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Curtate cycloids revisited


catnip

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There was a lot of discussion about curtate cycloids in the past especially about the mathematics of it ... generating the curve using parametric equations height and width as function of rotation, and height as function of width  (essentially eliminating theta) and about inflection points, recurves (extending a complete rotation by a small percentage)  Here are two articles that might be of interest that I found on www.savartjournal.org wriiten by R.M. Mottola.

 

The circle,curtate cycloid and sine curve are compared in the transverse arching of 6 cremonese violins.  The circle is used as a reference curve to show how the cc compares to it.  It seems that the cc fits the best but there are some differences or variations.

 

 

 

 

Cycloids_Savart_1.pdf

Cycloids_Savart_2.pdf

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He does some very dodgy things to make his case. First, he defines the end of the recurve not by where it stops and changes to something else, but by edge thickness. Second, he ignores what he defines as the recurve area completely. If you remove enough parameters, pretty soon a straight line would do the job. With a 1cm straight edge, you can demonstrate that any spot within the area he's looking at is a pretty good, if not exactly perfect, fit to a straight line.

 

As far as I know, a circle cannot define recurve at all, and on the sine wave he cuts it off because it doesn't fit. Remove all of the things that don't work and pretty soon everything works.Leave all the parameters, however, and only a cycloid even comes close. How inconvenient!

 

The consistent problem with inside-first, circles, bent sticks, and a lot of other strategies is that they really don't define the whole arch, most especially the crucially important areas all around the edge. That should be the whole point, and instead it gets completely ignored because it doesn't fit with various pet theories. Rather than solving the problems, how convenient to just chop them off and ignore them!

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One thing which seems evident from these papers is the rather high degree of variability in archings, even from a single Cremonese maker or shop, and despite some claims to the contrary in a concurrent thread:

http://www.maestronet.com/forum/index.php?/topic/332030-no-need-for-arching-templates/

 

I too have a hankering for a simple and original system, as long as I don't need to drop my pants, grab my ankles, while hitting my head with a rubber hammer to embrace it.

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What's interesting to me is the question of whether the variability is the result of a failure of execution (like late del desu scrolls vs the templates they were traced from) or whether there is no underlying architectural underpinning. I know I've had to remove wood from an arch I wish I didn't have to, in order to remove a chip in the curl or some such thing.

Interesting topic.

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What's interesting to me is the question of whether the variability is the result of a failure of execution (like late del desu scrolls vs the templates they were traced from) or whether there is no underlying architectural underpinning. I know I've had to remove wood from an arch I wish I didn't have to, in order to remove a chip in the curl or some such thing.

Interesting topic.

I'll venture that there was some kind of underpinning, and also that anyone who has made a fiddle or two has deviated from the original design to accommodate screwups .

Over the next 150 to 200 years, maybe some things about the original underpinnings were lost, but more important things were gained, as with Stradivari and Del Gesu compared to Andrea Amati?

(before violinmaking dropped into the big abyss)

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Perhaps I'm just too inquisitive or open-minded, but there's just something unsatisfying about accepting Cremonese arching as the peak of perfection, and then looking for some perfect mathematical construct to re-create it and thus achieve perfection on top of perfection.  This all bypasses the obvious (at least to me) questions of 

-What does the arching DO TO THE TONE?

-HOW does it do it?

 

That's just me, I guess... If I'm going to do something, I want to know why I'm doing it... not just because someone else did it that way, and some (not all) folks think it's really great.  There is nothing in the way of physics or acoustics that says a cycloid is superior, and if it is, then why isn't it used for the top long arch?

 

I've got nothing against cycloids; I think they look very nice and work fine.  I just don't worship them, or think that alternatives are worse by default .

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So, I hope a fundamental question here: We've all seen in paintings how a very beautiful organic form may take it's beauty from the way that it concordes with a geometrically (or otherwise mathematically) formed ideal - the really good example of this would be the Mona Lisa, because we have all seen the way that geometry has been transposed on that (rightly and wrongly) by different people. 

 

post-52750-0-88750100-1424035061_thumb.jpg

 

Accepting the limitations of curate cycloids generally, I contend that the presence of a mathematical framework that generally describes what is going on would have been as central to Andrea Amati as it would have been for anyone studying the art of painting. Part of the luthier/painter's art is to work within that rigid framework and produce something that is organically beautiful. 

 

Therefore, speculatively, a study and thorough understanding of curate cycloids would allow us to get inside the heads of makers from Andrea Amati to del Gesu, but that does not mean that they have to perfectly explain the forms of instruments made by them, just as you cannot describe every element of the Mona Lisa by a continued geometrical narrative. 

 

That may be the closest we can ever get to understanding how the Burgess rubber hammer works! 

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I too have a hankering for a simple and original system, as long as I don't need to drop my pants, grab my ankles, while hitting my head with a rubber hammer to embrace it.

 

David, have you considered dropping your pants to the level of your ankles, grabbing the crotch of your jeans instead of your ankles. That method would effectively restrain both ankles with just one hand, making it infinitely easier to hit your head with the rubber hammer using the spare arm. - why make life difficult for yourself? 

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"...That's just me, I guess... If I'm going to do something, I want to know why I'm doing it... not just because someone else did it that way, and some (not all) folks think it's really great. There is nothing in the way of physics or acoustics that says a cycloid is superior, and if it is, then why isn't it used for the top long arch?..."

 

But you use an established mold for the ribcage, accurate to the millimetre, and make the ribs 32 mm high at the bottom and 30 at the top, and size and place your f-holes as close to the model as possible etc,  Do you know why you're doing all that?  But on the arching you say nope!  I'm not interested in slavishly copying what Cremona did?

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David, have you considered dropping your pants to the level of your ankles, grabbing the crotch of your jeans instead of your ankles. That method would effectively restrain both ankles with just one hand, making it infinitely easier to hit your head with the rubber hammer using the spare arm. - why make life difficult for yourself? 

Brilliant, I'll try that first thing tomorrow. :)

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I'm willing to bet that you could find a strong correlation between glacial landforms and Curtate cycloids. But your career as a geomorphologist would be over if you claimed that glaciers used cc templates for creating landforms like drumlins. Just a thought...

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What's interesting to me is the question of whether the variability is the result of a failure of execution (like late del desu scrolls vs the templates they were traced from) or whether there is no underlying architectural underpinning. I know I've had to remove wood from an arch I wish I didn't have to, in order to remove a chip in the curl or some such thing.

Interesting topic.

There's one del Gesu we had at B&F that had an unusually scooped arch. We wondered why, and eventually someone noticed the tailings of what must have been a huge tear-out. Whoops! Better cut that out! And thus the strange arch. That's why I stress the idea of an underlying design--the idea that if you know what they knew, then you at least have a starting point for experimentation. If you don't admit to any underlying design, you're pretty alone. 

 

Also with some understanding of what the concept was, we can develop an understanding of how they morphed and changed, and the parameters they were willing to compromise on vs the aspects they felt always had to be there.

 

Regarding the superiority of their system, I've worked with several well-known makers who had problems like being commissioned to copy an instrument but wanting to reconstruct the arch. As one of them told me, "I've been making for 30 years, and this is the first one of my instruments that could be said to be 'Cremonese' sounding." After just one copy he was talking about completely revamping how he approached arching.  Others have noticed, on their own, how when they put more care into keeping to the templates, the results improved suddenly and radically. The reason I keep pushing on this is how definite the results have been for other people who've tried the strategy, in proportion to how well and accurately they work it--not just for me.

 

This isn't something I've been keeping to myself--I've taught quite a few people to use it, with good results for them. I think there may be a misconception that I'm in this alone. More to the point is the number of people who've told me that I should keep this all quiet. But as you can see from these threads, there's no threat to them. :-)

 

If you do it, and do it carefully not casually, the result can be remarkable. Part of the reason I believe that Stradivari violins are more consistently good is that my templates always fit his instruments more closely than they do on other makers' violins, makers who were being more casual about what they were doing.

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In terms of maths, there is an interesting paradox about the outline of a violin which may bear relevance to curtate cycloids..

 

We are, I think, all absolutely agreed that Francois Denis' geometry is correct, and therefore, the mould, the interior and the inside line of the purling reflect a geometrically perfect form. 

 

However, the total outline, includes the extra thickness of the ribs and the overhang. This results in a shape that we nevertheless find pleasing to the eye, but there is nothing geometrical or proportional to this form of enlargement. Therefore, although the violin outline is organically pleasing - the fact of the matter is that it is an arbitrary shape formed by wrapping aribitrary margins around a (mathematical) form. It is not of itself mathematical at all. 

 

Francois has also mentioned (I hope I am not misquoting him) that the precise curves of the pegbox has so far evaded any mathematical reasoning, even though its proportions are all bound up in how the scroll is laid out - another example - plausibly - of organic beauty not being rigidly governed by the underlying geometry. (But those rules being there nevertheless)

 

Both observations could act as a parallel to what could plausibly be the case concerning  rubber hammers curtate cycloids. 

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 Regarding the superiority of their system, I've worked with several well-known makers who had problems like being commissioned to copy an instrument but wanting to reconstruct the arch. As one of them told me, "I've been making for 30 years, and this is the first one of my instruments that could be said to be 'Cremonese' sounding." After just one copy he was talking about completely revamping how he approached arching.  Others have noticed, on their own, how when they put more care into keeping to the templates, the results improved suddenly and radically. The reason I keep pushing on this is how definite the results have been for other people who've tried the strategy, in proportion to how well and accurately they work it--not just for me.

 

My experience has been that various well-known and successful modern makers have subscribed to quite a wide variety of successful systems.

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Oh, that's absolutely true. A good advertising campaign is much more important that a good violin. One of your friends, I think he is, told me that he thought it was pretty funny that his reputation was built entirely on something that he's done very little of but got excellent press on. There are many paths to fame, for sure.

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Oh, that's absolutely true. A good advertising campaign is much more important that a good violin. One of your friends, I think he is, told me that he thought it was pretty funny that his reputation was built entirely on something that he's done very little of but got excellent press on. There are many paths to fame, for sure.

Doesn't sound at all descriptive of any of my personal friends. Since you express some uncertainty yourself, an easy way to find out would be to ask me. :)

 

When I mentioned "systems", I was talking about construction systems, not hokey-pokey marketing systems. ;)

 

But marketing systems might be an interesting topic for some people, and maybe worth a thread of its own?

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I'm inclined to see the 'variability' as being almost entirely in the long arches and the channeling.  

 

I believe the old makers exercised relatively greater freedom in handling some aspects of the channel.  There is significant variability, not just across different makers, but even really within one maker's work.  To be more particular, I see five major variables in the channel handling: 1) the line giving the bottom of the channel, 2) depth of channel bottom, 3) the line giving the point of recurve inflection from concave to convex, 4) the line where the arch raises again to equal the hieight of the working edge, 5) the height of the working edge. 

 

I suspect the channel was worked in relation to a marked channel boundary guide.  In earlier Amati family work, the various features of the channel seem to derive from a reasonably consistently placed channel boundary. Edge level returns at channel boundary.  Inflection occurs about the same place. Bottom of channel mostly occurs midway of channel boundary and some logical line in the edge work, etc.  But in later generations, these variables get decoupled from the channel boundary.  This might have been their way of experimenting to improve arching behavior, while adhering to a fairly traditionally set nominal channel boundary. 

 

My hypothesis is that they set an inner margin for the channel area (distance from outline) by comparatively consistent means, then worked the channels rather individualistically and comparatively freely.   I hypothesize that this is the source of the variations in archings.  However, the curves falling from the center line (top of long arch) to the margin of the channel form a very consistent family of curves.

 

The family of curves for this central portion of the cross arching is far too consistent to be 'at liberty'.  True enough, CCs can be made to give a fair emulation of this family of curves.  However, a much simpler workshop method gives the family.  1/2 the fall occurs in 2/3 the run.

 

post-30802-0-70907000-1424039103_thumb.jpg

 

Applies recursively for the central portion of cross archings. 'Run' is from the top of the arch to any point up to the channel boundary.   Provides very good (not perfect) fits very broadly for Cremona archings.  Given looseness of work and later distortions, hard to imagine any better fit to historical examples by any method.  Tops very often show slightly tighter curvatures through soundhole regions than this method predicts.  Can be understood as an initial shape by this method that is subsequently scooped in a bit further around the soundhole areas.

 

This method has the very great advantage of being easily implemented with nothing more than dividers and straight edge. 

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Perhaps I'm just too inquisitive or open-minded, but there's just something unsatisfying about accepting Cremonese arching as the peak of perfection, and then looking for some perfect mathematical construct to re-create it and thus achieve perfection on top of perfection.  This all bypasses the obvious (at least to me) questions of 

-What does the arching DO TO THE TONE?

-HOW does it do it?

These sound like rhetorical questions. 

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Just a few observations an and a question.

 

 

Whatever the method used by a working VM it has to be efficient .That would include the use of pre-made templates, the measuring of certain key hights at particular points with a caliper of some sort but almost certainly not measuring many points over and over as was suggested in the thread on using templates. 

 

 The strength of a recurved wooden arch is not dependent on the outside or inside curves but rather of a mean between the two. That is you may still be able to draw a simple arch from rib to rib through a section of the plate despite some recurve and the web thickness would be dependent on how much recurve there was with anything outside of that web adding more weight than strength. Sorry  I don't have the science to say that better but my point is that this gets truly complicated and  single line geometry doesn't tell the whole story.  

 

And now for the question. While The Amati obviously had knowledge of Euclidian geometry could curtate cycloids be derived from the maths that were known to them?   

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Nate,

 

Curtate cycloid templates can be made using some scraps of rib stock and a pair of dividers. I've done it many times - it only takes a few minutes.

 

It's my opinion that there is a lot of drawing things with a compass in violin design, but I would be wary of calling it "geometry". I think it's rather a system that uses a compass and rule.

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And now for the question. While The Amati obviously had knowledge of Euclidian geometry could curtate cycloids be derived from the maths that were known to them?

I was hoping someone would ask the obvious. :) just because we can approximate something with a given method doesn't mean it was derived that way originally. There are compass drawings in the Stradivari collection, but no Curtate cycloids I am aware of.

Anyhow, here are some landforms that remind me of Sacconi.

post-35343-0-25263700-1424056864_thumb.jpg

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There was an earlier topic on the Luthier's Library cc's VS the projected lines. I don't recall anyone being wildly enthusiastic about the cycloids. The few examples I've looked at (low sample size), the cc's were not a fantastic fit. Granted, one of those is the Stainer...

And "absence of evidence" leads me to the space alien hypothesis: if there is better evidence for the existence of space aliens than there is for your hypothesis, the term "robust" should be avoided. :)

Personally, I like arching templates derived from direct measurement: no gaps for extra-terrestrials to creep through...

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The Luthier's Library would throw a cycloid over the kitchen table if they had one. No one has said they fit every violin ever made, and even in Cremona maker quality and instrument quality varied quite a bit.

 

Direct measurement is fine if you like second or third (fourth? fifth, even?) generation resolution, but where are you going to get an undistorted 300 year old violin?

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