Why We Use Constants 128, 16, and 24 in Our Calculations

Introduction

In the art of violin making, achieving optimal sound quality involves a delicate balance between the physical properties of the instrument's components and their acoustic interactions.

Traditional methods often rely on tap tones and plate tuning, but an alternative approach described on these pages focuses on the weight ratios of the violin's various parts. Central to this method are the constants 128, 16, and 24, which are used to calculate the optimal weights of the bass-bar, top plate, and back plate, respectively.

Please read the full description of the method first if you haven't yet done so.

On this page, I particularly explore a subtopic of the method: the main takeaways why we are using these constants, linking them to the vibrational properties of materials, musical intervals, and harmonic relationships.

The Role of Mass in Vibrational Frequencies

The vibration frequency of an object is inversely related to its mass. In the context of violin plates, their mass affects how they resonate at different frequencies. Heavier plates tend to vibrate at lower frequencies, while lighter plates vibrate at higher frequencies. By carefully controlling the mass of each component, luthiers can influence the instrument's tonal characteristics.

Vibrational properties of the plates can be demonstreted for example by Chladni patterns that visually represent the vibrational modes of a surface. When a violin plate is excited at certain frequencies, specific modes of vibration emerge, forming intricate patterns. Aligning the mass of the violin's components with harmonic relationships can enhance these vibrational modes, contributing to a richer and more resonant sound.

Musical Intervals and Harmonic Ratios

Musical intervals are based on frequency ratios between notes. The perfect fifth, one of the most consonant intervals, has a frequency ratio of 3:2. This ratio can be applied to the mass of the violin's plates to create harmonious interactions between their vibrational frequencies.

Deriving the Constants

Constant 128 for the Bass-Bar

The bass-bar is a critical component that supports the top plate and influences its vibrational characteristics. To calculate its optimal weight, the method divides the planar surface area of the soundboard (SB Area) by 128:

Bass-bar Weight = SB Area / 128

Why 128?

• Mathematical Basis: The number 128 is 2⁷, a power of two, which is significant in octave relationships.
• Harmonic Considerations: Using a power of two aligns with the concept of octaves in music, where frequencies double with each octave.
• Scaling Factor: 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.

Constants 16 and 24 for the Top and Back Plates

Once the bass-bar's weight is determined, the weights of the top and back plates are calculated using the constants 16 and 24, respectively:

Top Plate Weight = Bass-bar Weight × 16
Back Plate Weight = Bass-bar Weight × 24

Harmonic Ratio of 16 and 24

• Ratio Simplification: The ratio of the back plate weight to the top plate weight is 24:16 = 3:2.
• Perfect Fifth Interval: This 3:2 ratio corresponds to the perfect fifth in music, suggesting that the plates will resonate harmoniously.
• Vibrational Harmony: Aligning the mass of the plates in this ratio promotes compatible vibrational modes, enhancing the instrument's tonal richness.

Practical Considerations

• Traditional Weight Standards: The weights resulting from these calculations align with traditional standards observed in historic violins known for their exceptional sound quality.
• Avoiding Overweight Components: By adhering to these constants, luthiers prevent the plates from becoming excessively heavy, which could dampen resonance.

Conclusion

The constants 128, 16, and 24 are more than arbitrary numbers; they are carefully derived values rooted in the principles of physics and music theory. By integrating mathematical relationships, vibrational properties, and harmonic ratios, these constants provide a systematic approach to crafting violins with good acoustic qualities.

Encouragement for Further Exploration

Luthiers and enthusiasts are encouraged to experiment with these principles, exploring how precise weight control and harmonic ratios can influence the sound of their instruments. Sharing findings and experiences can contribute to a deeper understanding of violin acoustics and the continual evolution of the craft.