Intonation Theory Question

Construction and repair of Classical Guitar and related instruments
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Michael Lazar
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Intonation Theory Question

Post by Michael Lazar » Sun Nov 05, 2017 3:16 pm

This is a question about saddle compensation. Does anyone here know how many cents a note fretted at the 12th fret will be flattened (lowered) for each millimeter of saddle setback using medium tension nylon strings tuned to A440, a 650mm scale and an action height of 4mm at fret 12? I can appreciate that the values may differ among the different string diameters and other variables but any sort of theoretic answer would be helpful.

soltirefa
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Re: Intonation Theory Question

Post by soltirefa » Sun Nov 05, 2017 3:33 pm

I'm not a luthier and know nothing about this topic, but I saw this lecture video one day from John Gilbert and Greg Byers discussing their respective intonation ideas. They approach it in different ways.

https://youtu.be/XCrGpExJRyc

OldPotter
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Re: Intonation Theory Question

Post by OldPotter » Sun Nov 05, 2017 3:40 pm

I understand that Trevor Gore has a complete description of saddle and Nut compensation in his book.

There is a starting point here;http://www.frets.com/FretsPages/Luthier ... pcalc.html
"When I was younger, I could remember almost everything, whether it happened or not." Mark Twain

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bacsidoan
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Re: Intonation Theory Question

Post by bacsidoan » Sun Nov 05, 2017 4:20 pm

Michael Lazar wrote:
Sun Nov 05, 2017 3:16 pm
This is a question about saddle compensation. Does anyone here know how many cents a note fretted at the 12th fret will be flattened (lowered) for each millimeter of saddle setback using medium tension nylon strings tuned to A440, a 650mm scale and an action height of 4mm at fret 12? I can appreciate that the values may differ among the different string diameters and other variables but any sort of theoretic answer would be helpful.
In theory, the mass of the string, the tension and absolute pich are irrelevant as things are expressed in ratio.

The frequency of a vibrating string is calculated by this equation:
mersenne_eq1.png
where:
l is length of the string
T is tension
Mu is mass per unit length

In this case l = 325 mm (at 12th fret)
Let's call f2 is the original frequency at 325 mm and f1 is the frequency at 326 mm
One can easily see that the ratio f2/f1 is 326/325 = 1.0030769230769230769230769230769

From this one can derive the cent calculation between two frequencies according to this equation:
cent3.gif
Plug in the number. The interval between f2 and f1 is 5.32 cents. In other words, each mm of saddle setback will drop the pitch by 5.32 cents at the 12th fret.

This is just a simplified calculation for 12th fret. If one goes up the register, the compensation will be progressively more and vice versa, progressive less for the lower registers.
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Last edited by bacsidoan on Sun Nov 05, 2017 9:44 pm, edited 1 time in total.

simonm
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Re: Intonation Theory Question

Post by simonm » Sun Nov 05, 2017 7:31 pm

In addition to what bacsidoan has outlined, there is a thread http://www.anzlf.com/viewtopic.php?f=33&t=7359 dedicated to the compensation calculations from the Gore and Gilet books.

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Trevor Gore
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Re: Intonation Theory Question

Post by Trevor Gore » Sun Nov 05, 2017 11:44 pm

Michael Lazar wrote:
Sun Nov 05, 2017 3:16 pm
Does anyone here know how many cents a note fretted at the 12th fret will be flattened (lowered) for each millimeter of saddle setback using medium tension nylon strings tuned to A440, a 650mm scale and an action height of 4mm at fret 12?
The simple answer, in the real world, is 3 cents per millimetre.
bacsidoan wrote:
Sun Nov 05, 2017 4:20 pm

In theory, the mass of the string, the tension and absolute pich are irrelevant as things are expressed in ratio.

The frequency of a vibrating string is calculated by this equation:
mersenne_eq1.png
where:
l is length of the string
T is tension
Mu is mass per unit length

In this case l = 325 mm (at 12th fret)
Let's call f2 is the original frequency at 325 mm and f1 is the frequency at 326 mm
One can easily see that the ratio f2/f1 is 326/325 = 1.0030769230769230769230769230769

From this one can derive the cent calculation between two frequencies according to this equation:
cent3.gif
Plug in the number. The interval between f2 and f1 is 5.32 cents. In other words, each mm of saddle setback will drop the pitch by 5.32 cents at the 12th fret.

This is just a simplified calculation for 12th fret. If one goes up the register, the compensation will be progressively more and vice versa, progressive less for the lower registers.
Doan's answer is technically correct and responds directly to Michael's (over specified) question. You don't need to know the action, the material properties or the tuning to figure out the answer, as Doan correctly points out. However, in practice, Doan's response is problematical. The reason is that when you apply compensation at the saddle, it applies to all the notes on the fretboard and the open string. So the compensation will flatten the open string as well as the note fretted on the twelfth fret. Usually, the open string will then be tuned back up to pitch, which will also sharpen all the fretted notes. At the twelfth fret (half scale length) you get exactly half the tonal change that you might have thought you were going to get, so the "real world" answer is half of what Doan calculated.

As expected, Frank gets it right here: http://www.frets.com/FretsPages/Luthier ... pcalc.html. Check his "extreme" example at the bottom of the page.
Trevor Gore: Classical Guitar Design and Build

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bacsidoan
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Re: Intonation Theory Question

Post by bacsidoan » Mon Nov 06, 2017 2:05 am

Trevor Gore wrote:
Sun Nov 05, 2017 11:44 pm
Michael Lazar wrote:
Sun Nov 05, 2017 3:16 pm
Does anyone here know how many cents a note fretted at the 12th fret will be flattened (lowered) for each millimeter of saddle setback using medium tension nylon strings tuned to A440, a 650mm scale and an action height of 4mm at fret 12?
The simple answer, in the real world, is 3 cents per millimetre.
bacsidoan wrote:
Sun Nov 05, 2017 4:20 pm

In theory, the mass of the string, the tension and absolute pich are irrelevant as things are expressed in ratio.

The frequency of a vibrating string is calculated by this equation:

mersenne_eq1.png

where:
l is length of the string
T is tension
Mu is mass per unit length

In this case l = 325 mm (at 12th fret)
Let's call f2 is the original frequency at 325 mm and f1 is the frequency at 326 mm
One can easily see that the ratio f2/f1 is 326/325 = 1.0030769230769230769230769230769

From this one can derive the cent calculation between two frequencies according to this equation:

cent3.gif

Plug in the number. The interval between f2 and f1 is 5.32 cents. In other words, each mm of saddle setback will drop the pitch by 5.32 cents at the 12th fret.

This is just a simplified calculation for 12th fret. If one goes up the register, the compensation will be progressively more and vice versa, progressive less for the lower registers.
Doan's answer is technically correct and responds directly to Michael's (over specified) question. You don't need to know the action, the material properties or the tuning to figure out the answer, as Doan correctly points out. However, in practice, Doan's response is problematical. The reason is that when you apply compensation at the saddle, it applies to all the notes on the fretboard and the open string. So the compensation will flatten the open string as well as the note fretted on the twelfth fret. Usually, the open string will then be tuned back up to pitch, which will also sharpen all the fretted notes. At the twelfth fret (half scale length) you get exactly half the tonal change that you might have thought you were going to get, so the "real world" answer is half of what Doan calculated.

As expected, Frank gets it right here: http://www.frets.com/FretsPages/Luthier ... pcalc.html. Check his "extreme" example at the bottom of the page.
Of course you are correct. I'm aware of that. I just tried to answer the OP's question directly. My calculation is based on the premise that if you keep everything the same and just move the saddle back 1 mm, the pitch at the 12th fret will drop by 5.32 cents. The pitck at the open string will drop half of it which is 2.66 cents. After the saddle compensation, if the player turn up the tension of the string to negate the 2.66 cent drop to keep the guitar open string back in tune then the fretted note at the 12th fret will drop 2.66 cents from the original frequency.

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Contreras
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Re: Intonation Theory Question

Post by Contreras » Mon Nov 06, 2017 2:19 am

bacsidoan wrote:
Sun Nov 05, 2017 4:20 pm
Michael Lazar wrote:
Sun Nov 05, 2017 3:16 pm
This is a question about saddle compensation. Does anyone here know how many cents a note fretted at the 12th fret will be flattened (lowered) for each millimeter of saddle setback using medium tension nylon strings tuned to A440, a 650mm scale and an action height of 4mm at fret 12? I can appreciate that the values may differ among the different string diameters and other variables but any sort of theoretic answer would be helpful.
In theory, the mass of the string, the tension and absolute pich are irrelevant as things are expressed in ratio.

The frequency of a vibrating string is calculated by this equation:

mersenne_eq1.png

where:
l is length of the string
T is tension
Mu is mass per unit length

In this case l = 325 mm (at 12th fret)
Let's call f2 is the original frequency at 325 mm and f1 is the frequency at 326 mm
One can easily see that the ratio f2/f1 is 326/325 = 1.0030769230769230769230769230769

From this one can derive the cent calculation between two frequencies according to this equation:

cent3.gif

Plug in the number. The interval between f2 and f1 is 5.32 cents. In other words, each mm of saddle setback will drop the pitch by 5.32 cents at the 12th fret.

This is just a simplified calculation for 12th fret. If one goes up the register, the compensation will be progressively more and vice versa, progressive less for the lower registers.
That's 'easy' for you to say! 😌
Put down the bagpipes ...
... and no one gets hurt.

amezcua
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Re: Intonation Theory Question

Post by amezcua » Mon Nov 06, 2017 4:41 pm

Why not simply place each separate fret for each separate note and check every note with an electronic tuner .Then you will all be happy .Unfortunately your religion states that every fret must be straight. So you will never,ever get to Heaven.

simonm
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Re: Intonation Theory Question

Post by simonm » Mon Nov 06, 2017 4:58 pm

amezcua wrote:
Mon Nov 06, 2017 4:41 pm
Why not simply place each separate fret for each separate note ….
Been done and before the advent of readily available electronic tuners. But as you can see from the vast number of such instruments for sale it was not exactly a popular option.

Alan Carruth
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Re: Intonation Theory Question

Post by Alan Carruth » Mon Nov 06, 2017 6:00 pm

...and we're back into the endless discussion of the deficiencies of Equal Temperament.

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bacsidoan
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Re: Intonation Theory Question

Post by bacsidoan » Tue Nov 07, 2017 1:36 am

Alan Carruth wrote:
Mon Nov 06, 2017 6:00 pm
...and we're back into the endless discussion of the deficiencies of Equal Temperament.
:)

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Michael Lazar
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Re: Intonation Theory Question

Post by Michael Lazar » Wed Nov 08, 2017 7:05 pm

Thanks so much for all of this. I really appreciate the time and effort along with your kindness. The question arose when a correspondent told me that three of six strings on a guitar he had made were flat at fret 12 by 5 to 8 cents. He insisted that he had his fret and bridge placements right and was wondering how much he should reduce the compensation for those strings. I suggested that all things considered and if all of his fretting and bridge placement was accurate the three offending strings would have to be defective in order to produce intonation errors of those magnitudes.

amezcua
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Re: Intonation Theory Question

Post by amezcua » Fri Nov 10, 2017 12:51 am

For Simonm .You sounded sad that a magic wand had not transformed every guitar in the world to a different tuning system .Many decisions are made to keep manufacturers happy .Happiness for them is the bottom line .It`s not always good for everyone else. Sometimes it`s very bad .Looking at guitarists statistically would we say they are overwhelmingly happy with intonation ? That would make any questions about it a rarity .
I wanted to separate the idea of "Frets out of line" from "Temperaments". They are both awkward but different .One is physically awkward and the other (temperaments) would create too many arguments as the choices are so numerous .
If a guitar was fitted with nylon and gut only would bone frets give players a chance to shape their own preferences just as they alter nuts and bridges ? Set the bone frets in wider slots and superglue any changes on top.

simonm
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Re: Intonation Theory Question

Post by simonm » Fri Nov 10, 2017 11:06 am

amezcua wrote:
Fri Nov 10, 2017 12:51 am
….
If a guitar was fitted with nylon and gut only would bone frets give players a chance to shape their own preferences just as they alter nuts and bridges ? Set the bone frets in wider slots and superglue any changes on top.

Lutes and other "fretted" instruments generally had tied gut frets with glued on frets of various materials on the top of the soundboard after the neck join. With that method you can move (most of) the frets to change the intonation.

The moveable frets I was referring to above can be seen here https://issuu.com/orfeomagazine/docs/orfeo_2_an on page 29. It is Daniel Friederich's no 437 (1977). He is not the only one who has done such fretboards but they have never become popular. (Would be easier today with CAM techniques). Other incarnations of the same kind of concept are the versions with variously curved frets. Doing that has been facilitated by laser/CAM techniques but is also not exactly popular. Then there are the various fan fret options too. The latter seem to have made some marginal headway in electrics and maybe steel strings but not in the classical world.

As for letting players loose with superglue on on guitar fretboards … I don't think even a factory making bottom of the barrel guitar shaped objects for hypermarkets would risk the potential damage to their reputations ... :lol:

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