At fret 12: The G str. isn't sharp. The A str.is flat !!

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Alan Carruth
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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by Alan Carruth » Mon Mar 12, 2018 7:00 pm

Again, look up some threads on 'intonation'. In particular, look up my recent post in the thread: "What is an acceptable level of intonation error?".

Intonation issues are often attributed to string stiffness, with the G string being the prime example. Trevor Gore looked into this rather exhaustively in his books, and came to the conclusion that its actually the way the tension changes when the string is fretted that causes the problem. Fat strings tend to see more tension change, so the G is the biggest offender for that reason. They are also, of course, stiffer, but that's a red herring in this case. The intonation errors caused by stiffness and tension change are somewhat different, and Gore showed that the errors we see with strings are more like those caused by tension change than stiffness.

Although tension changes are the main issue the resonances of the guitar top itself can also cause intonation errors. This varies from one instrument to another, and has to be dealt with on a case by case basis. I have seen guitars where such issues have caused tuning problems. This gets complicated.

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guitarrista
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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by guitarrista » Mon Mar 12, 2018 9:07 pm

Alan Carruth wrote:
Mon Mar 12, 2018 7:00 pm
Intonation issues are often attributed to string stiffness, with the G string being the prime example. Trevor Gore looked into this rather exhaustively in his books, and came to the conclusion that its actually the way the tension changes when the string is fretted that causes the problem. Fat strings tend to see more tension change, so the G is the biggest offender for that reason. They are also, of course, stiffer, but that's a red herring in this case.
I am not sure why the "red herring" comment. Didn't Greg Byers do the calculation first, showing the nut and saddle compensation values - both theoretically and experimentally. And judging by the values, it is clear that you have to compensate a lot more for stiffness than for fretting tension - measured by the magnitude of the values for each. Conveniently, saddle compensation was entirely for stiffness, and nut compensation is almost entirely for fretted tension changes, so it is easy to compare the two sets of numbers. While G is the worst offender on both counts, the compensation brought about by stiffness is 2-3 times higher in value than for fretting. Am I reading this wrong? I don't have access to Gore's book so I'd appreciate micro-excerpt of relevant formulae if his results are the opposite of Byers's.
Last edited by guitarrista on Tue Mar 13, 2018 2:22 am, edited 1 time in total.
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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by MarkInLA » Tue Mar 13, 2018 2:19 am

Hi again. Not fully understanding all of your technical jargon above. And though I'm sure you are likely correct over all these physical properties, while not disowning I, the OPer my own original post, one thing I am seeing is: The the reason the G does go sharp at the 12th could very well be due to its thickness as it bends over and down the fret, these tiny extra measurements shortening the G string even more, in turn sounding a sharp-[ened] 'G' at the 12th. So,... for those of you who do have the #G, not the flat A, perhaps try not planting this G all way down to the board, causing the # condition, but downward just enough to insure a clean note/node, with no buzzing....
Anyway, again, I would like to hear from those in my camp; if nothing else but to prove there are 'A flatters' out there ! But maybe not..It's only my theory..not yet a statistic... M, Los Angeles
PS. Of course this would likely not work for a half or grande' barre.. :chaud:

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by Alan Carruth » Tue Mar 13, 2018 7:08 pm

I will say at the outset that I'm not in either Byers' or Gore's league in math skill. Still, when I read Byers' article I was uncomfortable; he lost me at some point, and no amount of study could bring me back to the trail. Note, too, that, in the end, both authors make similar changes in the nut and saddle locations, so the actual effect is similar. As Gore points out, in many cases it's better to simply do something that tends in the right direction, as the result will be 'closer' than it would have been, even if it's not 'perfect'. In this case, perfection is a will-'o-the-wisp that will constantly elude you, so in practice it might not make much difference which system you follow. Still, it's usually better to do things for the right reasons, if only because you're less likely to run into nasty surprises. ;)

In 'Contemporary Acoustic Guitar ; Design and Build' Vol 1, 'Design', on page 1-65, Gore writes:

"Two writers on the subject, Byers and French, assume a solution by Morse approximating a clamped boundary condition, anticipating significant inharmonicity. (snip [equation given]) However, this does not match our experience, which is that inharmonicity is rarely an issue, implying that strings as normally found on a guitar are better described by a pinned end condition." He goes on to describe a test which confirms this.

'Inharmonicity' is the condition where each higher partial of the string is relatively sharper than it 'should' be, due to string stiffness. This is certainly real: the question is whether it causes problems with intonation. The theory here is that the ear does not (as tuners do) home in on the strongest partial, and determine pitch from that. Rather they tend to use a sort of weighted average of a number of strong partials to determine the pitch. The closer to 'harmonic' those are the more secure the sense of pitch. In theory, partials that are shifted significantly upward in pitch will move that sense of pitch sharp.

A string with a 'pinned' end is free to rotate vertically around the fixed point of the saddle top, while 'clamped' end is not, and must bend in order for the string to vibrate. This bending in effect shortens the string a bit, and raises the pitch. You do see this on wound strings that break over the saddle with a sharp bend. The windings pack together on the bottom of the curve, and the string actually rises a bit in front of the saddle, which can be clearly seen. Even with plain strings we're talking a matter of degree here.

In 'The Physics of Musical Instruments'. Fletcher and Rossing give the equation for determining the change in tension of a displaced string. It is:

T = T0 + ((E*A/L0) * delta L), where
T = final tension
T0 = original tension
E = Young's modulus (a measure of how much force it takes to stretch something by a given amount)
A = the cross sectional area of the string (or the core in a wound string)
L0 = the original length
delta L = the change in length

What this says is that for a given string and displacement the rise in tension will be proportional to the cross section area of the of the string and the Young's modulus. Materials like steel that have a high E value don't stretch much when you push them aside, and the tension rises a lot, so they need a lot of compensation. Nylon stretches more, all else equal, so you see less tension change and pitch rise for a given displacement assuming the same string diameter, and need less compensation. However, nylon strings, because they're lighter, have to be thicker to carry a given tension, and the cross section area is greater, which reduces the advantage a bit. This is, of course, particularly true of a plain G string, which is why it needs the most compensation.

I used the term 'red herring' because, of course, the G string is also the stiffest string, by quite a bit. For a given material the stiffness of a round string goes as the fourth power of the diameter. That makes the G string about 4 times as stiff as the high E if the same material is used. That's a larger difference than in the cross section, which is about double for the G string. Intuitively an explanation that relies on stiffness rather than area seems more likely.

Because of it's high stiffness a plain G string is, in fact, much more inharmonic than the high E. This is one reason, I think, for the somewhat 'fuzzy' sense of pitch it can give. This gets worse as you fret the string higher, since the stiffness has even more effect on the inharmonicity for a shorter string. The whole issue would be moot if Classical players could get, and would use, wound G strings that held up reasonably well. It's rare for a steel string guitar to be set up with a plain G string, except for Blues playing, where they like to 'bend' the pitch a lot. Steel Gs use metal windings, which hold up reasonably well. On nylon they use plastic, which wears fast, but sounds pretty nice while it lasts. The other issue is, of course, 'zip', which is a matter of left hand technique.

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by guitarrista » Tue Mar 13, 2018 9:26 pm

Alan Carruth wrote:
Tue Mar 13, 2018 7:08 pm
In 'Contemporary Acoustic Guitar ; Design and Build' Vol 1, 'Design', on page 1-65, Gore writes:

"Two writers on the subject, Byers and French, assume a solution by Morse approximating a clamped boundary condition, anticipating significant inharmonicity. (snip [equation given]) However, this does not match our experience, which is that inharmonicity is rarely an issue, implying that strings as normally found on a guitar are better described by a pinned end condition." He goes on to describe a test which confirms this.
At least in Byers's case, the characterization above (in red) is incomplete. Byers didn't just assume or anticipate, he (1) worked out theoretically what the contributions from both are, and (2) actually tested this. Byers provides data from physical experimentation which show that the stiffness effect is 2-3 times the fretting effect for string G, based on the magnitude of required compensation. I guess Gore's experiment somehow contradicts that, though the details are not given.

The rest of what you quoted (not reproduced above) does not really contradict Byers; it just reiterates that the g string is the worst offender for both.

OK. Thanks for going to the trouble of looking this up!
Last edited by guitarrista on Wed Mar 14, 2018 2:51 am, edited 1 time in total.
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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by Lovemyguitar » Wed Mar 14, 2018 2:48 am

Hi Mark. Okay, I did what you asked, and I checked the intonation on my two guitars. Neither of them have notched saddles (nor nuts) for the G string, and in both cases, contrary to your experience, it is the G string that goes slightly sharp at the 12th fret, whereas the A string (and all the others) are perfectly in tune at the 12th fret. So, sorry, but I don't think you've stumbled upon a new universal theory, since my two guitars (which meet your criteria) provide counter-examples.

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by David Conti » Wed Mar 14, 2018 5:03 pm

If it bothers you try a violin or viola, ALL fretted instruments are a compromise.
Nigel North actually tweaks his frets on his lute until "it sounds right to him"
almost all concert guitarist I know tweak there tuning depending on what key they are in.

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by Alan Carruth » Wed Mar 14, 2018 5:26 pm

guitarrista:
As I say, I'm not in either Byers' or Gore's league wen it comes to the math. I'd like to give you more of Gore's argument and proofs, but it's a pretty big section of the book, with lots of formulae. Suffice to say that he, too, did experiments and ran numbers, and came to a somewhat different conclusion than Byers did. My own experience and (limited) understanding suggest to me that Gore did a better job than Byers did with it, but that's just my opinion. I'm sure both factors, stiffness and tension change, enter into it. Maybe the 'experts' will settle the exact proportions at some point.

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by MarkInLA » Wed Mar 14, 2018 10:08 pm

Lovemyguitar wrote:
Wed Mar 14, 2018 2:48 am
Hi Mark. Okay, I did what you asked, and I checked the intonation on my two guitars. Neither of them have notched saddles (nor nuts) for the G string, and in both cases, contrary to your experience, it is the G string that goes slightly sharp at the 12th fret, whereas the A string (and all the others) are perfectly in tune at the 12th fret. So, sorry, but I don't think you've stumbled upon a new universal theory, since my two guitars (which meet your criteria) provide counter-examples.
Well, it could be your guitar. But thanks for at least addressing my OP the way I wished. M

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by Trevor Gore » Thu Mar 15, 2018 2:09 am

guitarrista wrote:
Tue Mar 13, 2018 9:26 pm
Alan Carruth wrote:
Tue Mar 13, 2018 7:08 pm
In 'Contemporary Acoustic Guitar ; Design and Build' Vol 1, 'Design', on page 1-65, Gore writes:

"Two writers on the subject, Byers and French, assume a solution by Morse approximating a clamped boundary condition, anticipating significant inharmonicity. (snip [equation given]) However, this does not match our experience, which is that inharmonicity is rarely an issue, implying that strings as normally found on a guitar are better described by a pinned end condition." He goes on to describe a test which confirms this.
At least in Byers's case, the characterization above (in red) is incomplete. Byers didn't just assume or anticipate, he (1) worked out theoretically what the contributions from both are, and (2) actually tested this.
It’s worth taking a close look at what Byers wrote. He explains using his Figure 1 (a picture of string modes) how sharpening of harmonics occurs. This is well understood and rarely disputed. The sharpening occurs due to the bending stiffness of the string (proportional to diameter^4). However, the nodes at the end points of the string don’t necessarily have zero slope at the terminations. The string can rotate rather than just bend. In fact, Byers’ Figure 1 shows such a case. He has assumed pinned ends for his Figure 1, allowing the strings ends to rotate. But when he gets to using an equation for string frequency, Byers utilises an equation by Morse derived from boundary conditions of zero slope at the ends (i.e. clamped ends). I don’t see anywhere that he made a case for zero slope at the ends or tested for it. He jumped to using the Morse equation with bending stiffness terms because it explains string inharmonicity. When I was doing my work, I didn’t know what to use as end conditions for the string – fixed or pinned (i.e. zero slope or free to rotate). That’s why I did the experiments to find out, and it turns out that pinned is a much better approximation, tested by fitting various string frequency equations (including Morse, simple Mersenne and others) to measured data. The simple Mersenne equation was by far the best match. As we set up guitar intonation to make the fundamental play true, there is little point in having unnecessary bending stiffness terms relating to higher partials included in the subsequent analysis.

Byers then goes on to model some results using his method including bending stiffness and concludes that “saddle setback is mostly, but not entirely, necessitated by [bending] stiffness (inharmonicity), and nut setforth results entirely from stretch". This is a conclusion derived only from how he did his modelling, not from his observations of string behaviour. It is also a conclusion unique to Byers, as far as I am aware.

Byers proceeds to compare measured results to his modelled results (Table 6 vs. Table 7). His measured results (Table 7) for saddle compensation generally fall between those of Table 3 (stiffness only) and Table 6 (stiffness and stretch), whereas his measured results for nut compensation generally fall beyond the modelled results in Table 6. So not a totally convincing affirmation of his techniques.

Some part of the difference Byers saw between real and modelled behaviour can also be explained by his adoption of manufacture’s data for the Young’s modulus of the string material. Actually measuring that and the cross sectional area of material in a floss core is a challenge in itself. I get around that by calculating the actual string stiffness from the rise in frequency caused by a “fretting” displacement on a test rig, after the string has been tensioned to pitch and has stretched out (because the stress/strain curve is not a straight line).

Byers also notes: “There is nothing theoretical about the experimental results, and they are the ones to be applied toward a practical solution”. And that may be why Byers’ results align fairly well with some of my modelled results.

Full details, of course, in 'Contemporary Acoustic Guitar, Design and Build'
Trevor Gore: Classical Guitar Design and Build

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by Alan Carruth » Thu Mar 15, 2018 3:13 pm

Thanks Trevor.

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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by guitarrista » Thu Mar 15, 2018 5:01 pm

Hi Trevor,

Thanks for chiming in. It is impossible to argue in detail when one paper is freely available and the other is not (too bad it is part of a book rather than a stand-alone published paper), but let me try to understand at least something based on what you wrote. When you say:
However, the nodes at the end points of the string don’t necessarily have zero slope at the terminations. The string can rotate rather than just bend.
What does this have to do with string stiffness, which is a property of the material? Since you are highlighting this (and based on what follows in your post), it seems you are saying that the ability to twist around itself at the string ends (as opposed to being fixed ends without that ability) somehow compensates for string stiffness so the harmonics' progressive sharpening is not evident in experiments. What would be the physical mechanism to explain this?
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Trevor Gore
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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by Trevor Gore » Thu Mar 15, 2018 10:26 pm

guitarrista wrote:
Thu Mar 15, 2018 5:01 pm
Hi Trevor,

Thanks for chiming in. It is impossible to argue in detail when one paper is freely available and the other is not (too bad it is part of a book rather than a stand-alone published paper), but let me try to understand at least something based on what you wrote. When you say:
However, the nodes at the end points of the string don’t necessarily have zero slope at the terminations. The string can rotate rather than just bend.
What does this have to do with string stiffness, which is a property of the material? Since you are highlighting this (and based on what follows in your post), it seems you are saying that the ability to twist around itself at the string ends (as opposed to being fixed ends without that ability) somehow compensates for string stiffness so the harmonics' progressive sharpening is not evident in experiments. What would be the physical mechanism to explain this?
I didn't say anything about "the ability to twist around itself at the string ends (as opposed to being fixed ends without that ability)". My use of the term "zero slope" should have given you a hint. I was referring to rotation like a seesaw, so the string tilts (rotates) rather than bends at the terminations. Bending gives a shortening effect to the string, whilst rotating doesn't.
guitarrista wrote:
Thu Mar 15, 2018 5:01 pm
It is impossible to argue in detail when one paper is freely available and the other is not...
I totally disagree. Sometimes you have to make an investment to gain the knowledge, like I had to do in order to write the book, which is readily available.
Trevor Gore: Classical Guitar Design and Build

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guitarrista
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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by guitarrista » Thu Mar 15, 2018 10:34 pm

OK, so to repeat my question with that clarification:

What does this have to do with string stiffness, which is a property of the material? Since you are highlighting this (and based on what follows in your post), it seems you are saying that the ability to [tilt (rotate) rather than bend at the terminations] somehow compensates for string stiffness so the harmonics' progressive sharpening is not evident in experiments. What would be the physical mechanism to explain this?

Once the string is displaced from its equilibrium, the force pushing it back towards equilibrium is still a bit larger than the purely restoring force of an ideal string, so progressive sharpening of higher partials would still occur.
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Re: At fret 12: The G str. isn't sharp. The A str.is flat !!

Post by prawnheed » Thu Mar 15, 2018 11:08 pm

There are two different effects. The sharpening of higher harmonics due to the stiffness of the string (inharmonicity) and the effective shortening of the string due to the way in which the string is constrained at the ends (mostly the bridge).

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