Intonation Ponderings

[johnallg]

I wasn't sure where to post this but decided it was a technical issue so here it is.

I've always wondered about bridges and intonation. Doesn't matter if we're talking bass, guitar, or whatever brand instrument. As I understand it, you are properly intonated when the distance from the nut to the 12th fret equals the distance from the 12th fret to the bridge saddle.

Here's where I get confused. Since the nut and 12th fret are parallel, why do the saddles all end up in different positions with the lower, fatter strings having a further distance from the 12th fret than the smaller, higher strings? String diameter is stated as one reason, but thinking about it, irrespective of diameter, they all hit the nut at the same place (distance) and also the saddle. Besides the differences in diameter would not account for the amount of distance differences of the saddles. So why is the proper places for all 4 (or 6) strings staggered?

Anyone have an answer to this?

[kiramdear]

That's a real good question. I too would like to know how that works.

[cjj]

Because the equal measurement point is just a starting point. The proper tuning is when the fretted note at the 12th fret is the same pitch as the harmonic at the midpoint of the string, which is only theoretically at the 12th fret. In reality, the harmonic point will not always line up with the 12th fret. It will be at the midpoint node of the vibration frequency, which will depend on the tension, diameter, flexibility, etc. But to get this frequency to align with the same tone at the fixed 12th fret, requires a bit of adjustment of the length of the string...

[johnallg]

That makes sense, but I'm still having difficulty resolving the distance differences with what I know about wavelengths and lengths. That would mean the string would be vibrating the same over the greater distance of the nut to 12th fret than it does from the shorter distance 12th fret to saddle. I'm having a hard time envisioning that. What am I missing?

[cjj]

Oh, man, it's been over 30 years since I took that acoustic physics course, I'm treading in thin ice (or is that hair??? Or maybe brain cells? ) as it is...

Maybe in the morning...

[cjj]

That makes sense, but I'm still having difficulty resolving the distance differences with what I know about wavelengths and lengths. That would mean the string would be vibrating the same over the greater distance of the nut to 12th fret than it does from the shorter distance 12th fret to saddle. I'm having a hard time envisioning that. What am I missing?

OK, it's morning, but I haven't finished my coffee yet. Anyway, without making any measurements or even thinking about calculations, Let's see what I can dream up...

It's probably best to think of this as two different string systems, which in reality, it is.

First, you've got the open string. The frequency of this string will be based on all the stuff we know about, tension, mass, flexibility, etc. Now, if you touch the string and damp it at the exact mid point, you can cause it to vibrate at twice it's fundamental frequency. This is the octave harmonic, which makes sense because all things being equal, halving the string length will double the frequency. This works regardless of the tuning of the string or the distance from the nut to the bridge saddle. It also has no relation to where the frets are since they are not involved.

OK, now let's take the string fretted at the 12th fret. This is supposed to be the octave point of the open string, but, this only works for a limited range of string lengths. If you move the saddle 6 inches, the octave point will move 3 inches and will be nowhere near the 12th fret. What we've really got is a new string system here. It's got a different length (12th fret to saddle, the distance to the nut is now irrelevant) and a different tension (you stretch the string slightly when you push it down on to the fret.

So, when doing your intonation, you are now trying to get frequency of the fretted string to match the octave harmonic of the open string. As you move the saddle, you change the length of the fretted string, changing it's frequency. Unfortunately, this also affects the open string, changing it's frequency and moving the harmonic point. To stay perfectly in tune, you'll have to adjust the string tension along with the saddles. The position of the open harmonic is only approximately over the 12th fret, but the fretted note is exactly at the 12th fret. If you're really careful and use something other than fingers to damp the string for the harmonic, you'll find that it's probably not exactly over the 12th fret (but it will be at the midpoint between the nut and saddle). If the fretboard was designed properly, the frets will be close enough to the right place that you won't notice that things are out of tune, but in reality they will almost never be exactly perfect. People have come up with all sorts of things to fix this, like fanned frets.

Where the saddles end up in relation to each other has to do with the actual string characteristics. Smaller, higher tension strings will have a different change between open harmonic and fretted harmonic than larger heaver strings. Yeah, I guess I didn't answer the question of why higher strings tend to have the saddles closer to the 12th fret than lower strings, that's more thinking than I really feel like doing, but I'm sure it could be explained with enough equations...

[tennis_nick]

Oh, man, it's been over 30 years since I took that acoustic physics course, I'm treading in thin ice (or is that hair??? Or maybe brain cells? ) as it is...

Maybe in the morning...

Think of it this way.

The halfway point from the nut to the saddle SHOULD be your 12th fret. Unfortunately, when you fret the string, you're bending it slightly, causing it to ring a bit sharper, because of the distance from the string to the fret. TO compensate for this, we move the saddle on the bridge further back, thus making the distance marginally longer, so that your 12th fret will ring true and in tune.

[johnallg]

CJ, thanks for the long explanation, and Nick, thanks for the synopsis. I was not really grasping the influence of the smaller strings having different characteristics of construction and stretching then the fatter wound strings. Looking at the way explained, it makes sense. Extrapolating, if you raise the action, you would have to reset intonation as the increase in stretch fretting at the 12th would necessitate moving the saddle to compensate.

On my bass, I've also found fretting at the 19th fret and making the corresponding B E A D notes for the 4 strings accurate (with a tuner) makes it easier than at the 12th. For me at least. Maybe its a hearing thing - I do have a pretty loud ringing in my left ear.

ANYWAY, thanks.

[jps]

Maybe its a hearing thing - I do have a pretty loud ringing in my left ear.

WHAT?

[winston]

I am happy to say that CJ's detailed explanation is congruent with the explanation I gave you on Skype the other night John.

[johnallg]

Hey, Brian, I thought you were going to type up that long conversation we had to post here! When you explained how the characteristics of the different size and construction of the strings changes how the note per length comes out, it made a lot of sense and clarified it for me why the saddles don't line up.

[jdogric12]

The distance from nut to 12th is NOT the same as the distance from 12th to saddle. The difference is roughly half the distance from the string to fret at the 12th. Whenever you fret any note, you are chaning the distances and tension anyway, and creating a long flat triangle, as opposed to thinking of this as a bisected line segment.

This is why guitars with high action rarely never play in tune.

Much of this never bothers us since our Western system of music is not technically in tune anyway, it's just really close. This is why you can't tune adjacent strings by harmonics. For a perfect 5th to "sound" in tune, it must be slightly sharp. For a major third to sound in tune, it must be slightly flat. Instruments like pianos are tuned "equally out of tune" for every key. (See: Well-Tempered Clavier by Bach) This is also why you can tune a guitar so that it sounds great in G but sounds terrible in E.

There you go, Brian...



Edit: oh, yes, and I didn't even get to the thickness of the string itself. Google yourself the formula for pitch/tension/length/thickness. I did once and learned it was way more complicated than I expected, both the googling and the formula.