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# How Guitar Fret Distances are Designed

This study is surely one that a person wanting to learn about the properties of vibrating strings and its application to string instruments should take up, since it includes so much of value, especially as a concrete example of vibrating strings fixed at both ends , its properties and application to the guitar, a very familiar musical instrument. Whether you are a musician, a physicist or anyone who enjoys guitar music, have you ever stopped to think about how to compute the correct fret positions on the guitar fingerboard?

Peré Mersenne (1588-1648) asserted that the pitch of a string is inversely proportional to the length and the square root of the linear mass density, and directly proportional to the square root of the tension:

These mathematical laws obeyed by vibrating strings as outlined by Mersenne can be learned from experiment and this is sufficient for practical purposes.

In this hub, the properties of vibrating strings fixed at two ends is derived by reasoning from some general principles of wide application and the result includes Mersenne’s’ rules, not only would this manner strengthen the validity of the principles, but would give practice in applying them.

## Vibrating Strings Fixed at Both Ends

A guitar is a common example of a string fixed at both ends which can vibrate. When we pluck it, the string starts to vibrate in one or more loops. The vibrations of such a string are called standing waves and they satisfy the relationship between wavelength and frequency that comes from the definition of waves:

where *v* is the speed of the wave, *f*
is the frequency (measured in Hertz, Hz) and λ is the wavelength.

The
velocity of a wave depends on the properties of the medium it travels. The
velocity of a wave on a stretched string depends on the tension ‘*T’* in the string and on the string’s
linear mass density ‘*µ*’

These standing waves may be thought of as the result of waves that travel down the string in both directions, reflected at the end and re-reflected at the other, and so on. The frequencies at which the standing waves are produced are the natural frequency or resonance frequency of the string.

The fundamental frequency, the main pitch you hear, is the lowest tone and it comes from the string vibrating with one big arc from bottom to top. The fundamental satisfies the conditionwhere λ is the wavelength of
the fundamental and *L* is the length
of the freely vibrating portion of the string, it correspond to one antinode
(or loop). Since the string is fixed at both ends, so any vibration of the
string must have nodes at each end. That limits the possible vibrations of the
string (‘mode of vibration’ in simple terms just means style or way of
vibrating). The corresponding frequencies at which the standing waves are
produced are the natural or resonance frequencies of the string.

In
order to find the fundamental frequency *f
*of the vibration, the equations

Thus, the formula for the fundamental frequency of a string fixed at both ends (Eqn 1).

Equation 1 is the basis for the design of stringed instruments such as piano, violins and guitars. To ‘tune’ a string-instrument, the tensions in the various strings are adjusted until their fundamental frequencies are correct. This equation shows the relationship between the fundamental frequency, length and linear mass density of a vibrating string fixed at both ends.

It includes Mersenne’s rules and accounts for the different masses and tensions of strings of the same length to create different pitches of guitar strings. Individual pitches on the strings are created by pressing down beneath a fret, thus effectively shortening the length of the string.

The
lowest note on the piano vibrates at 27.5 Hz and the highest vibrates at 4,186
Hz. In guitar, the open 6^{th} string is the lowest note which vibrates
at 82.4 Hz (Table 1). This note is called E note denoted by E_{0}. If
you double the frequency, you get the note E_{1}, one octave higher
than E_{0} which has a frequency of 164.8 Hz. ‘Octaves’ of a note are
just multiples of the original frequency. In general

where F_{0} is any
given note’s frequency and F_{n} is the frequency of the note n octaves
above F_{0} where n=1,2,3…

The highest note on a guitar of twelve frets is the
note E_{3} (that is three octaves up from E_{0})_{. }From
one note up to its octave, the average human ear can distinguished only 12
different tones or notes. So when a chromatic scale (comprising all twelve
notes in an octave) is played, we perceived that all sounds between the first
note and the last note (the first note’s frequency doubled) are included.

^{b}, B, C, D

^{b}, D, E

^{b}, E, F, F

^{#}, G, G

^{#}. The equal-tempered tuning is the standard system in music and it assigns the corresponding frequencies to the notes. Tuning is not linear. One has to double the frequency to get the next octave up; one has to quadruple it to get the next octave after that. Consequently, the notes within the chromatic scale are not equally distributed in frequency (or in length). So in general practice, musicians universally assigned A

_{3 }to a frequency of

F_{A3}=440
Hz and therefore A_{4} will be at F_{A4}=880 Hz. We then take
logarithms of A_{3} and A_{4} frequencies. Next, we mark in 11
equally spaced points between

We
then apply arc-logarithms to those points and arrive at the half-step ratio *r* of notes in the equal tempered tuning.
Therefore each successive note in the chromatic scale is *r* higher in frequency.

On a guitar, different strings have different linear mass density. The first string is like a thread, and the sixth string is wound much thicker and heavier. The tension in the strings is controlled by the tuning pegs. The length of the open string (also called the scale length) is the distance from the nut to the saddle. When you press down on a string at a fret (assuming tension is constant), you change the length of the string, and therefore its frequency when vibrating. The frets are spaced out so that proper frequencies are produced when the string is held down at each fret. Fig 1 shows the guitar fingerboard.

Table 2 shows the frequencies of the notes in the first string and the twelve fret bar positions on the fingerboard based on scale length of 26 inches.

## Comments

This is an inneffective table as the string length is not always 26". My equation d=L(1-2^-n/12) is much more usefull as d represents the distance from to nut to whatever fret no matter what the length of the string. It is only derived from combining f=f0*2^n/12 and knowing that frequency is inversly proportional to Ld. the part in brackets also tells us the protion of the string at each fret and when 5 and 7 is plugged into n. the portion of the string is 0.33 and 0.25. which explains why there are such strong harmonics as you are forcing a node on relevant points.

that's pretty cool

Wow, I did not know about the complexity of design based on computation until reading this post.

Thanks for the information.

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