How to Calculate Optimal Weights for Violin Parts

In this section, I present a novel methodology for violin construction centered around the relative weights of the instrument's components.

From Tones to Weights

Traditional violin making prioritizes tap tones due to their perceived impact on the instrument's acoustic properties. However, I propose that by focusing on the mass (weight) of the individual violin parts instead, the creation of an exceptional-sounding violin is not only possible but also grounded in the science. As we know from physics, the vibration frequency or pitch of an object correlates directly with its mass. This relationship (the mass-frequency relationship) suggests that focusing on the weights of violin parts offers a logical and viable alternative for pursuing exceptional sound.

This perspective challenges the traditional view of violin making, advocating for a shift towards a mass-based and more mathematical approach. By reevaluating the importance of tap tones and instead leveraging the intrinsic physical properties of materials, this method proposes a significant paradigm shift in how violins can be crafted for optimal sound.

On this webpage, I elucidate the method I have developed for determining the optimal weights of the different parts of the violin. This technique is the culmination of over three decades of meticulous observation and mathematical analysis, during which I have constructed more than 30 violins. By adhering to precisely calculated weights, each component contributes to forming a harmonious and acoustically compatible ensemble, which is fundamental to the high-quality sound production of the instrument.

Workflow in a Nutshell

To briefly outline the entire process, it begins by calculating the planar surface area of the soundboard. This measurement forms the basis from which we then determine the optimal weight for the bass-bar. The bass-bar is a key component in the method, because once its weight is established, it serves as a reference point for deriving the optimal weights of the other parts of the violin, namely the top and bottom plates, ribs, fingerboard, and neck.

With the calculations for optimal weights for different parts in hand, we then meticulously craft each part to match the precise weights determined. During this phase, it is imperative to strictly adhere to these weight specifications, consciously avoiding or ignoring results from techniques such as utilizing tap tones, which could prevent the weights from meeting the exact calculated values.

At the same time, it is essential to work towards trying to match, as closely as possible, the thicknesses of the reference violin's top and back plates. This involves carefully following the thickness patterns of the reference instrument. However, the primary goal remains achieving the exact target weight for each part. While we aim to align with the reference violin’s plate thicknesses, we prioritize the final weight above all else, making educated adjustments to the thicknesses as necessary to achieve the target weight. Thickness measurements of top and back plates from renowned and highly regarded violins have been published in various sources, such as Strad magazine posters and books like Giuseppe Guarneri del Gesù by Peter Biddulph.

Mental challenges

While working with the parts, departing from familiar techniques and placing trust in weights often poses the greatest challenge for luthiers. It can be tempting to mix in familiar techniques, such as tapping the plates just a bit to hear how things are going. However, overcoming this inclination is crucial. Specifically regarding tap tones, it is important to remember that it's not about the pitch an individual component emits, but rather ensuring that the parts work harmoniously as a whole. This requires luthiers to adopt an experimental and open mindset, and to embrace trust in something new. Practically, this shift can be the most challenging aspect as it involves a significant mental adjustment.

However, if we overcome these mental challenges and maintain a methodical approach, we are on the right tract on ensuring that each component contributes optimally to the overall sound quality of the violin, embodying a perfect blend of scientific accuracy and artisanal craftsmanship.

Step 1: Calculating the Planar Surface Area of the Soundboard

At first, our goal is to determine the planar surface area of the violin's soundboard. The planar surface area, in the context of violin making, refers to the total area of the surface of the violin's soundboard when it is projected or represented as a flat plane. The practical implication of this is that we do not account for the soundboard’s arching, although it would slightly increase the actual surface area. Knowing the planar surface area is needed for subsequent calculations affecting the instrument's acoustical properties.

To calculate the planar surface area, we begin by establishing the dimensions of the violin’s body to be built. This includes:

• Length of the soundboard (S)
• Width of the upper bout (U)
• Width of the middle bout (M)
• Width of the lower bout (L)

Once the dimensions are established, we calculate the planar surface area of the soundboard. The formula is:

Soundboard Planar Surface Area = ( S + U + M + L ) / ( 41 / 27 )

Here, the sum of the length and widths is divided by the constant 41/27, approximately 1.5185185.

This approach bridges the gap between the need for practical calculations and the violin's complex geometry. The violin's shape does not conform to simple geometric figures, making direct area calculation challenging. The rationale behind the formula is that by summing the key linear dimensions (length and widths) and dividing by the constant 41/27, we obtain a reasonable approximation of the planar surface area. The use of the constant 41/27 is grounded in empirical evidence; in my my experiments comparing the calculated area to physical models (a sample of 25 Guarneri violins) I have established it to be an effective tool for approximating the complex area of the violin’s curved shape.

Step 2: Calculating the Weight of the Bass-bar Based on the Surface Area of the Soundboard

After determining the planar surface area of the soundboard, the next step involves calculating the weight of the bass-bar.

The bass-bar is the key element in the method because we use its weight as a baseline from which we calculate the ideal weights for the other components of the violin. Having a common starting point for all parts ensures that the parts work together harmoniously to produce the best possible sound.

To calculate the weight of the bass-bar, we use the soundboard surface area obtained from the previous step and divide it by the constant 128. The formula is:

Bass-bar WT = Soundboard Planar Surface Area / 128

We use the constant 128 because it serves both as an effective scaling factor and aligns with fundamental principles in mathematics and music. Dividing by 128 scales the relatively large surface area down to a practical weight for the bass-bar, ensuring it is neither too heavy nor too light. Mathematically, 128 is a power of two (2⁷), which is significant because powers of two are foundational in exponential growth and scaling. In music theory, powers of two relate directly to octaves—each octave represents a doubling of frequency. By using 128, the method ensures that the bass-bar's weight is proportionally scaled from the soundboard's surface area to a practical and functional weight.

Step 3: Calculating the Weight of the Top Plate Based on the Weight of the Bass-Bar

After determining the weight of the bass-bar, we next calculate the weight of the top plate, considering it with its f-holes open and the bass-bar not yet attached.

We determine this by multiplying the bass-bar’s weight by the constant 15:

Top Plate WT (without bass bar) = Bass Bar WT × 15

It's worth noting again that the bass-bar’s weight is not included in this initial calculation. To calculate the final weight of the top plate when the bass-bar is attached, we add 1 to the constant 15. This means we multiply the bass-bar’s weight by 16 to obtain the weight of the top plate with the bass-bar attached:

Top Plate WT (with bass bar attached) = Bass Bar WT × 16

The constant 16 is also a power of two (specifically 2⁴), important in musical harmony, and we use it to work towards the violin’s overall harmonic balance. The use of powers of two aligns with the principles of musical acoustics, as each octave corresponds to a doubling of frequency.

Step 4: Calculating the Weight of the Back Plate Based on the Weight of the Bass-Bar

After establishing the weight of the top plate, the next step involves calculating the weight of the back plate.

To determine this, we multiply the weight of the bass-bar by the constant 24:

Back Plate WT (without back button) = Bass-bar WT × 24

This calculation provides the weight for the back plate before the back button is added.

As we remember, the weight of the top plate (with the bass-bar attached) was calculated using the constant 16, and now we calculate the weight of the back plate using the constant 24. The ratio of the weight of the back plate to that of the top plate (with the bass-bar attached) is therefore 24:16, which simplifies to 1.5:1. This ratio corresponds to the musical interval of a perfect fifth, one of the most pleasing and harmonically rich intervals in music.

By aligning the weight ratio of the top and back plates with the perfect fifth interval, we work towards the goal that the plates will resonate in a manner that is harmonically compatible. Establishing such harmonic relationships among the violin's components is a central tenet of this method. This systematic alignment ensures that each component not only fulfills its structural role but also enhances the instrument's ability to produce harmonious and resonant tones.

This concept is visually demonstrated through Chladni patterns, which show the vibrational modes of a flat surface, such as a violin's plate. When a plate is vibrated at certain frequencies, it forms patterns of nodes and antinodes that correspond to specific harmonic frequencies. These patterns are often used to illustrate the physical manifestations of musical intervals and harmonics.

Step 5: Calculating the Weights of the Ribs, Fingerboard, and Neck

With the weights of the top plate and the back plate determined, we now proceed to calculate the weights of the other essential components: the ribs (including blocks), fingerboard, and neck.

The combined weight of the ribs (including blocks), fingerboard, and neck should equal the combined weight of the top plate (with f-holes open and the bass-bar attached) and the back plate (without back button). Furthermore, each of these three components should weigh the same.

It’s worth specifying that connected to the ribs are the blocks, which serve as internal supports that connect the violin's top and bottom plates to the ribs. When we discuss the ribs' weight, we assume that the weight of the blocks is included.

To calculate the optimal weight for each of these parts, we divide the total combined weight of the top plate and the back plate by three:

Ribs, Fingerboard, Neck WT = ( Top Plate WT + Back Plate WT ) / 3

This calculation ensures that the ribs (including blocks), fingerboard, and neck each receive an equal portion of the combined weight. By maintaining this equal distribution, we strive to ensure these components work together seamlessly.

Total Weight

The combined weight of all the parts we've discussed amounts to 80 times the weight of the bass-bar. This includes the top plate (with the bass-bar attached), back plate, ribs (including the blocks), fingerboard, and neck. However, the finished violin's total weight will be slightly higher due to additional components such as the bridge, soundpost, tailpiece, pegs, and strings, as well as the varnish applied to the instrument.

About Varnishing

Particular attention should be paid to keeping the varnish as thin as possible. A thick varnish layer can add unnecessary weight and dampen the vibrations of the violin's body, adversely affecting its acoustic properties. By applying a thin varnish, we ensure that the weights of the varnished parts remain as close as possible to their optimal values, preserving the harmonic balance we've carefully established.

Final Words

By meticulously calculating and adhering to these weight specifications—from the bass-bar to the final varnishing—we strive to craft a violin that not only meets structural and aesthetic standards but also excels in producing rich, resonant tones. This weight-based approach, grounded in both physics and musical theory, offers a logical and scientifically sound method for pursuing exceptional sound quality in violin making.

Through this method, we embrace a blend of traditional craftsmanship and mathematical precision, opening new possibilities in the quest to understand and replicate the exceptional tonal qualities of violins crafted by the old masters. As we conclude this exploration, I encourage you to consider how these principles might enhance your own violin-making endeavors, contributing to the enduring legacy of this remarkable instrument