Stringed Instrument Math

This page provides equations and formulas that correspond to the Geometry and Intonation Page of this website, which focuses on nut compensation.  There is a real wealth of additional math and technical information on the web.

Math Links are provided for those that would like to pursue related or additional related mathematics.

Scale Length

There is some confusion in the definition of scale Length.  The Scale Length of any stringed instrument is the design-calculated length of the scale – it does not include the lengths of saddle or nut compensation.

An accurate method to determine the scale length of your instrument is to measure the length from the left edge of fret 1 to the left edge of fret 13, and multiply that length by 2.119 (The scale length is twice the length of the first octave, 0 – 13, and the first octave is 1.0595 times the length of the 1 – 13 octave.

String Frequency Equation

The String Frequency Equation is the fundamental equation for relating pitch frequency to the tension, length, and density (mass per unit length) of the string.  This and most of the following equations assume perfect flexibility of the strings.

f = (1/2L)*√(T/μ)

where:

• f is the frequency
• T is the string tension
• L is the vibrating length of the string
• μ is the linear density or mass per unit length of the string

Let’s break down the equation and try to make sense of it   Notice the first part, f = 1/2L shows that (when tension and μ don’t change) that frequency increases twice as fast as length. for example: when length increases by half, frequency doubles.

Note that for a particular string, μ is constant and will not change.

The remaining part of the equation is a little more tricky. It has a name: Transverse Wave Velocity, transverse meaning over the length of the string.  Let’s take a closer look:

Transverse (or ‘traveling’) Wave Velocity Equation

V = √(T/μ)

where:

• V is the transverse wave velocity
• T is the string tension
• μ is the linear density or mass per unit length of the string

This shows us that, when length is constant, that V, and therefore frequency, increases by √(T/μ. (It happens when we turn the tuning key, but also, a little when we fret a note. T is the only variable in the equation for V.)

When we are compensating the saddle, we extend the string length, and then re-tune by increasing tension with the tuning key.

Increasing saddle compensation results in the vibrating length of the higher fretted note (desired). But, the lower fretted note will also be lengthened, but only by about half as much in proportion to its vibrating length as the higer note.

With more compensation, the lower pitch will flatten with respect to the highest.  Since the lengths have increased, tension must be increased for both to be in tune.

Once the saddle is compensated, the open-string tension is established; that is, when you re-tune to a fretted note, it will always result in the same open-string tension that followed saddle compensation. (Otherwise the fretted notes would not be in tune.)

Since the tension is established, we can use a simplified equation, with only length as a variable, to determine the nut compensation length.

When you finish compensating the nut (and returning to the same action height) you will still have the same open-string tension that followed saddle compensation.  Therefore, both T and μ can be treated as constant in the in the equation, and we can substitute V, now a constant for √(T/μ in the String Frequency Equation

f = (1/2L)*√(T/μ)

f = (1/2L)*V     (substituting V for √(T/μ)

f = V/2L           (rearranging)

For the purposes of nut compensation, V will be constant and we now have a simple equation for showing frequency as inversely proportional to string length (frequency increases as string length decreases).

String Length Equation

The String Length Equation is a re-arrangement of the String Frequency Equation

L² = T/4μf²

L = (1/2f)*√(T/μ)

L = (1/2f)*V     (substituting V for √(T/μ)

L = V/2f           (rearranging)

where:

• T is the string tension
• μ is the linear density or mass per unit length of the string
• V is Transverse Wave Velocity

Fret Positioning Equation

The space between frets declines from one to the next by a fixed ratio such that the 12th fret arrives at a position half of the length of the scale length.

Note that the fret position is measured from the fret to the end of the scale.

L_n = L_s/2^n/12

where:

• Ls is the scale length of the instrument
• Ln is the length from the fret number to the end of the scale

With n = 1 in the Fret Positioning Equation, the computed value of 2^1/1200 equals 0.94387 (rounding off many decimal positions) indicating that the fret 1 position (distance to end of the scale) is the scale length times 0.94387.

Similarly, the position of fret 2 is the length of fret 1 to the end of scale times 0.94387.

Likewise, the position of any fret is the length of the previous fret to the end of scale times 0.94387.

The following equation represents the above examples, showing the easiest way to calculate a fret position from the known previous position.

L_n = L_n-1 (0.94387)

Cents Position Equation

Just as the spacing between frets decline by a fixed ratio, from one to the next, so does the physical length of each cent decline by a fixed ratio over the entire length of the fret board.

Musical pitch intervals of a half-step (or a quaver) correspond to the space between each fret on the instrument fret board. The spacing between each fret is divided into 100 cents. Also, for example a difference between fret 2 and fret 5 would be 300 cents, and the difference in an octave would be 1200 cents.

Note that the position is measured to the end of the scale, not to the nut.

L_n = L_s / (2^(n/1200))

where:

• n is the cents number
• Ln is the length to the end of the scale
• Ls is the scale length of the instrument

Nut Compensation length Equation

We can use this equation to calculate the length of compensation needed, or the correction, +or-, of a previous adjustment.

To calculate the length of needed compensation, we can alter the previous equation slightly, in order to measure from the nut (cents = 0) to the number of cents-off that we measured with a strobe tuner.  We do that by subtracting the cents-off position (to the end of the scale) from the scale length.

L_c = L_s-(L_s/(2^(n/1200)))

where:

• Lc is the nut compensation length needed
• Ls is the scale length of the instrument
• n is the number of cents off

Bridge Compensation length Equation

Bridge saddle compensation logically is done early in the setup process, following neck relief, and it may also be used for later adjustments.

While this calculation is not needed for electric guitars (saddles are quick and easy to adjust) it is very helpful in preparation for making a new saddle for an acoustic guitar, after measuring the pitch errors with the old saddle.

We can use this equation to calculate the length of compensation needed, or for +or- correction of a previous adjustment.

Positive compensation (fret 12 plays sharp) requires moving the saddle release point away from the nut, which lengthens the vibrating string length of fret 12. However, this will also change the vibrating length of the open string. The vibrating length of the open string is twice as long as that of the fret 12. Any adjustment will have only half the pitch effect on the open pitch as it does on the 12th fret pitch.  Because of that, we need to adjust, at the saddle, twice the amount needed at the 12th fret.

Let’s first consider the mathematics of standard saddle compensation – that is, we will ignore the need for nut compensation.  We would attempt to get the fret 12 pitch in tune with the open string.  To do so, we would measure the cents error at fret 12 and compute the saddle adjustment using the following equation.

(Note: Fret 12 is at 1200 cents pitch above open, and at half the scale length from the nut to the saddle.)

L_c = (L_s/(2^{((1200 +n)/1200)}) – L_s/2) * 2

where:

• Lc is the 12th fret compensation length needed
• Ls is the scale length of the instrument
• n is the number of cents off at fret 12

The problem with this method is that, without nut compensation, the open string is a very poor reference. So, how do we calculate the saddle compensation length when we compare fret 14 to fret 2 pitch, where fret 14 is not at one half the scale length?

Fortunately, we can us a well-known characteristic of the equal temperament scale fret board: If there is sharpness in pitch-error for an octave, then the ascending pitch error increases evenly at each fret within the octave. In fact, the same incremental increase in cents occurs over the entire fret board (neglecting the slight variations in action height).

Therefore, we can measure the error, comparing fret 14 to fret 2 pitch, but compute the saddle compensation as if we had compared Fret 12 to open pitch, and use the formula above, using the fret 14 pitch error as fret 12 error.

In the same manor, we could compare fret 16 to fret 4 pitch, for better accuracy higher up the scale.

Furthermore, for Classical and other 12-fret instruments, we could measure error comparing fret 12 to fret 2 pitch. This is not an octave – it’s only 10 frets.  Problem solved by multiplying the cents-off by 12/10 (= 1.2), and using the result as the fret 12 cents off in the equation.

It is interesting that the per cents-off compensation length for the saddle comes out to be the same as that for the nut.  (For the saddle, we double the calculation, but the calculation is based on cents length at fret 12, which is exactly half the length of cents at the nut.)

Compensation Lengths for Cents-off for Popular Guitar Scale Lengths

Nut compensation is increased by extending the release edge of the nut toward the saddle, shortening the vibrating length of the open string (while not disturbing the tension and pitch of fretted notes).  Add a bit of compensation length to file away in final adjustments with a strobe tuner

Saddle compensation length is increased by extending the saddle release point away from the nut.    Subtract a bit of compensation length to file off in final adjustment with a strobe tuner.  For a 10-fret test span, remember to increase either the cents-off or the result by 1.2.

For any scale length not listed, the last table entry of Scale Length = 1 can be used. Just multiply the compensation values by your scale length.

Most of us would round these numbers to 2 or 3 significant digits.

Math Links

Online Vibrating String Links

Online Scientific Calculators

There are many full-featured online calculators, where you can enter an entire formula containing higher math functions.

Similar calculator apps are, of course, available for your cellphone or tablet.

• Calculator – Search Google or Bing for: ‘calculator’

  This really nice calculator will appear, fully functional, at the top of your search results page.  It works with Google search, and Bing search brings up one that is almost identical.

  Although you can use any of the displayed function ‘keys’, you can alternatively use standardized keyboard selections : +, -, x, *, /, =, ., (, ), %, Backspace, ^ (superscript or power), r (square root), s (sin), c (cos), etc.

  You can even use your browser’s address bar to enter complex formulas, and the calculator will appear with the result.

  With Google search only, select ‘More info’ link at the lower right corner of the calculator for information on significant extra features.

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