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Chris S

There is another option to geometry

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There is long list of possibilities how to aproximate curves to any given precision. Starting from zillion of linear segments ending with NURBS (non-uniform rational b-splines - that's what CAD uses) and all kind of curve types in between (polynomials, goniometric functions etc). What exactly they used we will never know, but with certainty they had dividers and could draw lines and circles. The arches (and body shapes) are affected by precision of original maker, distortion over time, arching repairs etc. so we cannot judge from how they look now. Even if you start with perfect circular shape and put it under tension the shape will likely slowly creep towards catenary...

There are numerous past discussions about these curves, take a time to read them.

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Geometry and all that clearly has something to do with the invention of forms way back when people were deciding on things. As far as I am concerned - and bearing in mind what a hullaballoo I am having on the other thread, this has a particular place in the evolution of ideas, but:

1) I don't think that ignoring geometry and mathematics in any way makes you a lesser maker. If you can sense what's beautiful in a good violin, it doesn't really matter if there's an underlying theory - but some of us think it's just another one of many useful concepts that helps to inform an overall judgement. By the same token, you don't need to understand the underlying geometrical construction of a painting to know that it is beautiful. 

2) Making a violin from your own mathematical schema does not make it any better than anyone elses. Frankly the violins I like that have been made from these schemes appeal to me because of their coincidence with other classic making - i.e. oops, you've accidentally designed a Sanctus Seraphin. In reality the classic makers had a criteria and judgement about which geometric forms they used, and there are thousands of variants of the same design that they simply rejected, or simply did not adopt. So you could argue that chasing after an Amati-system design that the Amatis never actually made, flies in the face of the wisdom and experience handed down to them. 

3) At the end of the day, what judges a good violin from a poor one has nothing to do with the study of complex Renaissance mathematical ideas. 

4) Complex  Renaissance ideas require you to step outside of the standards of today and explore them with a different kind of eye. Unless you are motivated to do it, there are better things to do with your time. 

Doesn't stop it being interesting in it's own way though. :) 



 

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14 hours ago, Chris S said:

Actually I was not suggesting that the the arching was catenary in shape.   

Only the outline  .....

Chris, the irony though is that Caternary curves were things that mathematicians studied solidly though the period of classical violin making, so inadvertently - if you are right - you would be demonstrating something that maths or geometry would be able to consider, and that mathematicians of the time would see as interesting. Just to be a total sod, during 1637-8 when Galileo was buying his Amati, he was also writing, Discorsi e Dimostrazioni Matematica (1638) which speculates on the curve of a hanging chain (which he thought was parabolic). So - who knows! That's certain to be irrelevant coincidence, but I think it's kind of amusing... 

Robert Hooke, being a pain in the neck, figured out the mathematical characteristics in 1671, but published the code in an anagram in 1675, the key to which was not revealed until after his death: ut pendet continuum flexile, sic start continuum rigidum inversum. In the spirit of Hooke, I'll leave it there. :)

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Catenary chains (curves) are used (with some adjustments) in laying out piano bridges.  They accurately describe an idealized progression of speaking lengths (the vibrating portions of consecutive notes).  We typically have to deviate from the idealized curve for some reasons I won't get into here.

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1 hour ago, Chris S said:

Ben,   that is true.  I know that mathematics can describe all sorts  of things .  From my own study of mathematics ,  which I am only moderately good at,  I know that lines and curves can all be described by maths.   I have every respect for it.  Also for geometry.   I have a copy of Euclids ' Elements.   Also Apollonius's   Conic sections.  I  look through them from time to time.     I just choose not to apply it to this discipline.    It does not suit me.  I want a different approach as I outlined above.  

And  it is ,  as you say ,  interesting for its own sake.    It is a good study in itself.

As for calculus though ...  I bought a book on it   so I could teach myself the basics.    That was ten years ago.    I am about to begin the first chapter. 

Frankly to me, I'm more interested in the broader fantastical ideas of the Renaissance. I'm not very good at maths, and this geometry stuff seriously makes my head hurt. More interested to talk about the ideas behind it :)

 

 

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2 hours ago, Ben Hebbert said:

I'm not very good at maths, and this geometry stuff seriously makes my head hurt.

Math, geometry, and the related physics and mechanics are not a problem for me.

However, I don't see that good sound and a beautiful shape require any mathematical basis in construction... in fact, it could be detrimental if the chosen curves give odd-looking or odd-sounding results.  Better to understand what looks good and sounds good with a practiced eye and ear.  Then it doesn't matter really how you got there, either with mathematically-derived shapes or by eye and experience. But it seems to me that you need the eye and ear anyway, even if you want to use math and geometry, if you want good results. 

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Don

I expect that these early makers had a strong empirical criteria for what they wanted to produce, based on the same experiences that you cite, but in circumstances where mathematics was a priority I imagine that they simply reconciled their "sketches" with a mathematical ideal.

Following in the same vein, when I do my analysis, I record an actual outline (which could be twisted, distorted, compromised in many ways), throw my circles on it to see what's coherent and then based on those insights I produce a mathematical model, and only after that I overlay it to see how the two relate to one and other. There is always some discrepancy. 

Nevertheless, you can only make a copy of the best Strad because Stradivari did the groundwork ... it doesn't matter how you approach it, you could not produce a copy of it unless it is unconsciously faithful to the rationale behind Stradivari's own ideas. 

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5 hours ago, Chris S said:

Ben,   that is true.  I know that mathematics can describe all sorts  of things .  From my own study of mathematics ,  which I am only moderately good at,  I know that lines and curves can all be described by maths....................As for calculus though ...  I bought a book on it   so I could teach myself the basics.    That was ten years ago.    I am about to begin the first chapter. 

The basic concept, as elucidated by Newton (standing on the shoulders of Descartes), is simple enough.  Basic two-dimensional analytic geometry shows us that curves can be generated from equations, and is mostly what we've been jawing about in the "geometry" topics.  Newton (and Leibnitz) spotted the cheap tricks of constructing a line tangent to any point on a continuous curve by dropping the powers of the variables in a curve's equation by 1 and solving the new equation (differential calculus), and of finding the area bounded by the curve on top, and X=0 on the bottom, by raising the powers by 1 and solving (integral calculus).   Like a lot of concepts, it took a genius to spot it, but it doesn't take a genius to do it.

Most textbooks on it are made dauntingly complicated by dragging students who only need to learn the tricks through the theory of limits first.  I recommend Calculus Made Easy by Silvanus P. Thompson as the best way to get started for folks who aren't math majors.  The thing is still in print, and copies are easily found. https://en.wikipedia.org/wiki/Calculus_Made_Easy

All that said, it should be obvious that anything invented in the late 17th. Century (rigorous solutions to the catenary curve) didn't contribute to a concept (the violin) invented in the early Renaissance over a century earlier.  Q.E.D.  :)

 

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I feel that an overlooked branch of mathematics for violin makers is statistics and probability.  

If you make enough violins  some of them might be pretty good (even a dumb squirrel can find a nut sometimes).  And if the various peaks of resonance frequencies are distributed (by luck or by skill) such the addition of these makes no note is overly strong or weak sounding then your violin is "even" sounding and is liked by players. 

 

 

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45 minutes ago, Marty Kasprzyk said:

If you make enough violins  some of them might be pretty good (even a dumb squirrel can find a nut sometimes).  

I think that the statistics show that certain makers make good ones more often, and certain makers make duds, so there's a lot less luck than a dumb squirrel just hunting around for a long time.  It's more talent, skill, and knowledge. 

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5 hours ago, Marty Kasprzyk said:

 If you make enough violins  some of them might be pretty good (even a dumb squirrel can find a nut sometimes). 

C'mon Marty, I do not think this is your approach to violin making, or at least I hope so.....:);)

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4 hours ago, Don Noon said:

I think that the statistics show that certain makers make good ones more often, and certain makers make duds, so there's a lot less luck than a dumb squirrel just hunting around for a long time.  It's more talent, skill, and knowledge. 

Yes, sure, but that's not what Marty said. He said "If you make enough violins  some of them might be pretty good (even a dumb squirrel can find a nut sometimes)".  In other words, even makers who make "good ones" more often sometimes make one which is exceptionally better for no reason one can see. There is something very interesting here - if this does not happen it means the maker does something not conducive to "more betterness".

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2 hours ago, Chris S said:

What a stupid response !

No, not stupid at all. On the other hand, your comment... It's moderately important albeit not essential to reconnect to one's brain before placing one's "thoughts" on MN. 

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17 minutes ago, Chris S said:

Sorry Marty .  Perhaps I was a bit quick off the mark   to take offense  and 

respond that way.  

Hi Davide,

20 hours ago, Marty Kasprzyk said:

I feel that an overlooked branch of mathematics for violin makers is statistics and probability.  

If you make enough violins  some of them might be pretty good (even a dumb squirrel can find a nut sometimes).  And if the various peaks of resonance frequencies are distributed (by luck or by skill) such the addition of these makes no note is overly strong or weak sounding then your violin is "even" sounding and is liked by players. 

 

 

 I'm searching for a physical or mathematical explanation why only some of Strad's violins were great while some were just so-so.  Something was changing from one violin to the next that even he was unable to control in his making process.

The attached write-up tries to show why resonance peaks shouldn't be an octave apart and why the valleys between those peaks shouldn't be an octave apart either. 

 

 

 

Measuring violin note evenness from frequency response curve.pdf

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5 hours ago, Marty Kasprzyk said:

I'm searching for a physical or mathematical explanation why only some of Strad's violins were great while some were just so-so.  Something was changing from one violin to the next that even he was unable to control in his making process.

Certainly each piece of wood is at least a little bit different.  If greatness is significantly controlled by the wood, I think it will be extremely difficult to extract that information from assembled old instruments.

5 hours ago, Marty Kasprzyk said:

The attached write-up tries to show why resonance peaks shouldn't be an octave apart and why the valleys between those peaks shouldn't be an octave apart either. 

While non-octave resonance spacing might be nice, I don't believe it's that big of a deal.  My best violin by far has A0 and B1+ at C#, an octave apart.  It may be more of a problem if there are higher-frequency peaks and valleys that line up, since that will affect multiple played notes.  The A0 B1+ alignment primarily affect just one note... C# on the G string.  And that's not so bad if there are good, strong overtones.

1 hour ago, Chris S said:

Science and our understanding of science is limited and it cannot grasp the complexities of the structure of a violin. 

I partly disagree with this... the science of acoustics and structures and vibration and sound could grasp pretty much everything that happens with a violin (given enough money).  It is very complex, so armchair science and garage experimentation can only pick at the simple parts.  Still, any  understanding that can be gained from science I feel is better than not understanding it, although there is the potential pitfall of being guided by partial or mistaken understanding that leads in the wrong direction. 

What science can't do is decide what is good, bad, or great, although some statistical surveys could give some vague clues about that, too.

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2 hours ago, Don Noon said:

What science can't do is decide what is good, bad,

That's a real question....may be the first

The good violin is alway the violin you love, difficult to make good science with that matter, I hope so...

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7 hours ago, Chris S said:

OK ..  fair enough   , you do not accept my reply  .  That is a pity.   

However  ,  fundamentally I do not think violin making is a science.  Science and our understanding of science is limited 

and it cannot grasp the complexities of the structure of a violin.   It is a sculptural form  like any other sculpture  and it is 

essentially and principally an art form .     There is also an intuitive response that is too hard to measure.  

I'm sorry I didn't quickly thank you for your response. I hope you accept a late one.

I very much agree that science and art are separate and one can not completely explain the other.  

However after doing a lot of oil painting I found that most of my blue-red-yellow combinations came out looking like brown mud and there must be a physics reason why this  happens.  One of my better paintings has a plowed field which shows art and science can at least occasionally help each other.

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