation between factor of viscous damping δ and the three parameters m, b and k:

It is evident from expression for factor of viscous damping δ that an increase of m and/or k will result in a decrease of δ. An increase of b will result in an increase of δ. In our case, parameter b can be denoted as a measure of impedance of the analyzed mode (see SALI, Samo, KOPAC, Janez. Positioning of braces on a guitar soundboard. V: WICKS, Alfred L. (ed.), SINGHAL, Raj (ed.). Proceedings of IMAC-XX: a Conference on Structural Dynamics, Los Angeles, California, February 4-7, 2002, Proceedings of the International Modal Analysis Conference & Exhibit, 20. Bethel, Connecticut: SEM, 2002, vol. 1, p. 709-715). As already mentioned, too high δ is not a “good” (or quality) characteristic of a concert guitar. One can see that extremely low m and k on one side, and extremely high b on other side will result in extremely high factor of viscous damping, which of course is not desired. In general, lowering of height of braces or decreasing the thickness of sound board on places with relatively high intensity of bending of a certain mode results in larger decrease of modal stiffness k in comparison to modal mass m. In contrast, for places with relatively low intensity of bending, reducing the thickness of brace or soundboard leads into larger decrease of modal mass in comparison to stiffness (see C.M. HUTCHINS: Plate tuning for the violin maker, 25-32. Catgut Acoustical Society Newsletter, May 1983). The mode frequency of the most important modes which are low-damped modes is proportional to the ratio of k to m. Therefore, a relatively low resonant frequency of the mode means relatively low (too low) values of modal mass and stiffness. If the coefficient of viscous damping b (or conditionally said - impedance) is relatively high, this will also result in too high damping of the mode. Alternatively, a relatively high mode frequency means relatively high modal mass and stiffness. It is logical that too high modal mass represents a redundant impediment to string vibrations. If this situation is combined with relatively low parameter b for the analyzed mode, then damping δ will most probably be too low. A consequence of this will be low amplitudes of vibrations (due to high modal mass) and too long response time of the musical instrument. Both of these two characteristics are undesirable.

Through my research, it was noted that minimizing parameter b (impedance) is a primary aim in sound optimization. In addition, lower than higher modal mass and stiffness are also desirable. Finally, to achieve relatively low factor of viscous damping and relatively low resonant frequency of the main modes in the FRF of a guitar, one can see that m, k and b should be low (a relatively low b in comparison to m and k results in a relatively low δ ). We can conclude that for each mode there is a certain range of optimal values of frequency and factor of viscous damping. A thorough analysis of these two parameters can tell us much about (i) the ratio of modal stiffness to modal mass which can be affected during the procedure of guitar building, and (ii) impedance of the mode. In addition, adjusting of mode frequencies is necessary to optimize a FRF of a guitar in order to achieve high portion of consonant intervals and low portion of dissonant intervals in guitar tones (see rule of consonance-dissonance).

An example of modal behavior at about 300 Hz for a Sali guitar without braces on a sound board, without back board and without bindings is shown below. The dark areas indicate places with small displacements, whereas white areas mean the largest displacements of vibrations. We can see that for that frequency the area below the fingerboard has relatively small displacements. We can also see that displacements are not perfectly symmetrical according to a geometrical axis of symmetry. If we want to build a guitar with optimal FRF then we have to consider many such FRF's (for many frequencies). This means that positioning of braces on a guitar sound board is a very delicate and complex procedure. Namely, if something is optimal for one mode (frequency), the same can be non-optimal for another one. It is assumed that this phenomenon is the biggest problem for amateur luthiers and luthiers with insufficient knowledge of physics. The algorithm for optimal bracing of a guitar is not described in any of my publications.

The last phase in the sound optimization procedure is (i) calculation of the effect of the back board with braces on a guitar's FRF, and (ii) its mounting on sides. This is done after the last brace on a sound board is glued. Several years of experiments showed that the effect of the back board with braces on a guitar sound can be accurately predicted. Thus, before a final gluing of the back board on the sides, additional modifications on a sound board are still possible in order to adjust the guitar's sound (FRF).

The following figure shows different FRF's via some comparisons between different guitars. The horizontal axis represents the frequency in Hz, and the vertical axis represents the ratio between the guitar response at 1 meter from a guitar (output) to mechanical impulse at a saddle (input). The dimension of the vertical axis is therefore Pa/(m/s²). High-quality concert guitars are made by Olivier Fanton d'Andon and Thomas Humphrey. Yamaha C-60 is an extremely low-quality school guitar produced in large scale production. From the graphs it is evident that the relatively high frequency of the lower modes (resonant peaks) is not a favorable characteristic. In addition, amplitude of the second peak (first mode of the sound board) for a Yamaha guitar is relatively low, and its damping is high. One can see that the level of frequency response function for a Yamaha guitar is lower for almost all frequency range of interest in comparison to other guitars.