Fretboard, Frets spacing, Scale length of a Guitar

Ever wondered why the space between the frets of a guitar gradually decrease the further you go down the neck?

Its simple physics actually - I have explained the basics in this chapter of the BasicMusicTheory book : Chapter - 5A : Octaves and Temperaments (Cents). Have give it a read and come back, if you need to.

First some basics:

• The length of the string vibrating in open position is termed as the Scale Length of the instrument. It is roughly the distance between the bridge and the nut.
• The frets correspond to all the notes of the chromatic scale in that order.
• Pressing a string at a particular fret determines the vibrating length of the string, and therefore the pitch of the note produced (frequency).
• The neck generally accommodates a certain number of frets corresponding to about 1.75 to 2 octaves (21 or 24 frets respectively). Certain guitars even have 3 octaves (36 frets). This number determines the pitch range of the instrument.
• All length measurements stated here are with respect to the bridge of the guitar.
  

You see, the frets are positioned in a particular way, so that they produce all the notes of the octave(s) that it is designed for. Consider that the scale length of an instrument is L. The frequency of the note produced when this string is played open is then, say,F. Now the 1st fret will have to produce the next semitone.

Now, we know that the next semitone will have a frequency of (12th root of 2 or nearly 1.06) times F. Hence, the length of the string (at a constant tension) must again be L divided by 1.06 - derived from the basic formula:



So the next fret goes at a distance of L/1.06 from the bridge, and so on and so forth till all the frets are placed.

N.B.: The length of the actual neck of the guitar is variable due to the make and model and the kind of string intended to be used.

Again, one may infer, that the nature of strings is quite suited to the guitar. So, if this be correct, each guitar will have its individual suited brand of string. This, of course, is not the case as the strings tension can be adjusted (T in the above equation). Thus various types of gauges can be used on one guitar alone, just by varying the tension of the guitar strings (by using the turnkeys).
http://basicmusictheory.blogspot.com/2009/07/fretboard-frets-spacing-scale-length-of.html

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Fretboard, Frets spacing, Scale length of a Guitar