Here is some (probably too much) detail about the ratios I did find:
The ratios actually observed in Cremona classical making seem to be based on nothing grander in concept that the notion of ‘a part’ and differences based on a part.
These appears consistently in the ratios for almost every ( or actually all?) feature.
Let’s look at some examples:
The span across the upper eyes (between for celli) is executed as ‘the Stop Unit’ less a part’. So this can mean anything from 7/8, 6/7, 5/6, 4/5, 3/4. Notice also that if the ‘part’ is ‘none’, then this pattern includes the StopUnit itself.
The upper eye diameter then is in turn simply ‘a part’ of the EyeSpan. In practice this is usually 1/8, 1/9, or 1/10. But in ‘theory’, we can see that this series of options natural expands as 1/7 or 1/6 for a bigger eye, or 1/11 or 1/12 for a smaller one. The patterns couldn’t be simpler or more practical in use.
The lower eyes in turn are ‘a part’ bigger than the upper eyes. In practice, 2 to 3 is most typical, but we see many other choices like 3:4, 5:6, etc. --Always on the same consistent principle.
We also see the same sorts of ratios used in the ‘guide rules’ that help decide execution choices, but are not actually used to carry out the work. We see this example of this in the EyeSpan choice. In earlier generations of Cremona, the eyes usually don’t stray far from 1/4ths across the CenterBout width. And, in a good number of instruments from later Bros Amati through early Strad, the upper eyes general don’t stray far 1/3rds of the UpperBout width. These relationships to bouts have a nice appealing logic in considering the horizontal spacing and sizing of the upper eyes. Yet, we observe in the actual example instruments that true fits to these ‘guide rules’ are only sporadic. Nor are the eyes consistent placed symmetrically to the sides. While still a bit loose in accuracy, the eyes are more aligned to the center line of the instrument then to the naturally asymmetric bout and corner lines. And, in a far from immediately obvious relationship, the EyeSpan consistently derives from ‘a part less than the StopUnit’. This is interesting, because it builds in a relationship between the string and stop lengths and the soundhole system. However, in most cases, the ‘part’ chosen, i.e. 7/8th versus 3/4th is chosen to create an EyeSpan close to the 1/4th cBout guide, the 1/3rd UB guide, or both.
Let’s consider relationships in the head and scroll. The sizings here key from the scroll height. The scroll height derives again from the StopUnit. As it turns out, the StopUnit is a very important key in the Cremona instruments. Once again, we see the ratio for this relationship is based on the difference of ‘a part’. Most typically, the Volute or head height is ¾ the Stop Unit. However, particularly in violas, we also see the choice of 2/3 ratio. Again, the pattern could be logically extended if a making a special instrument that somehow presented motive for a size significantly larger or smaller than the standard choice of ¾ verus 2/3 allowed.
In turn, the volute frame, length:height, is typically in Classical Cremona work almost always simple 3 to 4, or 4 to 5. Notice also, that the turns and eye of the Cremona volute can be looked as presenting two more nested frames around the mid and inner volute turns. These frames also present the 3:4 or 4:5 ratios. And, in Cremona work, the reduction of diameters from one turn to the next tend very strongly to be based on 1/3rd reductions. In contrast, in volutes outside of Cremona, ¼ reduction are a reasonably common alternative.
So still, we are seeing a pervasive use of ratios with a ‘difference of a part’ concept running through all.
Now lets look at the stop unit, since this is such an important key. Also, there are interesting complications here that illustrate how Cremona work tends to stretch rather than break a habitual principle.
So, in most violin instruments, the basic body stop to neck stop ratio is 2 to 3. And, as with many of our modern numeric standards, the old ratio still lurks within. The often quoted modern standards for neck and body stop of 130 to 195 are in fact in a 2 to 3 ratio. Of course, the body stops and neck stops varied more in actual measure in the old Cremona work, but the used the 2 to 3 ratio.
So what do I mean by the ratio unit? The 1 in for example 2 to 3. In the modern numeric standard, the unit is 65mm. x2 65= 130. and x3 65 = 195.
But what happened when the old makers wanted a shorter or longer neck. For example in a piccolo, you likely want a longer neck than the normal ratio to fit the hand better. And in a big tenor, you likely want a shortened neck.
Give what we looked at so far, it shouldn’t be too surprising that the practice we can observing them using was to adjust the neck length by ‘a part’. They used ‘a part of the Stop Unit’ for this purpose.
So we see things like (2 + ¼)::3, or (2 – 1/3)::3.
Notice that the fairly common modern neck stop ratio of 7 to 10 equates in terms of the old system to: (2 – 1/10)::3
The there is more we could look at, all again illustrating the same basic ideas, lets leave this by looking at some of the ratios arising in the body outline work.
First, lets notice that the typical Cremona ratios for Body Length to Width can be viewed in the same ‘part less’ light. These ratios amount to: ‘Take the body as twice the width, less a part’. So, if the width is 4, twice would be 8, and less a part gives the traditional/observable 7, so the ratio is 4::7. We can see that all the Cremona width to body ratios follow this: 2::3, 3::5, 4::7, 5::9.
Now, as with the neck stop, we again can see that in special cases that can’t fit, the preferred to derive a solution from the standard ratios, rather than just choose a special ratio for the special case.
So, Strad’s Pochette can be observed to be based on the standard 3::5, used commonly in stout shaped celli, violas, and viols, etc. But to get the skinny Pochette, it is used as (3/2)::5.
Now let’s consider the Vesici divisions of the bout lines. These have more significance than just the vesici, because commonly the initial arcs radii rising from the bout lines toward the corner areas are based on the units and logical locations created on the bout line by the vesici. Similarly, the circle diameters for the outer corner circles tend to be based on the Vesici division of the bout line.
The basic conceptual or ‘guide’ idea of the lower bout Vesici is 1::1::1, the vesica piscis where the overlapping circles divide the line equally in three parts.
However, the most common variants observed in classical work for this are 4::3::4, and 5::4::5. But how are this continuous in a simple ‘difference of a part’ pattern? While we have no way of knowing, we can consider that possibly they work or thought of 1::1::1 in terms of the equivalent 4::4::4. This idea certainly did not occur to me in the early stages of my research. But, as I moved along and accumulated more and more examples and details of their choices, the possible suggested itself.
For one thing, the ‘bodega’ book, which is clearly not about the methods observable in the generations of Cremona work, but is about a system not greatly unrelated; that book often mentions ratios in unreduced form. So, not inconsistent with 4::4::4 instead of 1::1::1. Perhaps that book opened my thoughts to this notion.
But then there are other points. 5::4::5, 4::4::4, 4::3::4 do form a pattern based on differences of ‘a part’. So that is consistent. Further, initial riser arcs in Cremona work, from both upper and lower bouts general are centered in one of three ways: on the bout center, coinciding with the vesici circle itself, or centered and with radius ‘one vesici unit larger than the vesici circle’. However, viewed as 1::1::1, a number of the Vesica Piscis cases appeared to present the risers anomalously as centered mid way from bout center to vesici circle center. But, if we understand these cases as 4::4::4, then the risers present the normal ‘one vesici unit larger’ situation seen most commonly in all the Cremona work.
Adoption of the notion also help in understand the corner circles in a consistent way. The last circles giving the corner circles appear to be based on overlapping sets of concentric circles. The circles on the outer edge and circles on encompassing the purf are the main circles in this. And they meet and overlap cleanly in forming the corner. The corner designs involve two pairs of these circles overlapping, one pair outside the cBout area, one inside the cBout area. In each pair, one circle will use a ‘key’ radius. The key radii in the out circles generally derive from vesici divisions of the bout line. The key radii for the cBout corner circles generally derive as ‘a part’ of the radius of the main cBout circle.
Once again, understand the 1::1::1 vesici examples as 4::4::4 helps us see the consistency of the observable practices. Otherwise, a common sizing for the lower outer corner circles appears to be ¼ the lower bout, and not obviously related to division of the bout as 1::1::1. But, using 4::4::4 as the idea of the division puts these 1/4LB key corner circles in relation to the vesici divisions, just like all the other examples of key outer corner circle radii.
Ok. Enough uber nerdy ratio detail for now.