Design of Custom String Gauges

Why Custom?

Was there ever a time in your life where your creative energy was only exceeded by your spare time, like experimenting with alternate stringed instrument tunings? I remember trying to figure out the beautiful Restless Heart song, "Long Lost Friend", and using that tuning. I also have tuned my electric guitar like a dobro to play with a slide. In each case, I found it hard to tune higher or lower than 2 or 3 semitones. The string would either break or be so loose that it had poor tonal quality.

While my curiosity of string gauges matured, I have yet to find any books or documentation on this topic. I took the physics law of simple harmonic motion and compared the results with the tension data provided from string manufacturers. The results were very predictable. I have gathered statistical data for many different sets of standard string gauges and have found that this mathematical basis supports of all variations.

This article describes the physics of a vibrating string and methods of calculating a string diameter for a specific pitch. An Excel spreadsheet is provided that includes a string gauge calculator, k-factors, and some statistics of common string sets.

Vibration of a String

The fundamental frequency, fr (in Hertz) of a vibrating string is a function of three fundamental factors.

Where:

• l = length (meters)
• T = tension (newtons)
• m = mass (kg-sec2/m)

The length is easy. This is the distance from the nut to the bridge. A Fender Telecaster has a nominal scale length of 25.5 inches. Some pedal steel guitars have a scale length of 24 inches.

As tension increases, the vibrating frequency increases. This holds true up until the point where the tensile strength of the string is exceeded and the string breaks.

The mass is the volume of the string times the mass density of the string material. The density of most steel and nickel used to manufacture strings is 7800 kg/m³.

k Factor

The mass/tension relation described above is straightforward with plain strings. When wound strings are considered, their effective volume is not simply the cross sectional area times the length. The string is made with an inner core and a small winding of nickel or bronze around it. The ratio between the effective mass volume of a wound string and its outer diameter solid equivalent may be calculated by the following expression:

This uses the pi/4 relationship between the volume of a unit square and a unit circle. Realize however that the density of the winding (like phospor bronze or nickel) is different than the steel inner core. The difference is hardly worth using a different density. The densities are fairly close to steel and the winding is not usually the significant contributor to the mass of the composite vibrating matter.

The typical core to outer diameter relationship for a standard electric guitar set is shown in the table below.

Design of Custom String Gauges

In order to design a string gauge you must know the following:

• Scale Length (l)
• Note Frequency (f)
• Tension (T)

As mentioned earlier, the scale length must be measured. If the instrument has a staggered bridge, use the average length. The following table shows the pitch frequencies for an equal tempered scale based on the geometric progression of the twelfth root of 2. These are the pitch frequencies that a chromatic guitar tuner would tune your strings with. A 24-fret guitar ranges from E3 to D7.

Musical Note Frequencies (Hz)

Equal Tempered Scale

The tension varies with instrument and playing style. A normal electric guitar string set tension is usually about 13 to 15 lbs. This allows ease of string bending. Where acoustic guitar tension is set to have minimum fret rattle, a tension of 20 to 25 lbs. may be used. You can determine what is best for your instrument by looking up the tension for a standard set. The D'Addiaro company is one of the few or only string manufacturers that list the tension of the string on the product packaging. It is usually documented in pounds. 1 Newton equals 0.225 lb.

Using the equation for the frequency of a vibrating string above and solving for diameter, substituting the mass density of steel results in:

By substituting the constants of the density of steel and pi, and performing the conversion of units to pounds and inches, the equation may be further simplified to solve for the string diameter d (inches):

Where:

• f = frequency (Hz)
• T = tension (pounds)
• l = length (inches)

When calculating a wound string, divide the result by the k factor. If you don't know the k factor for the desired size, use 0.9 as an approximation. In general, you will find that string selection to the closest one thousandth of an inch is more coarse then most of the variation in the constants.

Custom String Gauge Design With a Scientific Calculator

Let's design the string gauge of the first string on a Fender Telecaster guitar. The scale Length is 25.5 inches. Find the frequency of the high E string on a guitar. As mentioned earlier, the low string on a guitar is the E3 note at 82.40689Hz; the high string on the guitar is two octaves higher (E5) or 329.6276Hz.

The tension for a standard electric set as mentioned earlier is about 13 pounds. To calculate the string diameter in inches, use the scientific calculator key sequence below:

 25.5

 8405.5038

  1.189696684213e-4

 0.002481707283268

  3.605551275464

  0.008947947291579

The result rounded to the nearest one-thousandth is a "nine" or 0.009 inches diameter. If your diameter result is above a 0.022 string, you would probably convert this to a wound string. You should divide the diameter by 0.9 to get the wound string diameter. The example below is for an A string on the same setup as above. We will use 15 pounds for the tension as D'Addario did using the A note frequency.

 25.5

 2805

   3.565062388592e-4

  0.007436720142602

   3.872983346207

   0.0288022932627

    0.03200254806967

Notice that without the k-factor, the string would be a 0.029" diameter instead of the desired 0.032".

Custom String Gauge Calculator

Included is an Excel v5.0 workbook that contains a string gauge calculator for steel strings, a musical note frequency table, k-factors, and more. Bear in mind that this spreadsheet uses a constant of 7800 kg/m³ steel density. This calculator will not work with nylon or gut strings. Refer to the equations above using the appropriate density of the desired string material.

Download an Excel Custom String Gauge Calculator

©1996 Terry Downs Music™