How fretting adds string tension and its effects on tuning

ReignOfRatch said:

The other night while adjusting my intonation, I was struck with a curiosity.

When you fret a note, the majority of the change in pitch is because the length of the string has changed, but there is also added tension because you've moved the string from its resting state and this should also have a very small effect on the pitch.

The questions this has me curious to solve now are:

1) How big is this effect anyways?
I know if you fret too hard you change the pitch a perceptible amount so I would assume this is also a perceptible amount.
The Nut height will add even more tension on the first Fret compared to the rest, and a low action should have less effect.

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The pitch error caused by stretching the string down to the fret is large enough to be audible on an uncompensated string. The actual error can be measured with a strobe tuner.

It can also be calculated with reasonable accuracy from the string core diameter, relief height, string height, fret position, and scale length. For example, with an E string height of 0.078" at 12th fret, 0.012" relief height, on a 34" scale bass, the pitch might be about 10 cent sharp. At the 20th fret, the pitch might be about 15 cent sharp.

The common procedure of intonating the 12th fret can reduce this error to zero at the 12 fret, and minimize it at the other frets. This works by lengthening the speaking length (fretted length) of the string, by moving the saddle back. Even though this lengthens all fretted notes (and the open string), it lengthens the higher notes more in proportion to their uncompensated length.

ReignOfRatch said:

2) Are frets locations ever altered ever so slightly to account for this?
I've heard of the rule of 18 (17.8ish in reality) but I don't know what all it takes in to consideration.

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There have been fretboards with individually intonated frets, and those are pretty wild looking. But most all fretboard constructions, I believe, are based on that rule of 18 and place the frets at their mathematically calculated positions.

Not sure about the nut though. Some seem to be a tiny bit closer to the 1st fret than it would be at it's theoretical zero position. This can help with intonation at the lowest frets.

ReignOfRatch said:

3) Would the gage, material, and winding style of a string make a difference in the added tension?

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Absolutely.

With a solid string, things are simply dependant on the cross sectional area of the string. With a wound string, a tight wrap can make the string behave as if it had a larger core, and less compliant. This increases the sharpening of uncompensated notes. A flexible core can make a string more compliant, with less sharpening of uncompensated notes due to fretting.

So both the core type and wrap type can effect the pitch error, and therefore the amount of compensation required.


These are interesting questions. I had them myself, and so calculated the errors along with the effects of compensation.

This simulation is the result, and shows calculated errors (in cents) for an uncompensated E string on a 34" scale bass:




https://www.talkbass.com/attachment...4/?temp_hash=053bc22930c4da3848e4c779ad62f37a

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What exactly is a cent, percent it's off in relation to the next fret? If so it seems like kinda a lot. And you would assume the picture should be set like a bass is, the 12th fret would be a 0 as well as the open string. True?

WardEarth said:

What exactly is a cent, percent it's off in relation to the next fret? If so it seems like kinda a lot. And you would assume the picture should be set like a bass is, the 12th fret would be a 0 as well as the open string. True?

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A cent is a 100th of a semitone, but it is not 1 percent. It is much smaller.

Each of the 12 semitones in the scale are related by a factor of 1.059... . This number is the 12th root of 2.

If you take any tone, and multiply its frequency by that value, you will get the next higher tone, 1/2 step up. If you repeat this eleven more times, on each new tone, you end up at the octave, which is twice the frequency of the original tone.

So that factor value, 1.059..., is used to divide an octave into 12 tones, with each tone having the same relation to its adjacent tone.

A similar thing happens on a very coarse scale, where each octave is related to its adjacent octave by a factor of 2. Each A for example, is twice the previous A: 55 Hz, 110 Hz, 220 Hz, 440 Hz etc.

If you notice, you don't get the next tone or step in the series by adding a constant value to the frequency; you get it by multiplying the frequency by a constant value.

The same principle is used to divide a semitone into 100 parts. In this case, the factor value is the 1200th root of 2, which is 1.000578... Notice that 1 percent would be a factor of 1.01, much larger than 1 cent.

Hope that made sense.


Referring to the illustration, yes, you would not actually want it like that. Its purpose is to show the errors caused by fretting before intonation is adjusted.

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Some of it made sense, maybe half of it.
I'd say a cent is basically what I thought it was. a semitone is a half step. a cent is 100th of a semitone. So I really don't see how its much smaller/larger than 1 percent. Like you said its a relative thing.

A picture that was set up showing both sides of the 12th fret from the perspective of the average bass guitar would be a bit more informative. Open string intonated to the 12th fret. I'm guessing the 5-7 frets and maybe 17-19 frets are the most out? But I really don't know.

If it's based on 100, then 1 is the same as 1%
In Thailand they say 5% as "Roy la Ha" = for every 100 there are 5

WardEarth said:

Some of it made sense, maybe half of it.
I'd say a cent is basically what I thought it was. a semitone is a half step. a cent is 100th of a semitone. So I really don't see how its much smaller/larger than 1 percent. Like you said its a relative thing.

A picture that was set up showing both sides of the 12th fret from the perspective of the average bass guitar would be a bit more informative. Open string intonated to the 12th fret. I'm guessing the 5-7 frets and maybe 17-19 frets are the most out? But I really don't know.

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If by percent, you mean 1 percent of a semitone, then there is very little difference between 1 percent of a semitone and a cent. As a percentage of tone frequency, 1 percent would be much larger.

The difference is in the way cent and percent are defined. For example, cent is defined as a factor of the 1200th root of two. Applying this factor to any tone or frequency will raise the pitch by 1 cent.

A percentage though, is not based on a factor between two percetage points; the percentage values are based on an additive difference between two points. This is a fundamental difference, not just in the math, but in the way we perceive pitch.

Also, how would you define a percentage? A percentage of which half step? A percentage of a specific tone and the tone one step lower? Or a specific tone and the tone one step higher? Each yield different results.


Anyway, here is the original illustration of the uncompensated E string, along with the compensated string:




https://www.talkbass.com/attachment...8/?temp_hash=ae3cf1be29931b7baa6e8c5ccae63c15

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Depending on how high your action is set... the increased tension just to REACH the fret would become a factor.

ReignOfRatch said:

2) Are frets locations ever altered ever so slightly to account for this?

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For a variety of reasons.

This is why your action affects intonation.
High action can cause your intonation to be sharp.
It occurs to me that was the problem with one of my basses.

The 12th fret was sharp, even with the saddle all the way back.
Lowering the action a little solved the problem.

There is a spectacular luthier near me, and he addresses this. Only one I've ever seen. On a bass, it's not worth it, but on a guitar, where it's very hard to get the first couple frets all singing sweetly together because of this issue, it's magic.

I've had it done to an acoustic and an electric (Washburn parlor and a Gibson les Paul). He takes a 10,000s of and inch level survey of the fret placements, works out his thinking, and moves the nut closer to the bridge by a tiny amount, which varies by a tiny angle.

It's amazing.

Every time I started researching the subject of tuning accuracy it seemed I was getting in way over my head and the rabbit hole goes down even farther than I can imagine.

Thank you @megafiddle for your thorough explanation.
It is truly an interesting academic pursuit, beyond how to play, and into why playing works like it does. A lot of science happening under our fingertips!