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I am aware that this might be a slightly unusual question, but just today I was wondering about the physical volume (meaning the capacity, NOT the loudness) of some musical instruments. In particular, I'd be interested in the string family - even though I know that, at least for viola and double bass, there might be fairly large discrepancies. Also, I'd mostly be interested in the volume of the body of the instrument (and yes, considering it is filled. So the air inside does count as part of the instrument).

For a violin and a viola I have found an estimate by violist Franz Zeyringer, stating the volumes to be around 1968cm³ and 3045cm³ respectively - the latter seeming oddly precise, given how much the length of a viola can vary. However, taking some quick measurements of my own violin, the first number seems very plausible.

Now, before I start approximating stuff with weird functions and throwing some multi-dimensional integrals around, does anyone here know anything or has run across something and knows where to find information about that topic?

Some Math Student
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    Interesting stuff! What do you want to use it for? Didn't find a lot but [this](http://lisafea.com/pdf/JMC-Helmholtz_Resonance--Re-ct2.pdf) and [this](http://americanhistory.si.edu/blog/2009/11/digital-stradivari-computer-models-of-violins-reveal-master-luthiers-techniques.html) might be of interest? Also just a thought, you could try filling a violin with water and seeing how much goes in ;) **EDIT** This [luthier forum](http://www.maestronet.com/forum/index.php?/topic/323914-rib-height-air-volume/) seems useful... – ChristopheLynch Aug 31 '16 at 23:12
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    @ChristopheLynch better yet, plug all the holes in *any* instrument and force it under the surface in a bathtub. Total displacement = volume. (Oh, and best to shrink-wrap first). – Carl Witthoft Sep 01 '16 at 11:26
  • OK, in all seriousness, there is a lot of variation in (string) instrument size, so the best you can do is define a general range. Ideally, a 3-D laser scanning system could get you the external volume pretty easily, tho' if you want the internal airspace, youll need to measure the body thicknesses as well. – Carl Witthoft Sep 01 '16 at 11:28
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    I've heard many suggestions of measuring internal volume directly with dry sand, small beads, etc. (Dry rice even, but I think it might have inconsistent voids.) Never seen anyone actually **do** this potentially messy thing though... – Andy Sep 01 '16 at 13:34
  • Are you interested in the volume, measured from the outside of the instrument in, or the volume of the chamber where air resonates? Not too much of a difference for string instruments, but it's a significant difference in comparing, say, an oboe and a clarinet. – Michael Scott Asato Cuthbert Dec 16 '16 at 18:37
  • I'm terribly sorry, I seem to have overlooked the comments so far. There is no particular urgent reason why I need to know this, the question just popped up in my head and wouldn't go away immediately. It is a fair point, I could technically fill an instrument with water/sth else and see how much goes in, however, I seriously doubt anyone owning, say, a cello or double bass would let me. Neither do I own a sufficiently advanced laser scanning system, unfortunately. I am more interested in the resonating chamber, but would not mind either. – Some Math Student Dec 16 '16 at 23:25
  • I don't think anyone actually measured the volume of all the string instruments, but you might find some info on a website of a company that makes them. –  Dec 30 '16 at 06:39
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    If by any chance your interest in the volume is because of its connection to the free air resonance frequency (something that instrumentmakers do pay attention to), then you also need to measure the area of the soundholes, which are also a factor: the smaller the soundholes, the lower the frequency. – Scott Wallace Jan 06 '17 at 19:46
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    @CarlWitthoft: If it's a Viola, there's no need to shrink-wrap it first :P – naught101 Feb 02 '17 at 23:40
  • @SomeMathStudent: you could fill it with rice, or small beads, or similar (something incompressible). That'd get you a pretty good estimate without screwing up the instrument. – naught101 Feb 02 '17 at 23:42
  • @naught101 you just had to go there, eh? :-) – Carl Witthoft Feb 03 '17 at 12:19
  • What about a harp? It has no resonating chamber at all. Is that why it's so quiet?!? – L3B Feb 06 '17 at 21:14
  • @Scott Wallace: That wasn't actually the reason, and I was aware of that, but thanks for pointing it out anyway. – Some Math Student Feb 15 '17 at 21:04
  • @L3B: A harp does actually have a soundboard, which acts as a resonating chamber. If you ever get close to one, you might be surprised how loud it actually can be - at least, I was. – Some Math Student Feb 15 '17 at 21:04
  • @some math student-- Well yes, I know it has a soundboard, But the OP was asking about "capacity" A board by itself does not have much capacity! – L3B Feb 15 '17 at 22:19
  • @L3B- with the exception of some folk and medieval instruments, most harps, including concert harps, do have a soundbox: the body of the harp is closed, with soundholes behind and sometimes on the sides or front. – Scott Wallace Feb 16 '17 at 11:10

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If you trace the outline of a guitar onto graph paper, count the number of squares that are not crossed by the perimeter tracing and that are inside the area (COUNT), and then count the number of squares that are touched by the line (PERIMETER), the formula for estimating the area of an irregular shape is:

AREA = Count + (.5 * PERIMETER)

For many guitars, the depth is reasonably even for an estimate, and so simply multiply the area by the depth for a volume. The neck can be treated separately and then summed with the body volume.

VOLUME = AREA * DEPTH

For "curved-back/curved-front" instruments, "my gut" says, to compare the irregular shape of the depth to the ideal rectangular bounding box of the depth, calculate the percentage, and then use that percentage:

VOLUME = AREA * DEPTH * PERCENT

"My other gut" says this is an a bad gut reaction, that may work for "simple swells" but not for complex depth measurements.

In any event, you can probably use this method even if you only have photos and one or two measurements to establish scale. The finer the grid, the closer the estimation.

This does not require 50 lb bags of rice.

Yorik
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  • Indeed, for a guitar that method would work fairly well. Even having to calculate the are of the guitar quite precisely would not be an issue - however, once you do get to the violin family or any more curved instruments, getting the percentage you talk about may not be as simple as described, at least I tend to be off by a lot with such guesses. Especially talking about irregular instrument-esque shapes. So, I'd rather go with your "other gut". Thanks for the suggestion, though. – Some Math Student Mar 08 '17 at 23:19
  • The suggested approach is somewhat unaware of technology. I propose to use mobile phone camera, masking the background and do some statistics, i. e. color similarity computation by pixel. The result just needs to be multiplied by the real world area per pixel factor. – guidot Apr 07 '17 at 11:48
  • @guidot: agreed. I left that out so as to keep the method easy to understand. While pixels are point samples, not "squares," they are usually presented in a grid form and are amenable to machine methods for counting – Yorik Apr 07 '17 at 14:32
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Ok this is very approximate but it would work for some instruments.

Let get something we can use - rice has been the best suggestion so far in my view and use it to fill up the case that the instrument travels in. My cello has a bespoke hard travelling case and its a tight fit. I would be unhappy tipping rice into my cello but I might just about accept putting it into the case. Then transfer the rice to something we can measure it in.

Seems to me that if we are careful we can use this for many string and woodwind instruments, even perhaps brass instruments, providing they have well fitting cases and we use a bit of common sense regarding levels etc.

Having said that, why you would want to do this is still baffling me.

JimM
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  • The volume of the hard case would probably be 3-4 times the volume of the actual instrument (e.g 1.5-2 times the width and depth, maybe 1.2 times the length). – naught101 Feb 06 '17 at 00:52
  • @naught101 Not sure that I agree with you. The internal measurements of my made-to-measure cello case follow the contours of my cello. Its a tight fit. I don't intend to try it but I think it would be really quite close - much less than 1.5 times and perhaps even as close as 1.25 times. – JimM Feb 06 '17 at 09:18
  • fair enough, but even if it's only 1.25 in each dimension, that's nearly twice the volume. Still not a very accurate measurement... – naught101 Feb 06 '17 at 23:21
  • True, that would likely work. And a case would probably get me a fairly decent approximation, with only a few calculations. Don't have all these instruments handy, though, and it's not urgent enough to warrant an active search for people willing to put their instruments through that. Even though it would probably not harm them. – Some Math Student Feb 15 '17 at 21:01
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You can use some modeling SW such as SolidWorks to figure it out. The actual precision will depend on the amount of time you invest on details, but I guess it is (way much) easier than manual calculation of integrals.

Alexis
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