## The Physics Of Tuning A Bass Guitar String

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The physics of strings has been known for several hundred years meaning that a mathematical formula has been derived which specify their behavior.  And as it turns out the strings of a bass guitar obey the same formulas as that of other kinds of strings.  Surprisingly these formulas hold independent of material type, and independent of whether its an upright bassdouble bass or electric bass.  Of course, the pitches are, as any bass guitarist knows, dependent on the length of the string, the tension it’s under, and the thickness of the string.  The curious guitarist will now ask, what kind of behaviors does the formula describe?  The formula describes the frequency of the string’s vibration, or the pitch in laymen’s terms.

The pitch of the string is given by the formula

$\frac{1}{2L}\cdot\sqrt{\frac{T}{\mu}}$

Here the L  is the length of the string, T is the tension, and μ is the linear density (given in units such as pounds per foot or kilogram per meter).  Guitarists will instantly recognize that the thickness of the string must be related to the linear density.  The thicker string has a higher linear density, and therefore by the formula its position in the denominator makes thicker strings correspond to lower frequencies.  But how are the strings tuned?  Certainly the length as defined by where the strings touch the bass guitar at the two ends doesn’t change.  And certainly the linear density doesn’t change.  The answer lies in how the tension is dialed up or down by turning the tuners.

Reliably tuning a bass guitar follows one of several well known techniques.  First one string must be tuned to a reference.  You have two choices.  EIther find a friend with perfect pitch and ask him or her to hum or sing one of the open notes, preferably the E at 41 Hertz.  If this fails, you’ll have to get a tuning device that vibrates at a single frequency.  A tuning fork is a good example.  A tuning fork is a piece of metal whose primary harmonic is of one note.  Strike it lightly and it will produce the needed note.  If a string were a fixed length and never lost its shape, it would also always have a single note.  If you strike a tuning fork too hard, you will produce higher harmonics that are related to the primary but at the wrong frequency.

Once one string is tuned, the 5th fret tuning method is used to adjust other strings.  The technique is very simple.  Putting your finger on the string at the 5th position generates the same pitch as the next higher string in open position (no frets pressed).  For example, if your E string is tuned, then fret it in the 5th to generate an A, and adjust the neighboring string until its pitch matches the A.  Continue doing pairs sequentially until all strings are tuned.

What if you don’t have a reference?  Well, it turns out the situation isn’t too bad.  Since all the strings will be tuned appropriated relative to each other using the 5th fret method, music will sound reasonable good though off by some absolute amount for all notes.  Given this, tuning against an imperfect reference such as a piano is also better than having no reference at all.

The modern bass guitarist who has a smart phone or a computer doesn’t have to worry about tuning because guess what, there’s an awesome app that acts as a bass tuner which all the guess work and cumbersome reference tuning forks out of the picture.  We highly advise our physicist readers to check out the app and website!

(Photo Credit: soapbeard / Creative Commons)