# Is it a coincidence that the length of the body of a violin is pretty exactly one fifth of the wave length of its lowest note?

The standard body size of a violin is 35 centimeter in length. Its lowest note is G3, which has a wave length of 175 centimeter, five times 35. 35 cm is the wave length of B5, which is a major third above G. Is this a conincidence or did anybody calculate this when designing it?

• The second half of your question is not a coincidence: a major third can be seen as being defined by being five times (ignoring octaves) the frequency of the lower note, and thus its wavelength would be five times shorter. A major third has a frequency ratio of 5/4. Jan 30, 2020 at 9:27
• @00 or rather one fifth as long. The phrase "n times shorter" (or smaller, or less, etc.) is inconsistent and confusing. For that matter, so is "n times longer" (or greater, or more, etc.). It's better to use "shorter" and "longer" for additive comparisons, and "as long as" for multiplicative ones. Jan 30, 2020 at 21:31

It is a coincidence, because there is no simple relation between the wavelength of sound in air, and the wavelength in the structure of the instrument (either in the wood or the strings).

The fact which is critical to the sound of a violin is the lowest vibration frequency of the air inside the body, which is about an octave higher than the lowest note. That depends mainly on the volume of the air and the cross section area of the F holes. The exact shape of the instrument (and the shape of the holes) are not so critical.

For a "half size" instrument the volume is smaller which would increase the frequency of this air resonance, but reducing the area of the F holes decreases it.

• Kind of, sort of, like the volume of air in a flute and how many finger holes are open? Jan 29, 2020 at 23:56
• Note that I recall a recent study to find the optimal geometric shape for sound holes that showed that violin F holes actually came empirically pretty close to the optimal. Might not be a "critical" part, but it helps. Jan 30, 2020 at 14:21
• @CamilleGoudeseune the volume/cross section ratio is more like an ocarina or other vessel flute en.wikipedia.org/wiki/Vessel_flute . In linear flutes the length of the pipe is in fact half the wavelength of the lowest mode of vibration. Linear flutes can be overblown to multiples of the fundamental frequency; it's harder to do that on a vessel flute Jan 30, 2020 at 19:16

Yes, it's coincidence, mostly. Consider that 200 years ago A was maybe 415 Hz, not 440 or 444 Hz. It is "mostly" true that there's no real value to a resonant chamber more than 1/4 wave of the lowest tone desired, but there's also a lot of art bordering on black magic to create a body which produces a "pleasant" resonant strength at all wavelengths (including overtones) in the normal playing range of the instrument.

It might be interesting to do that same calculation for a few dozen other bowed string (or for that matter, plucked) instruments. However consider that half, and quarter, and even 1/8 size celli and basses are available for child players. Some tonality may be lost but overall they serve their purpose. For that matter, there are certainly "non-standard" fullsize violins & cellos & basses.

• When calculating - be aware that a 'half-size' instrument is actually nowhere near half-size!
– Tim
Jan 29, 2020 at 16:03
• @tim, good point. I know that but should have made it clear that actual physical measurements on instruments should be done. Jan 29, 2020 at 16:10
• At a tuning of 415 Hz the respective length would be 37 cm, still a lot closer than other string instruments are to this ratio. That begs the question if other instruments could be improved by approximating this length. Jan 31, 2020 at 9:59

Being EXACTLY 1/5 is a coincidence (see Carl Witthoft answer). But the musical instrument size (or generally everything made to produce sound) is pretty much related to the wavelength of the sounds it produces. The laws of physics dictate that you cannot efficiently radiate sound (or any other waves, say, electromagnetic waves) using something much less than 1/4 of the wavelenght.

That's why cello and bass are bigger.