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On the question Do 1/2 size violins sound at normal pitch? I found out that 1/2 violins are actually ~86% of a full size violin.
What do the fractions 7/8, 3/4, 1/2 etc for violin sizes represent, if they don't represent the body size or the scale length of the instrument?
It's just a naming convention for the scale length. The numbers don't actually equate to anything. See the quote below:
Smaller scale instruments are used extensively to teach younger players. The size of these is described by a "conventional" fraction that has no mathematical significance. For example, a 7/8 violin has a scale of about 317 mm, a 3/4-size instrument a scale of 307 mm, a half-size one 287 mm, and a quarter-size one 267 mm. 1/8, 1/10, 1/16 and 1/32 and even 1/64 violins also exist, becoming progressively smaller, but again in no proportional relationship. (A full-size instrument is described as 4/4.) - Wikipedia
While it doesn't have a math reason, it is more practical remember simple a fraction over a seemingly random number.
Try this. It's to do with volume. Most people know the volume of a box is length time width times height. Using your numbers, if each of those measurements is 0.8 of the full size then the volume 0.8x0.8x0.8 times the size, and 0.8x0.8x0.8 is 0.512 - pretty close to half size. I haven't checked the other fractions perhaps you could see if my hunch works.
