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PostPosted: Tue Jul 31, 2007 9:05 pm 
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Cocobolo
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Hello All,

I know the definition of luthiery, or what a luthier is, has been the discussion of many topics. But,
I’ve been wondering about something lately, and it’s mostly due to the several topics I’ve read in
the past few of weeks. To name a couple, “Tone bar placement” by Hodges Guitars. And, “Sound
ports suck?” by SteveS. Although one of my favorite subjects is Physics, that particular thread
(As Paul Woolson said) made my brain hurt. However, it did make something very clear to me.
Even though my first guitar turned out very well, I realize now, I have so much more to learn
about building stringed instruments.

After reading posts by folks here at the OLF that are obviously well versed in Lutherie. It seems
to me that there is another level to stringed instrument building, that goes beyond the act of
construction, so perhaps the title used for this topic “The science of lutherie” is appropriate.

When I read things like how a top along with it’s particular bracing is a “System”, and, how
“thermodynamics” are involved with a dual ported guitar body, that tells me that there are things
to guitar building, that I haven’t quite got a grasp on yet. Simply put, I believe it’s generally
excepted, that there is a relationship between design, and the sound a guitar produces. Of course,
it’s not just that simple, and so enters, the element of science.

So, I’ve been wondering, sort of on a personal level, how did you folks acquire this scientific aspect of guitar building?
And, at what point on your journey to become a luthier, did you realize that there is a science to luthiery

I don’t mean to pry, and only if you felt inclined to share your personal experiences, I thought that
your stories could be very enlightening, and encouraging to those who aspire to be a highly skilled luthier.

Sincerely, Robert

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PostPosted: Tue Jul 31, 2007 10:12 pm 
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Koa
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First name: Bob
Last Name: Johnson
City: Denver
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Status: Amateur
The REALLY good luthiers are really magic nomes


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PostPosted: Tue Jul 31, 2007 11:33 pm 
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Koa
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    There is the Science of Luthierie and then there is the Art of Luthierie. The mechanics of the guitar are important and you need to understand that when you are building.
     The one area of a guitar that I do go by the numbers is the neck angle and set up. I do want on a steel string to have about .500 to .550 inch height at the top in fromt of the bridge so that is the total height of my bridge saddle.
   
     If you do the numbers of resultant forces the is about the perfect height for the most energy to be transformed to sound. You don't need to go higher as then you tend to over torque the bridge and saddle and that takes energy off the top.
    Too low and you don't get the optimum energy transfer. The above books mentioned will help you understand the forces applied to the body so you can plan on how best you want to use the information.here are so many variables in the construction of a guitar that it can stagger your mind. The wood thickness and weight to strength ratio. Gluing techniques and bracing to name a few. Finish type and thickness , glues. Once you learn to how the applied techniques affect the overall guitar you can decide what you are looking for.
good luck and have fun.


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PostPosted: Wed Aug 01, 2007 12:04 am 
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Cocobolo
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Robert

I think what you are really asking is not so much how does one "science", which is really just an investigative technique, but rather how does one acquire an understanding of how the structure of a guitar effects the tone.

I think there are several ways to acquire this understanding. Traditionally builders learn this through long term experience. Trial and error with different woods and different structures. Over time our intuition develops and we just "know" what to do to get the desired results. This approach works but it does take a long time.

A scientific approach, where careful measurements are made and the tonal outcome is objectively correlated with structural changes also works. A great deal of understanding can come from that. It is not always easy, though, to devise experiments that isolate the structural feature in question well enough. The guitar is a very complex system and everything seems to be interrelated with everything else.

Another way is what I would call a "technical" approach. Where the structure of our guitars is measured and records are kept. Over time, the data base makes it possible to correlate the structure itself with the tone of the finished guitar. Keeping Chladni records, or deflection measurements, are examples of the kind of data I'm talking about. Once a link is found between measured properties of the guitars components and the tone of the guitar, it becomes possible to use these measurements as milestones as we work our way through the construction process. They become a road map of sort to keep us on track to the desired result. This method isn't much different from the intuitive method except that the measurements are made by some mechanical means rather than just by "feel" and the results are written down rather than just committed to memory.

I think most of us use all three of these methods in differing proportions. Some rely more of the scientific approach and others rely more on intuition. The trick is to find the mix that is right for you. Any way you go about it, it takes a while, but it is a fascinating journey !!

Mark



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PostPosted: Wed Aug 01, 2007 1:05 am 
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Brazilian Rosewood
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[QUOTE=Blanchard] Another way is what I would call a "technical" approach. Where the structure of our guitars is measured and records are kept. Over time, the data base makes it possible to correlate the structure itself with the tone of the finished guitar.

!!Mark
[/QUOTE]

That's the scientific method at work, right there

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PostPosted: Wed Aug 01, 2007 2:23 am 
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Koa
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    The art of building to me is not the technical end as the bracing bridge action have to be to a determined location of you will have a wooden box. The fact is you have to understand the mechanics and the physics of the guitar.
     The object is to control the forces applied so you can turn the energy stored in the strings to work ( music ). The more efficiently you learn to do this the better you guitars will sound.
    Balancing all the variables is not easy and comes with experience and learning. Keeping building logs is a great tool in that you will learn more from a failure than you will from success.
     Keep the wood dry and the strings on the outside.
john hall
blues creek guitars


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PostPosted: Wed Aug 01, 2007 2:41 am 
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Koa
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    There will always be as many methods and opinions as they're are
luthiers discussing them and all are just what works for each builder in
their shops and for their own guitars.

    Rick Turner has had his obvious and illustrious experiences in so many
areas that enable him to approach lutherie from some directions that
many of us would have difficulty doing. In the "Soundports suck" thread
Rick gave a a few tidbits of his revelation that closed some loops for me
and helped a few things that had plagued me for years make full sense.

    I've always taken a very old school approach to my building by choice
and just because I really enjoy the process that had become regular to
me. Even though I'd made moves to some more high tech and scientific
methods and equipment for a few years, I made the decision to swing
back to my old school ways and gear for the charm of them as well as to
keep the craft and process as enjoyable for me as possible. It's important
that the customer enjoys the completed guitar, but just as important that
we enjoy the process of creating it, in my opinion.

    I'd built as many as 52 guitars in a year at one time, but found myself
regretting that short walk to my shop on more mornings than I wanted.
Those feelings made me realize quickly that my business had taken a turn
to one of numbers and deadlines and production from one of inspiration
and passion and simple joy of being in it. I'd spent some time in a very
high pressure guitars building production environment and have never
wanted go back into it in any way.

    Science and the facets of it that can be applied to lutherie are
intriguing and challenging to understand and a background in them can
certainly be an asset to any luthier's work. I'm not a engineer by trade or
education, by any means, but have studied much of what an engineer will
during their course of study. On top of those studies, I've always been in
close contact and communication with engineers in several fields as part
of my effort to become a better luthier and to more deeply understand
what makes these wonderful things that we build work and how we can
make them work better.

   Working as a toolmaker and journeyman machinist also allowed me to
see that the practical application of an engineers work on paper to an
actual product was a valuable asset simply because it was the proving
ground for the engineers work that made sense on paper. Many times, I
would catch a problem that would prevent to product from working or
even from being produced that the engineer had overlooked or not
considered so both sides of the coin in that relationship proved to be of
equal importance.

    I think the more appropriate term would be "Science applied to
lutherie" or Science in lutherie" as Filippo stated above, but I don't think
lutherie itself is a science. Just as fine cabinet or furniture making
requires the application of science, they're not science themselves.

   By all means, understand the science that can be applied to and is
revealed in lutherie, but never forget that it's all about the craft of lutherie
for bith builder and player.

Just my opinion,
Kevin Gallagher/Omega Guitars

   


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PostPosted: Wed Aug 01, 2007 2:45 am 
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Brazilian Rosewood
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To me, it is truely a science. The big difference is you throw in the variable of vibration into the mix and then reverse engineer from there to determine which wood, which brace or tone bar or what part of the finishing process made the difference.


I really appreciate people like Tim McKnight who arent afraid to think outside of the box and come up with innovative ideas. From what I can see, McPherson is another one along with Babicz that are expanding their thinking outside of the box and growing the art and science of lutherie. At some point I fully expect someone to come along and say "what box??"


As far as my personal lutherie skills, I am one who is daily trying small improvements in the standard instruments. With each completed instrument I will be able to tell what difference each change from the normal made in the final outcome of the instrument. Recently I was able to play a new build without telling anyone that it was any different than any other of the number of guitars they had seen and heard. It certainly didnt look any different. When I looked up in the middle of the song to get a reaction from the crowd, I noticed that nobody was talking and everyone was staring intensely at me and the guitar. Most of them had heard me play before, so that wasnt something new to them. Afterward I  had quite a number of people seek me out and ask "what did you do that is so different in that guitar?"....I was beaming from ear to ear because I knew that I had made a difference.


My desire in both the art and the science of this trade is to make a difference.


 


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Ken H


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PostPosted: Wed Aug 01, 2007 3:13 am 
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Brazilian Rosewood
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[QUOTE=ToddStock] 'Left Brain Luthiery' by David Hurd is a good place to start[/QUOTE]

I bought this book recently and was disappointed.
Though some of the concepts are useful and may point one in a new direction, I found little that I could use.

First, most (if not all) of the examples are from the ukelele world, and there is very little data directly from guitars, and virtually all of that refers to nylon-string data, mostly extrapolated from ukelele research (not experimental).
If you are interested in experiments on 'Royal Hawaiian' Soprano ukeleles, you'll love this book!
On page 3, Hurd states that the scale length for a guitar is 26 3/8 inches (670mm) which gives you an idea of where the book is headed.
It's also very overpriced: $55 + shipping for 160 pages of photocopied material with a plastic snap-on binding.

Hurd does introduce some of the fundamental concepts, so it may save you a bunch of searching and research on the internet and in the physics section of your public library.

Though it has its flaws, I found 'Tap Tuning' by Siminoff to be more interesting- though it has a 'mandolin focus'. It has some historical info about Loar and some practical info about instrumentation and measurement techniques. You could take the Siminoff book to the basement and actually get to work with it.

There's little published info in this area- I'm still waiting for the Al Carruth book!

Cheers
John


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PostPosted: Wed Aug 01, 2007 3:30 am 
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Koa
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Location: Amherst, NH USA
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Good science is hard. The science of musical instruments is very hard. But, I don't believe that it is impossible. Does that mean that we will "solve" the guitar? I doubt it. We can, however, apply scientific methods to the analysis of guitars and perhaps gain some insight as to what is really going on. From there we can start to apply those insights to our building.

Some people have argued that the guitar is chaotic and will always defy analysis. That may be true but chaotic systems are not random systems and we can make some general observations about them. For example, you have a waterfall and downstream from that waterfall is an island. Some of the water goes around to the left and some of the water goes around to the right. If you drop a cork into the water above the waterfall it is almost impossible to predict if which branch the cork will take. You can drop two corks at as close to the same spot as you can and still not predict the ultimate path. However, the cork will always go down stream and not back up the waterfall.
Al Carruth has noticed that the guitars that display a certain Clhadni pattern tend to sound better. So, he builds them to they have that pattern. For the most part, his guitars get good reviews and have a certain consistency.

Did this require a full understanding of the guitar? No. But it did use scientific methods to measure the guitar and reproduce instrument that have the same or similar measurement.

We have to be careful, however, that we don't fool ourselves. As Richard Feynman said "The easiest person to fool is yourself". There was this guy who had a horse that could do arithmetic. He'd demonstrate his horse a county fairs. This remarkable horse was studied, however, and it was found, that the horse only could do it's math when the owner was round. And, more importantly, when the owner made an error, the horse did too. What appears to have happened is that the owners body language would change when the horse was stomping out the answer just as the horse was approaching the correct value. The owner probably wasn't trying to defraud anybody and was just as surprised an anyone when the flaw was uncovered.

We have to be care as luthiers when we set up experiments. Builder X, who has build over 1 Godzillian guitars, does many things as part of his process that he no longer is aware of. Suppose he wishes to try new bracing pattern. Does he grab a set of top wood randomly from his stash? I suspect not. His experience with bracing patterns could very well lead him to choose a top set that would work well with this new pattern. I don't know this for a fact, of course, but I can't imagine that he would be able to stop himself. This might lead to some disappointing results if the new pattern is then placed into production where that tops are not so closely matched.

I was talking to a tennis shop owner who said that he almost always sells a racket that he loans out for a trial. The player, when trying out the new racket, will compensate for the different feel of the racket by paying just a little bit more attention to the fundamentals and, guess what, he hits the ball very well with the new racket. A few months later the old habits come back and the guy is back in the shop looking for yet another racket. But I digress.

I think that a scientific approach to studying the guitar is ultimately going to lead to some changes in the way we build. But it is not the approach that every builder should take. Only a few like to view the world that way and that should be enough.


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PostPosted: Wed Aug 01, 2007 4:39 am 
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Brazilian Rosewood
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robertD asked:
"So, I’ve been wondering, sort of on a personal level, how did you folks acquire this scientific aspect of guitar building?"

Tourist in New York: "How do you get to Carnegie Hall?"
Musician on sidewalk: "Practice, man!"

Lutherie is a craft. It's not 'pure art' because we're not totally free to make what we feel like making: as has been said, you have to keep (at least some of) the strings on the outside. It's certainly not a science, since there is no objective measurement that everybody can agree on as defining a 'great' guitar. But it is a 'high craft', with a lot of artistic leeway and sensibility in it, and we certainly use scientific methods to further our craft.

In fact, the designs we most commonly use are the outcome of a long process of 'science', even if it was not mathematically sophisticated. Luthiers came up with ideas about how instruments worked, tried things on the basis of those ideas that they thought would be improvements, and some of those things worked. Any time somebody, like, say, Torres, or C.F. Martin I, came up with something that worked much better than the usual stuff, everybody copied it, and the bar was set a bit higher. As a result the designs we have are remarkably good. That's not to say they can't be improved, but it does suggest that it will be difficult to make _major_ improvements without the injection of new technology. That's happening, of course, with carbon fiber and Nomex, for example, but it may take a while to perfect those things.

Much of the more 'technical' sort of stuff that's being done now is one more product of the computer revolution. Twenty five years ago, when I was working with Carleen Hutchins, she had $20,000 worth of equipment from B&K in her lab that she could use to produce a spectrum chart of an instrument. Now you can do all of that with a PC,a good sound card, some free software, and less than $50 worth of equipment. And you can still play 'solitaire' on the machine when you're not measuring stuff!

The elephant in the room is, of course, that just because you can measure something doesn't mean it's important. Conversely, it may not always be easy to figure out how to measure the things that are important. The biggest problem we have is to objectively define just what it is that makes a 'good' guitar. Once we do to our own satisfaction we have at least some hope of finding something we can measure that will correlate with that.

The complexity of the thing has been mentioned: and that's another issue. As Mark said, it can be tough to isolate variables. A good example of that is the whole issue of saddle heaight and break angle: does the higher saddle give more sound because it's higher, and has more leverage, or because there is more down pressure of the strings on the top? How can you isloate those things on the same guitar, since it's nearly impossible to make perfectly matched instruments? The list of such conundra is long.

Dave Hurd's book is very useful, but less so, perhaps, than one might hope. The concepts are certainly good, and if you applied his techniques for a while you'd start to get a data base that would serve you well. I can't fault him for sticking with the data he's got, and forcing the readers to develop their own darn data. but the concepts could have been the subject of a magazine article (and I need to read the GAL one carefully): IMO the math that backed it up was an expensive luxury given the intended audience, however good it made Dave feel.

If you're not so much into math, check out Benade's "Science of Sound", which is out in a Dover reprint at a good price. He was a wind guy, so there's more in there about clarinets than you might like, but there's a lot about hearing and room acoustics that's very useful, as well as much of what was known at the time about guitars.

Fletcher and Rossing's "The Physics of Musical Instruments" is very useful. The math is a bit heavy for my skill from time to time (well, let's face it, most of the time) but after a few readings you'll still have gotten a lot of the concepts.

One very interesting publication that you can download is Howard Wright's 1996 thesis, given at UWales/Cardiff: "The Acoustics and Pschycoacoustics of the Guitar". It was a computer modeling study, where they generated gutiarlike sounds on a computer based on the actions of the model, and varied different things like the mass of certain modes to find out what changes people could actually hear. Even though no wood was involved beyond the guitar that they modeled, and the model was very limited, the results have proven quite 'true to life'. Again, skip the math and try to understand how the thing works in a qualitative way.

It is sad that the Catgut Society folded itself into the Violin Society of America, since the VSA is not interested in anything but bowed strings. The Catgut 'Journal' used to run good articles on guitar acoustics from time to time, and if you can find copies, they're worthwhile.

Get in touch with Tim White, and get a reprint of the old 'Journal of Guitar Acoustics'.

Most of all, think about ways that _you_ think the guitar might work. It almost doesn't matter what those might be: maybe touching the strings tickles leprechauns. In that case, you'll want to figure out how to count leprechauns, so you can see what might get more of them to camp out in your guitars. After a while you'll either get good at counting leprechauns, or else decide that you can't. Then you can make up some other story. The human mind is a wonderful story telling machine, and the objective of science is to try to figure out which of those stories in more likely to be true. Eventually somebody will hit on the 'right' story.


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PostPosted: Wed Aug 01, 2007 5:21 am 
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[QUOTE=Alan Carruth]Most of all, think about ways that _you_ think the guitar might work. It almost doesn't matter what those might be: maybe touching the strings tickles leprechauns. In that case, you'll want to figure out how to count leprechauns, so you can see what might get more of them to camp out in your guitars. After a while you'll either get good at counting leprechauns, or else decide that you can't. Then you can make up some other story. The human mind is a wonderful story telling machine, and the objective of science is to try to figure out which of those stories in more likely to be true. Eventually somebody will hit on the 'right' story.
[/QUOTE]


I was about to think "yada yada yada" and then you come up with this profound statement


If your approach to guitar building is totally technological, then you have lost the race before you even started. While I agree that some portion of technology has it's place and may be of some use, it is nothing compared to the power of the human brain to come up with a new idea and then try it. How many of today's technologies were the result of people trying to achieve one thing and then accidentally coming up with something totally different? Dynamite is one of those things that comes to mind.. Who would have ever thought it would be the one thing that heart patients take to overcome heart attacks and heart pain?


Keep searching for the magic... it does exist!


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Ken H


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PostPosted: Wed Aug 01, 2007 5:52 am 
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Koa
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"The elephant in the room is, of course, that just because you can measure something doesn't mean it's important. Conversely, it may not always be easy to figure out how to measure the things that are important. The biggest problem we have is to objectively define just what it is that makes a 'good' guitar. Once we do to our own satisfaction we have at least some hope of finding something we can measure that will correlate with that."

Thanks, Al. That is the biggie that I see. The perfect example in my life has been the invention of the transistor and the early development of transistor amps which measured near perfect but didn't sound as good as high quality tube amps that had ten times the measured THD (total harmonic distortion).   "What?" said the engineers..."These things are perfect, they have to sound better."   Well, it turns out that THD measurements just didn't matter all that much to the human brain in qualifying good tone from bad.   In about 1970, Walt Jung came up with a far more important set of measurable criteria...Transient Intermodulation Distortion...a factor which did indeed seem to correlate much better to human ears' and brains' perception of musicality.

So irrelevant measurable data is a real issue. You can get scientific as you want, but if you're precisely measuring the wrong thing, it just doesn't matter.   It's about as relevant as saying that your guitar has a scale length of 25.5" when someone asks you how much it weighs.

My own explorations right now are more in the engineering of the structure of the instrument. I'm trying to tease apart as much as I can the issues of tone as separate from structure. So far I'm extremely happy with where the structural design elements are going, and I'm quite happy with the final tonal results, though I'd like for my guitars to open up faster than they do.   Having the first of my tilt necks on hand right now...a guitar that's now about eight years old...I can see that what I did was right, but I'd rather it didn't take eight years for the tone to develop like this.   I'll have that guitar at H'burg, battered and scratched a bit, but in all it's post Antarctic glory.   


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Sorry, no time to talk right now. I'm building a guitar.   

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PostPosted: Wed Aug 01, 2007 9:21 am 
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[QUOTE=Alan Carruth]

Most of all, think about ways that _you_ think the guitar might work. It almost doesn't matter what those might be: maybe touching the strings tickles leprechauns. In that case, you'll want to figure out how to count leprechauns, so you can see what might get more of them to camp out in your guitars. After a while you'll either get good at counting leprechauns, or else decide that you can't. Then you can make up some other story. The human mind is a wonderful story telling machine, and the objective of science is to try to figure out which of those stories in more likely to be true. Eventually somebody will hit on the 'right' story.
[/QUOTE]

Funny how the internet perpetuates that. I've had to wade through a lot of fairy stories and misinformation on the web with regards to this craft.


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PostPosted: Wed Aug 01, 2007 10:05 am 
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Where the rubber meets the road is in being able to land well within the parameters that you decide establish "your tone" time after time after time. Traditionally that involves what I would call learned intuition...which is a kind of scientific method, even if you cannot exactly translate into words what your hands are doing based on what they feel and your ears hear. That learning process has a lot in common with the scientific method. We try things and they work or not. We reject the things that don't work and we keep going forward. Some hypotheses hold up to repeated experiments and become theories, and some don't.

Experienced hands can determine appropriate density/stiffness qualities, and while the builders may "reject" scientific lutherie, in fact they are practicing it, just not with numbers from a Lucci meter or deflection test data or closing the ring mode as Al finds so useful.

To utterly reject the science is to reject one's own learning of lutherie.   




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PostPosted: Wed Aug 01, 2007 11:21 am 
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Very nicely stated rick !!!!!

Mark



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PostPosted: Wed Aug 01, 2007 11:22 am 
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Correction:

Very nicely stated Rick !!!


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PostPosted: Wed Aug 01, 2007 12:44 pm 
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Cocobolo
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I would have to agree with Alan's statement.

[QUOTE=Alan Carruth]
Lutherie is a craft. It's not 'pure art' because we're not totally free to make what we feel like making: as has been said, you have to keep (at least some of) the strings on the outside. It's certainly not a science, since there is no objective measurement that everybody can agree on as defining a 'great' guitar. But it is a 'high craft', with a lot of artistic leeway and sensibility in it, and we certainly use scientific methods to further our craft. [/QUOTE]


I think we can all say that there are excellent guitars, in appearance, playability and sound available on the market today and that each is built remarkably different. But I also think that there are a great number of similarities in those instruments that we as guitar builders build into our own instruments or at the least try to.

For instance, Soundboard thickness of a spruce top, we would generally agree that a thickness of greater than 2.5 mm would be too thick and less that 1.5 mm too thin. We have general dimensions of a guitar that we adhere to, none proven scientifically but more through trial and error. Thus is the nature of this craft.

Tap-tuning well there’s another story where the leprechauns have yet to tell. Though I like the idea and going through the process because it’s fun to do and it produces some benchmarks and is a great conversation piece, I could never say that tap tuning can produce a better guitar than adhering to my dimensional specifications and the location of my bracing. Which is generally driven by what I see and feel.

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PostPosted: Wed Aug 01, 2007 2:00 pm 
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I haven't had time to read all the post, put I will later tonight, i need to bind a guitar now but find this topic interesting.
The thing that I find so great about lutherie, is that to me it's a collision of art and science.
And if you saw my shop right now you'd think a collision happened!

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PostPosted: Wed Aug 01, 2007 3:02 pm 
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    This is what makes this forum great. Mr Carruth as always thank you for your time and experience. I am rereading some posts.
    As you get more involved in this craft the science becomes the art. Experience , and time proven methods seem to survive the evolution as you gain knowledge . It is true often the advice is worth just what you pay for it.
    Hopefull we all learn from each other and to all
thank you
john hall


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PostPosted: Wed Aug 01, 2007 3:22 pm 
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"For instance, Soundboard thickness of a spruce top, we would generally agree that a thickness of greater than 2.5 mm would be too thick and less that 1.5 mm too thin."

And there are weirdos like me who graduate their "flat tops" from a little over 3 mm in the center out to just under 2 mm at the edges... That works for me...and I can prove it!


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PostPosted: Wed Aug 01, 2007 3:38 pm 
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[QUOTE=tippie53]    This is what makes this forum great. Mr Carruth as always thank you for your time and experience. I am rereading some posts.
    As you get more involved in this craft the science becomes the art. Experience , and time proven methods seem to survive the evolution as you gain knowledge . It is true often the advice is worth just what you pay for it.
    Hopefull we all learn from each other and to all
thank you
john hall[/QUOTE]


I couldnt agree with you more John ! And I forgot to say thanks to Mr Carruth also. It is true that some of what I read is (or sounds like) hocus pocus or counting leprechuans, but I sure do love the magic when one of these magic spells comes to life.


Great thread!


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[QUOTE=Rick Turner]And there are weirdos like me who graduate their "flat tops" from a little over 3 mm in the center out to just under 2 mm at the edges... That works for me...and I can prove it![/QUOTE]

OK Rick, question for you.

Do you build a truly flat top as in not to some dish/radius/dome/sphere? Are you saying that your "flat top" tapers from the middle out to the edges and that's it.

Just an inquiring mind

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