That's why the saddle is laid into the bridge on an angle... so that all of the strings will have the same vibrating length (dashed line in the graphic above). String Length Compensation is essential for an instrument with a fixed scale length and fixed fret spacing, but different string thickness (stiffness).
My first grandchild was born on September 27... his name is Eli. Since the day of the birth, no one has gotten much sleep. That was 5 days ago. All is good with the baby and the new mom...Woody is still in recovery.
String Vibration and Harmonics
Many readers might know how to tune a guitar by matching the string harmonics. These vibrations occur at the nodes of the vibrating string and are simply called the 'harmonics'.
First, some note naming to get everyone on the same page. All of the open string notes (frequencies) on a guitar, except 1st string E, are below middle 'C' on a piano. Middle C is designated as C4 because it is the first note in the 4th octave from the bass end of the piano keyboard. On a guitar, all open string notes (except E4) have a vibrating frequency of less than 262 hz (Middle C).
I will be traveling to Atlanta to see my new grandson later in October. Hope to see you back here next month.
Next, play the harmonic at the 5th fret on the A string (again). While the note is still ringing, match it with the harmonic on the D string at the 7th fret.
Now play the harmonic at the 7th fret on the A string and match the harmonic at the 5th fret on the low E string. This note is an E4. The open 1st string can also be matched to this same E4 note. At this point four of the guitar's six strings are in tune. The last two (G and B) are tuned similarly.
The diagram below is a graphic of how the harmonics match up across all 6 strings.
To understand guitar string vibration some basic components of the vibrating string need to be noted. Essential among these is the Vibrating Length of the string which, without string compensation, is not the same for each string. The Total Length of the string, between nut and saddle, is made up of the Vibrating Length plus a Dead Length at each end. These terms are explained in the graphic below.
Finally, if you are guitar repair person, harmonic tuning is the easiest way to tune a guitar that is laying flat on its back on the workbench.
First, let me say this: If one is not familiar with the harmonic tuning method, actually tuning this way is waaaaaay easier than explaining how to do it. One might have to read these instructions a couple of times.
Start with tuning the A string. Strike an A-440 tuning fork, then match the tuning-fork A with the harmonic at the 5th fret on the A string. This note is an A4 (440 hz).
On a guitar, an open string vibrates between the nut and saddle at a specific frequency determined by the player using the tuning gears on the peghead. This is called the fundamental frequency of the vibrating string. The open vibrating string has only two nodes (points of minimum amplitude) and they are at the nut and the saddle. The point of maximum amplitude is called the anti-node and the open-string fundamental has only one anti-node.
The guitar string is also vibrating in many sub-fundamental string lengths that have an integer number of nodes and anti-nodes. The node at the 12th fret is called the 2nd harmonic because it divides the string into 2 equal, vibrating string lengths, thus, two anti nodes.
There is another node at the 5th fret (4 anti-nodes), and another node at the 7th fret (3 anti-nodes). There are more nodes but, for now, this will do.
Don't take it for granted that every note played on a guitar fretboard will be at perfect pitch. The pitch (frequency) of a vibrating string is determined by 3 factors: String Length, String Mass and String Tension.
> We change string length by shortening the string as we move up the fretboard... raising the pitch.
> We cannot change the string mass, so be it.
> We increase string tension mainly with the tuning gears in the peghead.