In addition to the four traditional instruments of the violin family, and various intermediate instruments proposed from time to time, such as Carleen Hutchins' Violin Octet, mentioned above, violin family instruments in a number of other sizes exist.
These, however, although they differ in size, have their strings tuned to the same pitches as the regular instruments to which they correspond.
Thus, you may see advertisements for violins which, instead of having the standard size, called 4/4 size, are in sizes ranging from 7/8 down to 1/32. The fractions are conventional names for the sizes; a 1/2 violin is not actually one-half the length of a regular violin. These are intended to allow young children to have instruments suited to them as they learn to play the violin.
A Sears, Roebuck catalog from 1912 notes that one of the violins offered is also available in 3/4 size, "especially adapted for children from 9 to 14 years of age", and another is also available in "ladies' or 7/8 size"; the 1/2 size and 3/4 size for violins are also referenced in a 1917 catalog of John Friedrich and Brother, so this fractional notation for violin sizes predates the introduction of the Suzuki method of violin teaching in the postwar era.
A French book, "L'Art du Luthier", from 1903, gives an extensive table of recommended dimensions for different instruments of the violin family, in millimetres, from which I will give the value for the length of the body:
Violin, 1/4 size 297 mm Violin, 1/2 size 302 mm Violin, 3/4 size 332 mm Violin 358 mm Viola (large model) 473 mm Cello, 3/4 size 653 mm Cello 770 mm Double Bass 1m 130 mm
An even older source, from 1895, gives 14 1/8" as the body height of a full-size violin, 13 7/8" as that of a ladies' violin (not explicitly referred to as 7/8 size), and 13 1/8" as that of a three-quarter size violin, and I have now seen references in English to the three-quarter violin as early as 1866.
A German book, from 1892, gives these dimensions for a number of instruments from the violin family:
Violin, 1/2 size 320 mm Violin, 3/4 size 335 mm Violin 360 mm Viola 400 mm Cello 760 mm Double Bass, 1/2 size 1m 105 mm Double Bass, 3/4 size 1m 108 mm Double Bass 1m 110 mm
Here, the viola is definitely one based on a small-size model, Stradivarius' small model being 411 mm long.
Another source gives the body size for a 7/8 violin as 343 mm, but it gives that for a full-size violin as 355 mm rather than 358 mm.
A web site giving modern values for different fractional sizes of the violin, however, gives considerably different values for the sizes, and so does a second modern web site, and then I found slightly different values in a PDF document on yet a third site, and significantly different ones on a fourth site in French, and I give the sizes from all four below:
Violin 356 mm 355 mm 356 mm 355 mm Violin, 7/8 size 345 mm 345 mm Violin, 3/4 size 335 mm 335 mm 335 mm 330 mm Violin, 1/2 size 310 mm 320 mm 320 mm 317 mm Violin, 1/4 size 280 mm 285 mm 285 mm 292 mm Violin, 1/8 size 255 mm 255 mm 260 mm Violin, 1/10 size 235 mm Violin, 1/16 size 230 mm 230 mm 216 mm Violin, 1/32 size 215 mm
and the first of these sites also notes that Suzuki sizes for 1/10 and 1/16 violins are significantly smaller than the traditional sizes, being similar to 1/16 and 1/32 sizes on that scale respectively; the sizes given on the fourth of those sites seem to correspond to that.
The dimensions from the third source at least seem to reflect the best dimensions available from the first two, but I'm not decided as to whether I should regard it as authoritative, or possibly derived from an inconsistent mixture of sizes from other sources. Perhaps plotting these sizes, and the old French ones, on a graph might help me see if there is some system to this.
Of course, neither 302 mm nor 310 mm nor 320 mm is anywhere near half of 358 mm; if we were to divide 358 mm by the cube root of 2, the result is 284 mm, so the fractions do not even represent volume.
Since 1/4 size is 285 mm in some cases, I suppose one could try the sixth root of each fraction. That would give:
4/4, full size 356 mm 7/8 348 mm 346 mm 3/4 339 mm 336 mm 1/2 317 mm 3/8 302 mm 299 mm 1/4 283 mm 3/16 269 mm 267 mm 1/8 252 mm 1/10 242 mm 238 mm 1/16 224 mm 1/20 216 mm 212 mm 1/32 200 mm
These sizes don't correspond too badly to the modern values given in the previous table, except that the "1/32" violin would have to be called a "1/20" violin instead, which perhaps splits the difference between "traditional" and "Suzuki" sizes.
On further reflection, it seems likely to me that the fractions should be considered to be merely conventional sizes, for steps in a uniform logarithmic scale of sizes, with the additional 7/8 fraction indicating an intermediate step. So either 3/16 or 1/5 would indicate the same size, halfway between the 1/4 and 1/8 sizes. Sizes on that basis are shown in the second column; only those for the intermediate values are changed, generally becoming slightly smaller.
The values for 1/2 size and 1/4 size in the French source seem to be so close to each other (and to the size given as that of a 3/8 violin above) that it doesn't seem that those sizes could be the basis for a systematic scale.
In proportion to the ideal size for a viola, the actual size of the instruments commonly in use is equivalent to that of a violin 275 mm in length. Thus, a viola is really a "1/4 size" viola, or thereabouts, which makes it quite understandable that its sound quality is compromised. While the cello and double bass are also available in smaller sizes described in fractional terms, smaller size violas instead have their sizes given directly in inches of body size.
Incidentally, the usual size for a double bass is considered not to be the full size for a double bass, but instead its 3/4 size. Full size is considered to be 1 metre, 160 mm, and 3/4 size 1 metre, 110 mm, so the size for the double bass given in the old French book quoted above is intermediate between these two values. On the other hand, the old German book quoted above gives exactly the current 3/4 size usually used as the full size for a double bass.
The 1912 Sears, Roebuck catalog noted above advertises both 3/4 and 1/2 double basses, but no full size ones, so this would seem to corroborate that the usual size is designated the 3/4 size.
Also, Antonio Stradivarius made at least a few small violins. Two of them, the 'Cipriani Potter' from 1683, and the 'Belle Skinner' from 1736, have a body height of 339 mm, which would seem to make them 3/4 size violins (although I have seen the Cipriani Potter referred to as a 7/8), and another, the 'King Maximillian Joseph' from 1702, has a body height of 347 mm, which would seem to make it a 7/8 violin. The 'Gillott' from 1720, having a 267 mm body size, doesn't correspond to one of the more usual values, yet it is close to being midway between the size of a 1/4 violin and a 1/8 violin, which suggests that a standardized system of sizes for violins has long been in existence.
A paper discussing violin sizes which I encountered showed examples made by three different members of the Amati family, along with another example by Lorenzo Storioni. Another source notes that Andrea Amati, in addition to 355 mm full-sized violins, also made some of 342 mm, or 7/8 size.
I was surprised that the sizes Antonio Stradivari used for small violins were, almost to the millimetre, the same as those I had listed in my final attempt at laying out standardized values for the fractional sizes of violins.
Of course, the fact that the ratio between the sizes of a full-sized violin and a 1/2 size violin was the sixth root of two, and thus if one adds in the 3/4 size violin in a uniform logarithmic scale, the ratio between successive sizes is the twelfth root of two - which is, of course, a value very familiar from earlier pages in this section, as it is the ratio between frequencies, and also wavelengths, and therefore string lengths, for successive notes one semitone apart in the equal-tempered scale.
But at first this seemed to be an amusing coincidence, or due to perhaps some sentimental attachment on the part of violin makers. But perhaps it might simplify memorizing the appropriate placement of one's fingers when playing a violin and moving across different sizes of instruments.
Stradivarius also made a few guitars. And what would be more reasonable than that if there is a need for reduced-size violins for younger players, the same might apply to guitars? And why not use the same set of sizes for both instruments?
And then the obvious dawned on me. If the ratio between instrument sizes is the same as the ratio between string lengths for successive notes, then a single template or jig would suffice either directly, or for the making of jigs for the individual instruments, to control the placement of frets on the fretboard of a fretted instrument such as a guitar.
Of course, on a violin, it isn't the size of the body that determines the spacing between fingers on the fingerboard, it is the distance between the nut and the bridge. And an analogous rule applies to the guitar. In the case of the violin, one can see from the image above comparing the violin and the cello that not only is the cello proportionately thicker, but its neck is proportionately shorter. And some small-sized violins have necks that are larger in proportion than on a regular violin, to make fingering more convenient: the instrument may be intended as an easily portable one for an adult player instead of one suited to a child.
However, it may be that the fractional sizes I have worked out, while they were at one time the traditional sizes for smaller violins, are not the sizes used most often at present. After all, violins don't have frets, so the issue I've identified that would encourage the particular sizes I've proposed for guitars doesn't really exist for the violin.
I have seen the figure of 92.5% as the ratio between successive sizes quoted in a few places.
And if the fractions are merely conventional sizes, could it be that the steps between 4/4, 3/4, and 1/2 are the same as those between 1/2, 1/4, and 1/8?
Given that the sizes normally seen today are 4/4, 3/4, 1/2, 1/4, 1/8, 1/16, 1/20, and 1/32 - and also providing a nice explanation for the fact that when 1/20 was inserted for instruments used under the Suzuki system, it had the same size as 1/32 as used by others... could it be that 3/4 is no longer being treated as at a half step, and instead the size of a full step has been shrunk, so that things average out, making the sizes of at least some of the violins in the sequence still similar?
The image above compares violin body sizes, along a logarithmic scale, between, on the lower scale, the traditional system I have inferred, and on the upper scale, the possible modern system that I am now suspecting may exist as an alternative.
Fractional size instruments suffer in sound quality. But players of the viola often end their careers with injuries, and this is not even uncommon to violin players.
Given that the cello has a smaller body size than its pitch would call for, but its sound is still satisfactory, but with a sweeter quality, due to its additional thickness, it seems worthwhile to examine in detail how greater thickness can compensate for smaller body size.
Here, then, is a chart of how this could work out:
Violin Viola Cello A: 382 mm A: 764 mm 34 mm E: 68 mm 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 53 mm E: 70 mm E: 90 mm 7/8 A: 346 mm A: 692 mm E: 35.5 mm E: 71 mm 3/4 A: 336 mm A: 672 mm E: 36 mm E: 72 mm 1/2 A: 317 mm E: 37 mm 1/4 A: 274 mm A: 411 mm E: 39 mm E: 60 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 41 mm E: 62 mm E: 82 mm E: 105 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 42 mm E: 64 mm E: 128 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 42.5 mm E: 64.5 mm E: 129 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 43 mm E: 65 mm E: 130 mm
The existing size of the violin is shown in the 4/4 row, but that of the cello is shown in the 1/8 row. The existing body size of the viola is shown in the 1/4 row, but with the thickness increased to 60 mm from 37 mm.
Instead of the increase in thickness of the cello being proportional to the square of the departure of its body size from the ideal, it is less than the square root of the change in body size (the 2.33 root), so the demand for increased thickness as we reduce the body size of an instrument is modest.
Not all possibilities have been shown in the table above; just the top row, with the proportions of all the instruments matched to those of the violin, those with the proportions of all the instruments matched to those of the cello, and those others that seemed likely to be useful.
Thus, a 7/8 violin made slightly thicker is shown, and the violas in proportion to a thickened 1/10 and 1/8 violin are shown, as those appear likely to be ergonomically useful; as well, the cello sizes in the 1/10 and 1/16 rows are shown, as they would correspond to a 3/4 cello and a 1/2 cello, and, of course, younger players would play those.
For the intermediate instrument between the viola and the cello, tuned an octave lower than the violin, body sizes corresponding to some reduced sizes of the cello are shown, as they would be relevant to younger players as well.
A 1/12 size, halfway intermediate between 1/10 and 1/16, is shown, so that a viola with a body size equivalent to that of a 7/8 violin could be shown.
Above the full size or 4/4 violin, the body size of the Mezzo Violin from the Violin Octet is shown, as the intermediate instrument could be made larger in proportion and still be playable like a cello. However, the Alto Violin in the Violin Octet, which corresponds to the viola, has a body length of only 508 mm, not 573 mm, so I may not be following the correct rule in scaling up the instruments.
The Baritone Violin, which corresponds to the cello, has a body length of 866 mm but a rib height of 138 mm, so both the body length and the rib height are greater than that of the conventional cello, so I cannot use it as the basis for deriving a different scaling rule by treating it as the scaled-up violin to be compared with the normal cello.
In a description of the Baritone Violin, however, it is noted that while it has a powerful tone with respect to its lowest string, unlike the cello it does not have a more powerful tone with respect to the A string, which is the highest string on both instruments. Thus, perhaps the cello, like the viola, is undersized, rather than being, as I had assumed, like the violin, ideal for the pitches it plays. Therefore, in order to derive a scaling rule from the Violin Octet, perhaps the thing to do would be to scale up the Mezzo Violin, and compare the result to the Baritone Violin, treating the Baritone Violin, like the conventional cello, as the example of how to exchange thickness for increased body height.
However, it is not necessary to do that, as the Violin Octet was explicitly designed in accordance with the scaling rule given in J. C. Schelleng's paper The Violin as a Circuit.
Of course, this might just be exchanging neck injuries for arm and wrist injuries, at least in the case of the viola, but if a thicker instrument could be played without problems, this shows a way to extend the violin family while respecting the health of players in addition to sound quality.
And, of course, thickness is not an issue for the cello or for the intermediate instrument intended to be played in a similar manner.
The simplest thing to do would be to ignore everything but the main air resonance, using the formula for the Hemholtz resonance of a sphere with one hole. In this case, the wavelength is proportional to the square root of the volume, so doubling the wavelength only requires dimensions to be increased by a factor of 1.5874.., or two to the power of two-thirds. As for the front and back plates of the instrument and their area, their frequencies can be changed by changing their thickness; however, making them thicker raises the frequency, and it is not practical to make them thinner for larger instruments, as this would lead to them becoming fragile.
This implies that the basis for an enlarged instrument could either be one that has the body height increased by the basic scale factor, but the rib height unchanged, or one that has the body height increased by the square root of the basic scale factor, and the rib height increased by the basic scale factor. But then when the body height is reduced, the rib height would be increased so as to keep volume constant.
Using the latter alternative, the chart above would be modified to the following:
Violin Viola Cello 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 35 mm E: 35 mm E: 35 mm 7/8 A: 346 mm A: 692 mm E: 37 mm E: 37 mm 3/4 A: 336 mm A: 672 mm E: 39.5 mm E: 39.5 mm 1/2 A: 317 mm E: 44 mm 1/4 A: 274 mm A: 411 mm E: 59.5 mm E: 59.5 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 70 mm E: 70 mm E: 70 mm E: 70 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 79 mm E: 79 mm E: 79 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 84 mm E: 84 mm E: 84 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 89 mm E: 89 mm E: 89 mm
However, while this exacts a heavier penalty in increased thickness for reduced body size, it still leads to the cello turning out to have an ideal size which is actually thinner than its existing size. That is unlikely to be the case.
If we take the claim that the A string on a cello is the one that has its sound enhanced by the instument to indicate that its resonance is in fact a factor of 27/8 too high, that would make this quite impossible indeed. Note, though, that the A string on a cello is actually tuned a whole note higher than the lowest string on a violin, its G string.
At this point, one might wonder if one might as well give the cello three higher strings, and one lower string, than it presently has, thus with seven strings tuned in fifths to F, C, G, D, A, E, B, and thus with no need to have any other instruments in the violin family.
If we multiply 105 mm by the square root of 27/8, we get an ideal thickness for the conventional cello, with a body height of 759 mm, of 193 mm.
Violin Viola Cello 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 51 mm E: 66.5 mm E: 96.5 mm 7/8 A: 346 mm A: 692 mm E: 37 mm E: 70 mm 3/4 A: 336 mm A: 672 mm E: 39.5 mm E: 75 mm 1/2 A: 317 mm E: 44 mm 1/3 A: 289 mm A: 866 mm E: 53.5 mm E: 147.5 mm 1/4 A: 274 mm A: 411 mm E: 59.5 mm E: 87 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 70 mm E: 102 mm E: 133 mm E: 193 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 79 mm E: 115 mm E: 218 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 84 mm E: 122.5 mm E: 231.5 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 89 mm E: 130 mm E: 245 mm
Using that empirical scaling factor as the basis, the above shows revised recommended rib heights for an ergonomic extended violin family.
As a check on whether this makes sense, a 1/3 size violin has been added, sized so as to correspond to the Baritone Violin in the Violin Octet. Calculating by this method leads to a rib height of 147.5 mm instead of 138 mm, which is at least roughly in the same ball park - and, of course, my calculations made some rather drastic simplifying assumptions.
So, applying the appropriate correction factors:
Violin Viola Cello 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 49.5 mm E: 63.5 mm E: 90 mm 7/8 A: 346 mm A: 692 mm E: 37 mm E: 67 mm 3/4 A: 336 mm A: 672 mm E: 39.5 mm E: 72 mm 1/2 A: 317 mm E: 44 mm 1/3 A: 289 mm A: 866 mm E: 53.5 mm E: 138 mm 1/4 A: 274 mm A: 411 mm E: 59.5 mm E: 84.5 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 70 mm E: 99 mm E: 127 mm E: 180.5 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 79 mm E: 112 mm E: 204 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 84 mm E: 119 mm E: 216.5 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 89 mm E: 126 mm E: 229.5 mm
Thus, in this case, using the Baritone Violin as a guide, the thickness of a cello with its existing body height would need to be almost doubled, from 105 mm to 180.5 mm, to re-position its resonances to correspond to its lowest string.
And, initially, I thought that increasing the thickness of a viola, to allow it to have a good sound while being no larger than a violin, to 64 mm, would already lead to an awkward instrument; now, an even larger thickness, almost twice as great, of 112 mm (even greater than the 105 mm thickness of a conventional cello!) seems to be required.
However, the figures in these last two tables are based on the premise that the conventional viola, with body height of 759 mm and thickness of 105 mm is an instrument whose resonances are actually higher than those of the much smaller violin, which seems hard to accept.
The sizes given on this page for violins and other related instruments are quoted in either inches or millimetres. But neither English units nor metric units were in use in the Cremona of Stradivari's day, so possibly expressing dimensions in terms of the units which he had used might avoid rounding errors. One place where this might be significant is in the case of the ratio between dimensions D and E, since this would affect the precise angle between the belly and back of the violin.
From the book Stradivari by Stewart Pollen, it is noted that a harp made by Stradivari has markings indicating that he used a Cremonese foot of 483 millimetres. Other sources give such a unit as well.
An official document, approved by a decree of May 20, 1877, Tavole di Raggulaglio de Pesi e Delle Misure Già in Uso Nella Varie Provincie del Regno col Peso Metrico Decimale gives the length of the "Trabucco di Cremona" as 2.901233 metres.
This leads to the following system of units:
Trabucco (6 Piede) 2.901 233 metres Piede (12 Once) 483.538 833 333 mm Oncia (12 Punti) 40.294 902 777 mm Punto (12 Atomi) 3.357 908 564 814 814 mm Atomo .279 825 713 734 567 901 234 567 901 234 567 mm
where the Piede is another name for the Braccio da Fabbrica, the unit in question, and the figures are prolonged sufficiently so the repeating decimals are visible without any claim to significance.
Of course, this was long after Stradivari's day, but it shows that the unit in question existed for long enough that a precise value was eventually established for it, to seven significant digits.
In Milan, the Atomo was further divided into twelve Momenti; in Piacenza, it was instead divided into twelve Minuti, each of which was divided into twelve Momenti, with each of those further divided into twelve Scrupoli, so this constitutes a duodecimal system of measurement. If we subject the Cremonese units to the Piacenzan mode of division (the Braccio in Piacenza was longer, 675 mm), we get:
Trabucco (6 Piede) 2.901 233 metres Piede (12 Once) 483.538 833 333 mm Oncia (12 Punti) 40.294 902 777 mm Punto (12 Atomi) 3.357 908 564 814 814 mm Atomo (12 Minuti) .279 825 713 734 567 901 234 567 901 234 567 mm Minuto (12 Momenti) .023 318 809 477 880 658 mm Momento (12 Scrupoli) .001 943 234 123 156 721 mm Scrupolo .000 161 936 176 929 726 794 695 930 498 mm
and so we would only have to repeat the division by 12 three more times to get a unit about the size of a nanometre, but we don't have to go any further to get into the realm of modern microelectronics; it wasn't that long ago that integrated circuits had feature sizes on the order of a Scrupola. A Scrupola is about 1.62 microns, and the original 80386 was made with a feature size of 1.5 microns.
In terms of the marks one might make on a ruler, to have a smallest unit comparable to 1 mm or 1/16 inch in size, instead of stopping at the Punto, presumably each Punto should be divided into half, with marks representing six Atomi.
The effect of this on the shape of a violin can be significant. Take the example of Stradivari's 1720 violin given above, with a body 14 1/16 inches long/high, with the neck side 1 3/16 inches high/thick and the button side 1 1/4 inches high/thick.
In units of 1/16 of an inch, one has two heights of 19 units and 20 units, separated by a distance of 225 units; if the figure of the violin's ribs is a symmetrical wedge, then the two plates are tilted, in opposite directions, from the axis by the arctangent of 1/450, or about 0.1273 degrees.
Convert those measurements to millimetres, and round them to the nearest millimetre. Then the body is 357 mm long, the thickness at the neck is 30 mm, and the thickness at the button is 32 mm. Now, the plates would be tilted by an angle of the arctangent of 1/357, or about 0.1605 degrees.
If the actual dimensions of the violin were in units of 6 Atomi, about 1.68 mm, then we would have heights of 18 units and 19 units, separated by a distance of 213 units, so now the angle in question is the arctangent of 1/426, or about 0.1345 degrees.
What this really means, of course, is that measurements need to be made to a greater precision than one millimetre, whatever unit may be used, for the angles of the plates to be more than guesswork.