Wood Density, Tone, And Other Intangibles

László

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As some here know, I do a lot of work with classical instruments, and I'm always interested in how they generate sound, their tone, etc. In doing my reading, I stumbled on a 2008 scientific study the other day that I've been thinking about. Seems that a Dutch team that included a well known violin maker decided to study a CT scan of a variety of violins that included Stradivari, to try to discover why the early 18th Century Cremona violins projected better than more modern instruments. The physician member of the team specialized in evaluating lung density.

The reason for the CT scan was that it didn't make sense to saw up a zillion dollar violin to look into the secrets of the wood itself. The CT scan makes that unnecessary. Evidently, this is the first time this procedure was done to evaluate a fiddle.

Over the years, there has been speculation that it's got to do with the varnish finish formula, or the placement of the sound peg, or a variety of other factors. What was discovered was that Stradivarius and Amati were very, very good at finding woods of uniform density -- by that, not the most dense, but instead the most even growth:

http://www.sciencedaily.com/releases/2008/07/080701221447.htm

The scientific explanation given is that the sound waves will project better when the wood vibrates at a more constant rate resulting from the even growth rings.

While some electric guitar folks have advanced the idea that the tone comes from the pickups and the strings and the woods have nothing to do with it, experience shows that's clearly untrue. Play any two electric guitars of the same model, with an identical factory setup, and they sound different. It is obvious (and makes sense given the fact that the vibrations of the wood affect the way the string vibrates) that the wood plays a significant role.

Limiting the scope of this post strictly to PRS, I've been a PRS player since 1991. During that time, I've owned (and played) quite a number of them, and in several instances, more than one of the same model at the same time. In every case, they sounded different, and one would be more "lively" than the other, with the notes feeling like they were easier to produce and almost springing off the guitar.

Of course, not having access to a CT scanner when buying guitars, I have no idea why this is the case.

But -- the two liveliest guitars I've ever owned are the two limited run guitars I have, the Artist V with a Peruvian 'hog neck, and very densely ribboned 'hog back, and the Sig Ltd with the so-called sinker neck (the back of mine isn't translucent, so I have no idea what that's like at all).

Anyway, the article raises some interesting thoughts.

Whether these

Edit: Oops, i didn't finish my sentence and forgot what I was going to say. Heh. I'm getting really fuzzy headed in my old age.
 
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Wood (especially the top) is pretty much the amp for the strings..in combination with the bracing and the air blah blah...
A lot of people don't realize that hundreds of years ago when the European masters made instruments, they used old wood. When they harvested wood they didnt do it for their instruments, they harvested for their children or grandchildren's instruments. They would get wood and sink it in a river or lake for a couple or few generations.
All kinds of enzymatic and other processes take place and transform the cellulose and other structures in the wood to create better sounding instruments.

Native Americans also sunk wood that would be used to make ceremonial instruments (drums etc) sometimes hundreds of years later.
Stonebridge (and a Swiss company that has a patented process) ages some of their guitars' tops too.

SCGC has some wood (cant recall the species) that was felled and burried OVER 2000 years ago. They will be making very fer VERY expensive guitar tops out of that wood.

So, the "secrets" to great sounding instruments isnt all that secret. Builds can be copied and cloned and reverse engineered. People can build a Strad copy exactly like an original...and they can do it using 400 year old sinker wood. Lots of companies do..lots build pre-war D28s using sinker or old BRW and adi...but will they sound and feel and play like the old ones???
I don't know.

Interesting topic and food for thought. Thanks for posting.
Wonder how long it took for that fiddle to get the ctscan? Wonder how long it woulda taken it to get in for an MRI?
I know that if it was a Canadian fiddle waiting for an MRI (granted, it would be free)...well it would be 300 years old by the time it got-in for the imaging :-( lol.

Thanks for posting this. I too am a wood junkie.
Cheers.
 
Wood (especially the top) is pretty much the amp for the strings..in combination with the bracing and the air blah blah...
A lot of people don't realize that hundreds of years ago when the European masters made instruments, they used old wood. When they harvested wood they didnt do it for their instruments, they harvested for their children or grandchildren's instruments. They would get wood and sink it in a river or lake for a couple or few generations.
All kinds of enzymatic and other processes take place and transform the cellulose and other structures in the wood to create better sounding instruments.

Native Americans also sunk wood that would be used to make ceremonial instruments (drums etc) sometimes hundreds of years later.
Stonebridge (and a Swiss company that has a patented process) ages some of their guitars' tops too.

SCGC has some wood (cant recall the species) that was felled and burried OVER 2000 years ago. They will be making very fer VERY expensive guitar tops out of that wood.

So, the "secrets" to great sounding instruments isnt all that secret. Builds can be copied and cloned and reverse engineered. People can build a Strad copy exactly like an original...and they can do it using 400 year old sinker wood. Lots of companies do..lots build pre-war D28s using sinker or old BRW and adi...but will they sound and feel and play like the old ones???
I don't know.

Interesting topic and food for thought. Thanks for posting.
Wonder how long it took for that fiddle to get the ctscan? Wonder how long it woulda taken it to get in for an MRI?
I know that if it was a Canadian fiddle waiting for an MRI (granted, it would be free)...well it would be 300 years old by the time it got-in for the imaging :-( lol.

Thanks for posting this. I too am a wood junkie.
Cheers.

Actually, the sinker wood theory has been discredited in several scientific articles. There's no evidence to support that this was done by luthiers of the time.
 
There are some related articles linked to from that one, ones that are more recent discussing how they used those CT scans along with computer controlled machinery and some fungus to recreate the Stradivarius.

Here's an interesting quote from one of those articles:

Stradivarius himself knew nothing of fungi which attack wood, but he received inadvertent help from the “Little Ice Age” which occurred from 1645 to 1715. During this period Central Europe suffered long winters and cool summers which caused trees to grow slowly and uniformly – ideal conditions in fact for producing wood with excellent acoustic qualities.
 
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There are some related articles linked to from that one, ones that are more recent discussing how they used those CT scans along with computer controlled machinery and some fungus to recreate the Stradivarius.

Here's an interesting quote from one of those articles:

Yes, I've read that article, and that may be a very good reason that the instruments were so uniform in terms of the wood density.

It's notable that there was also a "little ice age" in the 19th century that (for example) caused the famous Potato Famine, but this period didn't produce any particularly well known instruments.
 
So if they're able to make a violin with a 5mm top on it sound better through fungal exposure, I wonder if they could do the same for thicker woods such as an electric guitar, and if the effect would be more or less pronounced. Also, they focused on spruce, right? I wonder how other tone woods would be affected.
 
treat the fretboard first and see if that makes the expected improvement.
 
I seem to recall Mr. Paul Smith hisself scandalized the world of electric guitar builders with his ideas that Wood Really Matters.

It seems somewhat obvious now, but when he hit the scene, it was super-hot pickups and fancy trems, brass everything and heavy headstocks that were considered the path to tonal nirvana.
 
I seem to recall Mr. Paul Smith hisself scandalized the world of electric guitar builders with his ideas that Wood Really Matters.

I always anger people when this subject comes up.

But.

The whole point of the electric guitar is to take the wood out of it.

When they first amplified acoustic-type guitars, the sound from the speakers vibrated the wood of the guitars when fed back. It was very tricky amplifying those types of guitars.

So.

No less a personage than Les Paul explained that his "log" guitar was meant to take the wood vibration out of the equation.

The point of an electric guitar is to support as rigidly as possible the vibrating string system. The more rigid the support, the more sustain you get, and the less the wood plays any part in the sound.

There is a youtube video that I cannot seem to find to save my life that points out how unreliable our ears are to make various determinations. The issue is that our brain uses any info it can latch onto to cheat. This is part of the mechanism that allows you to hear a single conversation at a cocktail party by the way.

At any rate, in this lecture, they play a Led Zeppelin tune backwards and ask if you hear the demonic lyrics.

And you don't hear anything but gibberish.

Then they play it again, but this time they superimpose "lyrics", and I'll be damned but I can now hear it all as clear as day. I will tell you that seeing this demonstration made me question my ability to judge what I hear. I was fairly stunned.

What this all means is that you just can't trust yourself when you know what the answer is supposed to be. And this isn't about integrity, or having a good ear, or experience or anything else. Your brain can't do everything and it takes shortcuts whenever it can to help you make sense about what's around you. What this means is that when two choices are even remotely similar, your brain will hear what you expect to hear. Period. You need to know this about yourself.

Double blind experiments are the only way to be sure.

I won't say that the wood has NO effect on the sound. Clearly the neck vibrates a little and this leaks a small amount of energy from the vibrating string system, but what I will say is that the better built the guitar is, the less the wood plays any part in the sound. A proxy for this is sustain. The more sustain your guitar has, the less energy is leaking from the vibrating string system and the less the wood imparts anything to the sound.

To be clear, I love the beautiful wood that PRS uses, but tonewoods essentially are stiff hardwoods. It's the stiff part that's important. Truth be told it probably won't be long until the highest end guitars are made from some graphite composite or other. It's already starting to happen to violins and other orchestral instruments.

I asked PRSh once whether he had considered making graphite instruments. He said that he had. But his basic point was that he felt he had such a good inventory of expensive, hard to get wood, that he was going to fight it out in that space as long as it continued to work for him.
 
I always anger people when this subject comes up.

But.

The whole point of the electric guitar is to take the wood out of it.

I can see that. You want a strong, light, rigid material, and wood has those characteristics, is fairly plentiful, and relatively inexpensive. We've been working with it for thousands of years and it can be very pretty. Good stuff. But for better or worse it's idiosyncratic, being a natural material.

Carbon composites can be like that too, and I was really surprised to hear how good a Rainsong carbon fiber guitar can sound. But it's pretty expensive and nontraditional, so can be a hard sell to guitarists.
 
No less a personage than Les Paul explained that his "log" guitar was meant to take the wood vibration out of the equation.

The point of an electric guitar is to support as rigidly as possible the vibrating string system. The more rigid the support, the more sustain you get, and the less the wood plays any part in the sound.

This is not a new argument.

These points were bandied about in the 70s and 80s, and lots of designs were created to address them. Steinberger went in one direction. Another was the whole hippie sandwich thing of Alembic and others, who used heavy brass parts, and sandwiched wood into very stiff necks and bodies to create a kind of plywood effect to eliminate the natural wood resonances.

Fortunately (the way I see it), players in the main weren't enthusiastic about the actual results, and the reason was not simply that guitar players are traditionalists.

If Les Paul's purpose was to eliminate the wood from the equation, he didn't accomplish anything other than feedback reduction. The wood is why two electric guitars of the same model will sound slightly different from one another. Especially in the case of the Les Paul, one can be a great one, one can be a dog, and experienced players know the difference.

The differences don't have to be huge to be heard. The ear is more sensitive than one gives it credit for, provided that it's in good shape, and that one is an experienced listener.

I have participated in MANY double blind listening tests of various kinds, going back to the beginning of digital v analog audio. I've participated in them in my studio with my studio tech, in other studios, and put on by manufacturers here in the US, and even in Europe. Double blind tests can be useful, but they also have weaknesses, as I'll get into shortly.

I participated in an Akai demo in Amsterdam in the 90s at Wisselloord Studio, owned by Polygram, where I was working on a project. It was double blind, the purpose being to show that the Akai time compression algorithm of the day couldn't be identified on more than a random basis. The audio was run through the studio's SSL J series console, into Genelec mains, soffit mounted, triple amped, that probably ran 30 grand at the time. So we had good sound. And Wisselloord has great sounding control rooms designed by professionals. Details were evident. This isn't always the case, and is a weakness of some controlled tests.

I was able to pick out the compressed track on 10 of 10 tries. I wasn't supposed to be able to do that, according to their other double blind tests.

The problem was, I knew what to listen for.

The point is, some people know what to listen for in things like instruments, too.

Sure, a carbon fiber guitar with a high modulus of stiffness would be "perfect" for sustain and the "perfect" electric guitar sound. It is also boring. Incredibly boring.

Ever play a Steinberger L Series guitar from the 80s? I had one for a short time. It worked exactly as one would expect. It was cool to look at, white, no headstock, and its small size seemed great to handle in the confines of my control room. I really liked not banging the headstock into my gear while working! And I bought into the argument that you (and the manufacturer I might add) make.

It was made of a mishmash of carbon fiber and graphite. The whole idea was to increase the modulus of stiffness, increase sustain, and do exactly what you're talking about in your post.

It was wonderful at what it tried to do, which made it a very, very boring, toneless instrument to play. I wanted to like it. But basically, there's a reason you don't see a lot of players with Steinbergers on the stages of the world any more. It had no personality. It was not inspiring to work with.

It is the "flaws" of a wooden instrument that give it that wonderful tone!

As for classical players moving to graphite instruments, actually, you see players like Yo Yo Ma using them for outdoor summer concerts, or for traveling so they can spare their very expensive antique instruments the rigors of travel. You have to remember that a really fine antique instrument costs as much as a house. I know orchestral players who have mortgages on their instruments!

So no wonder that for certain projects, they are reluctant to travel with them!

This is not too dissimilar to what we see with folks playing Ovations and Rainsongs on tours, gigging, etc. But I can tell you that these same players use wooden instruments in the studio to make critical recordings.

I know, because I work with orchestral musicians on recording projects.

As subject to the power of suggestion as our brains may be, experienced musicians are able to hear quite subtle details in their instruments, and are able to coax what they need to in the way of sound out of them in a way that perhaps the layman is not. This may be a difference between gifted musicians and the general population.

Now as to the problem with a double blind test:

With instruments, the player and the instrument are a feedback loop. That is, the player adjusts the note on the fly with various techniques including finger pressure, pick attack, amount of force, vibrato, pizzicato, sforzando techniques, etc, to produce the notes. This process isn't just striking or bowing a note, it's the pressure and modulation applied while the note is playing as the player is listening to it. There is a constant adjustment in most cases as long as the note is held.

So to merely listen double blind to instruments doesn't do the player any good, because he or she can't impact the note itself, and this is what distinguishes a responsive instrument from a dog of an instrument. Wooden electric guitars may be hardwood, but there are good ones and not so good ones.

Secondly, there is another problem with double blind testing, and that is the effect of ear fatigue during the test. After a few A/B switches, it has been shown that the brain effectively becomes somewhat dazed, and the ability to clearly analyze decreases the longer the test is in play.

Thirdly, double blind testing can't tell you long term results. For example, go listen to an A/B test of studio monitors. I've done this in tests run by Dynaudio, Genelec (my studio at the time was a beta tester for them), and others. Yes, you can determine how well a pair of monitors does in a given room (of course the room part of the equation skews the results also). What you can't determine in such a test is how it feels to listen to them for an extended period of time; days, weeks, months.

I can tell you from experience that some monitors are quite tiring and irritating to sit in front of for extended periods, even though they do very nicely on A/B tests. I don't like to work with certain Quested nearfield monitors for this reason, though they sound absolutely stunning in A/B tests!

In the case of instruments, living with the instrument and playing it can completely reverse the judgment one makes playing several in a store, for example. How many of us have taken home a guitar, and after only a few months, found that it was not what they were looking for in terms of the sound they wanted to produce?

Yet when we find a great one, that just suits our ear, it gets better the longer we own it, not worse!

Why? I think a big thing about the "it" factor isn't just how it sounds to the one-day player, but as we gain more experience playing it, learn its nuances, etc., the instrument keeps on giving, responding, and helping us achieve the sounds we want. That is, if it's a good one.

Granted, some players will achieve that differently from other players. Everyone's needs and tastes are different. There are probably still some headless Steinberger players out there.

Not me.

Finally, I don't want to give the impression that I'm somehow angered by your post. I'm not. Every person's experience is different, and people are absolutely entitled to believe what their experience tells them is true, be that from personally trying things out, or from reading and study. But it's also true that the mind is suggestible to a variety of theories, as you say, and if it makes sense to you, you're going to fit your personal experience into that suggestion.

I've been playing a variety of instruments since I was 4 (1954). I grew up playing piano, Hammond Organ, accordion, guitar, bass, and a
few others. I've made my living in the studio as a composer and producer since 1990. I spend an awful lot of time listening to instruments. This lifetime of experience has led me to certain conclusions that differ from the ones you draw.

And that's why discussion of ideas is a good thing!
 
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I can see that. You want a strong, light, rigid material, and wood has those characteristics, is fairly plentiful, and relatively inexpensive. We've been working with it for thousands of years and it can be very pretty. Good stuff. But for better or worse it's idiosyncratic, being a natural material.

Carbon composites can be like that too, and I was really surprised to hear how good a Rainsong carbon fiber guitar can sound. But it's pretty expensive and nontraditional, so can be a hard sell to guitarists.

I have mixed feelings about this. Part of me looks forward to the possibilities of the new. I've seen a couple of really interesting carbon fiber guitars that are different and flat cool. On the other hand, there's just something about the natural beauty of wood that's hard to give up.

In many ways guitarists are kind of Luddites. I mean, who else is still using tubes for anything?

Here are some examples of interesting carbon fiber guitars: http://www.carbonfibergear.com/6-sexy-carbon-fiber-guitars/
 
In many ways guitarists are kind of Luddites. I mean, who else is still using tubes for anything?

Well, we use them BECAUSE they distort in a certain way, not in spite of it. Tubes are very good at what they do, and their somewhat chaotic nature is exactly what produces the tones we like.

I don't see that as Luddite, I see it as sticking with something that works for sound production (as opposed to reproduction. if you want reproduction, try running your guitar through a hi fi rig, it'll sound, well, hi fi. Sounds lousy, doesn't it?).

When they come out with something that sounds more musical in its ability to distort, we'll probably switch. Turns out it wasn't the accurate reproduction of the guitar tone that we wanted, it was the inaccurate, distorted sound we glommed onto! Incidentally, that doesn't mean high gain all the time; even what we guitar players call a "clean" sound has far higher levels of distortion than we would consider acceptable in a hi fi rig.

And by better, I don't mean merely emulating tubes, like a modeler. I mean something that sounds different from tubes but still musical. That hasn't come out yet. Something will come out that will replace the electric instruments we now use.

But isn't it interesting that the ear craves not perfection, but individuality and even distortion, when it comes to sound production? This dovetails with my point about wood guitars - the individuality and lack of linearity is what makes them good! Not scientific perfection. We don't want that, any more than we want to listen to a violin that doesn't have the grit of the rosin on the bow.

Ever hear a bunch of violins playing exactly in tune? It sounds like an organ. We don't want that, it sounds really crappy. In fact, in the 70s, the days of the "divided down" string machine technology, the keyboard was perfectly in tune with itself, all up and down the octaves. Roland invented the chorus circuit so that the strings on its S (220, 330 etc) synths would sound a bit out of tune with one another like a real orchestra, with all its imperfections! You might be surprised to learn that the circuits used in the original chorus pedals were not invented for guitar. They were invented for synths. By the way, Roland owns Boss, and that's how the chorus pedal got started.

People use what serves their purpose. You'd be a luddite if you traveled in a wooden horse-drawn wagon to get to work. A metal modern car is a more efficient daily transportation device, gets us there faster and drier, and doesn't need hay or a barn, nor does it need us to shovel its poop every morning. So that's what most of us use for transportation on a daily basis. Sure, cars have their own problems and cause plenty of them, and when that becomes intolerable, whether for economic or environmental reasons, we will change what we use for daily transportation.

So far, we haven't found the graphite guitar or the transistor amp to be better at making music than wood or tubes. At some point that may change, but it isn't luddite to use what sounds good now. We should use what we think works best musically to create music, shouldn't we?

They routinely find combs for hair at archaeological digs, dating back to prehistoric times. We still use combs today. Is it luddite to use a comb? Most of us would say no, it isn't. Does it matter what the comb is made of? Not really. Go to the hair salon and they have plastic, metal, and wooden ones for different purposes.

But the comb is effective as a device. So we still use it, even though it dates from ancient times.

So is the leather shoe. They've been making leather shoes for a long time, too. I think the frozen cave man they dug up from 10,000 years ago was wearing leather boot things, stuffed with grass for warmth. Leather isn't waterproof, but it's comfortable to wear, shapes to your foot the way man made materials rarely do, and it lets the moisture out. They dig 2000 year old Roman leather footwear up at sites in Britain all the time, because it is preserved by the peaty soil. Is it luddite to wear leather shoes?

You can buy nylon shoes, too, like running shoes, but they still sell an awful lot of leather shoes.

There are materials and products that are no longer useful in today's world. And there are materials and products that are quite useful. Stone is a good example. We still use stone as a building material.

We still use concrete. It was invented by the Romans. People think that concrete is a modern invention. The Romans even had a concrete that cured under water, and used it for docks! It had a pumice they got from Italian volcanic deposits called Porcillana.

We still use bricks. Just like the ancient Sumerians. Bricks are still made of clay and mud and crap to bind them. They could use more modern materials if they wanted to. They're still useful. I live in a house that has bricks on the outside. It's 31 years old and has survived a lot of bad weather here in Michigan. So it's effective stuff. I'm not a luddite.

I could go on. But you get my point.

Incidentally, I should point out that I'm also a very experienced synthesizer player and programmer. I started with analog synths, and still use them, but I also use digital and modelers.

Synths are good at what they do, but they haven't replaced pianos or organs, devices that date back in time (in the case of the organ, the Romans used them), but are also good at what they do and sound different from synths. Something new doesn't have to outmode something older. Things that are useful can live side by side with one another.

Come to think of it, most of us like a good vocalist. Singing may even predate using rocks to bang on a log. ;)
 
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I always anger people when this subject comes up.

Hmmm...

The whole point of the electric guitar is to take the wood out of it.

When they first amplified acoustic-type guitars, the sound from the speakers vibrated the wood of the guitars when fed back. It was very tricky amplifying those types of guitars.

So.

No less a personage than Les Paul explained that his "log" guitar was meant to take the wood vibration out of the equation.

The point of an electric guitar is to support as rigidly as possible the vibrating string system. The more rigid the support, the more sustain you get, and the less the wood plays any part in the sound.

I don't think this is an entirely accurate reading of the events it purports to describe.

What they wanted to take out of it was the rather-free-to-vibrate top; the shaking of the top, along with the fairly microphonic pickups that were the norm in those days, made for a pretty uncontrollable mess when things got amplified beyond a pretty low level.

Ted McCarty, who had a lot to do more to do with the development of the "Les Paul Model" guitar than did Les himself, noted that they actually tried some non-wood materials to test the idea "what would happen if you did actually take the wood out of it and replace it with a much more rigid material." The resulting "guitar" sounded like crap, as did solidbody guitars whose bodies were all hard (Eastern) maple, at least to the R&D folks at Gibson, which is how they came to land on the mahogany+maple recipe they did.

What you're really trying to do with a solidbody guitar is find the right balance of wood with everything else in the system. Everything in there -- including the wood -- affects the outcome (in terms of sound produced) and, in the immortal words of my friend Ken Parker, "It's a long [expletive] equation, and if anyone says it all comes down to ONE thing, they're full of [expletive]."

Hey, Les, thanks for the well-written and thoughtful commentary. Always fun to hear your (well-informed) thoughts on this stuff. EXCELLENT point on the whole concept of the importance of how things sound over time (meaning, extended periods of time, not just hours but days-weeks-months-years) and how some particular pieces of gear prompt fatigue. That absolutely agrees with my personal experience. Talk to Steve Kimock about guitar speakers sometime...
 
That absolutely agrees with my personal experience. Talk to Steve Kimock about guitar speakers sometime...

You and I are often like-minded when it comes to these kinds of things. I have talked with Steve about guitar speakers - it was he who schooled me about absolute polarity being audible and somewhat important. You are right- he's a very bright guy, as far as I'm concerned, great pair of ears, too!

One day we need to sit down for that drink!
 
It depends on what your expectations are too. The electric was conceived as a louder acoustic, and if that had been possible to do back in the early 1930's, the developement of the instrument, and of popular music in general, might have been very different. What players thought they wanted back then was an acoustic guitar sound that could be heard over an orchestra--can you imagine Charlie Christian or Chuck Berry or Jimi Hendrix with an extremely loud acoustic? Neither can I!
We are Luddites to a certain degree, but I think it's that we get used to a particular sound from our instruments. An acoustic archtop guy at the dawn of the electric guitar would have rejected the best electric available at the time because it didn't sound like his Gibson L-5, and wouldn't have been much louder either--it took a while for the amplification technology to catch up. Then when rock 'n' roll came in, and guys started driving their amps a little harder, the older jazz guys (who might have been the sonic rebels of their generation) didn't like it--although guys like Charlie Christian used a grittier sound than some of the later clean mellow players like Johnny Smith. As late as the mid-'70's, I'd run into older cats who felt that Eric Clapton's and Jimi Hendrix's use of distortion was a gimmick to cover up what these guys thought was bad playing. I hope I never get that narrow-minded--although I admit that I can't stand machine-made techno music! Those crazy kids...hey, get off my lawn!
 
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Very insightful posts, Les - and right on, in my view. Thank you for sharing your knowledge and experience.
 
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