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Violin Design

March 23, 2015

There's an Internet meme in which site commentators offer to play a sad song on the world's smallest violin in response to some previous comment. If you type "world's smallest" into Google search, the first auto-complete choice is "world's smallest violin." This concept was used as early as 2002 in the episode, "Squilliam Returns (season 3, episode 8b)," of the cartoon series, SpongeBob SquarePants.

In that episode, Mr. Krabs (my favorite character in SpongeBob SquarePants) plays the world's smallest violin as background music to several sad monologues.[1] An unusual feature of the small violin is that its notes are in the tonal range of a regular violin, and they're just as loud. Unless the strings are made from some exotic material, these smaller strings couldn't produce the same frequencies as violins of the standard size.

The fundamental resonance frequency f of a string depends on the length of the string L, its linear density ρ (which is the string mass m per unit length), and the applied tension T. All these variables should be familiar to even those who have played with toy guitars, since the thicker strings give the lower frequencies, and tightening any string leads to a higher frequency.

string resonance equation

So, a violin a tenth the size of a standard violin, presumably with strings a tenth the diameter (giving a linear density one hundredths that of a normal string), would sound notes at a hundred times the frequency for the same tension. Since the vibrational amplitude is likewise reduced, the notes would be much softer.

Violin fabricated by Jakob Stainer in 1658

A violin fabricated by Jakob Stainer in 1658.

Stainer's instruments were popular in chamber music settings, but the Antonio Stradivari instruments were louder and preferred in orchestral settings.

(Via Wikimedia Commons.)


I wrote about the evolution of the shape of the violin in an earlier article (Violin Evolution, October 31, 2014). One feature that you'll notice from the above example are the "f" holes in the body of the violin. This shape is not just decorative, as a recent paper in the Proceedings of the Royal Society A shows. Earlier string instruments, such as the lute, had simple, circular holes. The "f" holes produce a louder sound.[2-3]

Mechanical engineers from MIT teamed with a Boston violin maker for this study in which an analysis was done on hundreds of violins built in the tradition of Cremona masters.[3] They isolated the design features that contribute to the acoustic power and fullness of sound of these violins.[3]

Acoustically, the sound production mechanism of a violin is simple. The vibrations of the strings are coupled by the bridge element into the body, which acts as a sounding board, and also as a Helmholtz resonator. All the sound energy is derived from the vibration of the strings, but the placement of the bridge and the construction of the body determines how efficiently this energy is transformed to useful sound waves.

Stradivarius violins appear to have special sound quality, and there have been many scientific studies to determine why this should be. Although their sound quality has been attributed to the mechanical properties of the wood used in their construction, or the varnish and other chemical treatments, the major determinant of violin sound is its design.

This study came about by a question posed by a lute player to Nicholas Makris, coauthor of the study and a professor of mechanical engineering at MIT. The question was whether the carvings within a lute's sound hole change the sound quality. In collaboration with fluid mechanics expert, Yuming Liu, an MIT research scientist, he found that air flow was fastest at a hole's periphery, while its interior, whether open or partially blocked by lacework, was not a significant factor affecting the air flow. [3]

In their studies, the researchers acquired technical drawings of historical violins of the 10th through 18th century from various sources, as well as images of the instruments made using X-ray and CAT scanners. They correlated the instrument dimensions with the acoustical properties.[3] The acoustic conductance of the sound holes, that is, the flow of sound waves from one side to the other, was dominated by the air flow at the perimeter of the hole. As a consequence, violins with more elongated holes produce more sound.[3] These elongated holes also take up less area, another factor that increases sound intensity by allowing more sounding board area. Other factors, such as having a thicker back plate, influence sound intensity.[3]

Figure caption

Evolution of stringed instrument holes from the 10th to 18th centuries. (MIT image, reformatted.)


The question arises as to whether the change in hole shape and the thickening of the back plate were done intentionally. The authors believe that the changes evolved by natural mutation; that is, craftsman error. These "errors" propagated to subsequent designs since they improved the sound.[2-3] Says Makris,
"You're cutting with a knife into thin wood and you can't get it perfectly, and the error we report is about 2 percent … always within what would have happened if it was an evolutionary change, accidentally from random fluctuations... Whether they understood, ‘Oh, we need to make [the sound hole] more slender,’ we can't say. But they definitely knew what was a better instrument to replicate."[3]

This research was funded in part by the U.S. Office of Naval Research.[3]

Figure caption

Modeled air flow velocity shows that maximum velocity occurs at the perimeter of the holes. (MIT figure, reformatted.)


References:

  1. My maternal grandmother had a recipe for Krabby Patties long before the cartoon show. She would mix the contents of a can of tuna (oil-packed) with an egg, bread crumbs and optional seasoning, shape the mixture into a patty, and fry. My children used to enjoy these, but I would make them with water-packed tuna, drained, with added olive oil. There are various other recipes for Krabby Patties on the Internet, some of which sound better than this.
  2. Hadi T. Nia, Ankita D. Jain, Yuming Liu, Mohammad-Reza Alam, Roman Barnas, and Nicholas C. Makris, "The evolution of air resonance power efficiency in the violin and its ancestors," Proceedings of the Royal Society A, vol. 471, no. 2175 (March, 2015), DOI: 10.1098/rspa.2014.0905. This is an open access article with a PDF file available, here.
  3. Jennifer Chu, "Power efficiency in the violin," MIT Press Release, February 10, 2015.

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