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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 length 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.)

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 simplifies equation with only length as a variable, to determine the nut compensation length.

When you finish compensating the nut (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, 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 position is measured to the end of the scale.

Ln = Ls/2n/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 21/1200 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.

Ln = Ln-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 fretboard.

Musical pitch intervals of a half-step (or a quaver) correspond to the space between each fret on the instrument fretboard. 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.

Ln = Ls / (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 above 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 cent position (to the end of the scale form the scale length

Lc = Ls-(Ls/(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
 

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 maintaining the pitch of fretted notes).

Scale Length 10¢
24.75"  (628.65mm) 0.014203" 0.14255"
25.5"  (647.7mm) 0.014725" 0.14687"
650mm  ((25.59") 0.37535mm 3.7437mm"
660mm  (25.98") 0.38112 3.8013mm
1" or 1mm 0.000577456 0.00575858

For an unlisted scale length, the last table entry of 1 can be used. Just multiply the compensation values by your scale length.

The length of correction gradually diminishes for each cent of error, but it diminishes too slightly to be significant in the first 10 cents of error.   I rounded the lengths to 5 significant decimal digits to illustrate this.

Most of us would round off 2 or 3 more decimal digits, making the 10-cent value = to the 1-cent value times 10.   Therefore, the 1-cent value times the number of cents off would be pretty accurate.  Add a bit of length to file off in final adjustment with a strobe tuner.

 

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 aps 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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