by Ian Noyce
Previously published in Sonics Oct-Dec 1982

THE VARIOUS relationships between string length, string mass per unit length (usually given in kg per metre) and resultant pitch (or 'frequency' as our technical brethren call it) are fundamental - sorry! - to understanding why various string length is taken as the length between the nut and the bridge.

PITCH depends on string length and string tension.

1. If the string's mass per unit length remains constant, the longer the string, the higher the tension required to achieve the desired pitch.
2. If the string's length remains constant, the higher the string's mass per unit length, i.e. the heavier the string, the higher the tension required to achieve the desired pitch.

The equation which ties all this together is:

where

f = fundamental frequency
(first mode of vibration see fig 1)
L = string length between fixed
points (i.e. bridge and nut)
T = string tension
P = string mass per unit length

Figure 1. first four vibration modes of a string fastened at both ends.

This formula may look pretty formidable: just take it for granted (it's actually derived from the fact that the note produced by a string is proportional to the speed that sound travels in the string, and that speed in turn depends on how tense and how heavy the string is.)

By cutting up a new set of Dean Markley regular gauge electric strings, weighing a measured length (640mm) on an electronic balance to get the "mass per unit length", and doing the number juggling on my calculator till I nearly wore out the battery, I came up with the results in fig 2.

The formula actually produces answers in units of force called "Newtons", and since you might be happier thinking about "kilograms", I simply divided the formula by 9.81, because there happens to be 9.81 Newtons in a kilogram of force. So the results for T in Fig 2 are actually obtained by plugging the appropriate figures into the formula:

Figure 2. Relationship between pitch, mass and tension for Dean Markley strings of 640mm length. 640mm is in fact a midway between a Les Paul and a Strat scale length and is the scale length we use on most Noyce Guitars.

STRING GAUGE FREQUENCY STRING MASS PER UNIT LENGTH TENSION     f P T     cycles/sec (Hz) kg/m kg E or 1st .010/.254 329.63 0.401 x 7.28 B or 2nd .013/.330 246.94 0.708 x 7.22 G or 3rd .017/.432 196.00 1.140 x 7.32 D or 4th .026/.660 146.82 2.333 x 8.41   (.014" core)       A or 5th .036/.914 110.00 4.466 x 9.03   (.015" core)       E or 6th .046/1.168 82.407 6.790 x 7.71   (.016" core)             TOTAL 46.96

COMMENTS

As may be seen from the measurements set out in Fig 2. the tension required to achieve the correct pitch is reasonably similar for all strings (or at least it will be in a balanced set). This is of course necessary to avoid unequal stress being inflicted on the body of the instrument. It's also the reason why all your strings end up feeling much the same to play.

• Wound strings are no exception in this relationship, because in these, the tension is applied primarily to the core, with almost none at all to the winding.
• We should at this point also introduce the concept of "stress". When we exert a force across a section of something such as a guitar string, stress is produced, in that section. When the forces pull on that string, the result is called tension (as distinct from when they push, which creates compression.). So if we pull the string, we increase the tension, thus increasing the stress, and it the stress becomes excessive the string breaks. It will break earlier if it is a thinner string (i.e. having less cross-sectional area).
• Since a heavier string would require greater tension to achieve a particular pitch, this increased tension would make the string "stiffer". Now, since a stiffer string enhances the upper modes of vibration (harmonics), with less movement in the lower, if you were after a "tighter", brighter sound, you might use a light top, heavy bottom strings.
• Acoustic guitars with medium strings carry almost double the tension of electrics with .010"-.046" strings: 80-90kg (180-200 lbs) depending on the the scale and gauge of the strings.
• Putting .009"-.046" strings on a Gibson Les Paul, as the Les Paul's shorter scale reduces tension about 6% compared to Fender's longer scale, just as going down a step in string gauge gives a proportional decrease in tension.

Editor's note. Just thought we'd let you know that we checked out these relationships practically in a rather novel (and rather gross) way. Hanging a B-string from a nail over our managing editor's door, we clamped it at the free end approximately 640 mm from the nail and hung one of our our mail bags off the clamp: Filling the mailbags with (wait for it!) Sonics 1982 Yearbooks to a total weight of approximately 8kg, we proceeded to pluck the string and lo and behold a sound not too far from a B was heard.

Ain't that groovy! Try it yourself (if you've got enough Sonics Yearbooks!).

HOW DOES IT FEEL?

Every guitar has its own feel and amount of 'give' under the hands when bending strings, etc. Although many aspects of this 'feel' relate to properties of the instrument and the hardware on it, it is also greatly influenced by what goes on between the bridge and tailpiece.

Example 1: If you look at the length of a string on a Strat between high E machine and the nut, and that between the saddle and the back of the body, it represents 30% of the effective string length (i.e. from nut to bridge saddle). The same observation on a Les Paul represents 15% of scale length. This is a very significant factor in the string-bending feel of each guitar. To simplify this, let's look at a more exaggerated example.
Fig 3a and 3b show two strings identical in everything except overall length, but mounted such that the length between the nut and the bridge is the same for each. Since for tuning purposes this is the critical length, both strings can be seen as identical as far as what is needed to tune them to the same pitch is concerned - and will thus require equal tension. However, the string in Fig 3a will be much easier to deflect (bend) than the one in Fig 3b, as the increase in tension that happens when you deflect the string is to some extent distributed over the entire length of the string.

Example 2: The string break, or down bearing at the nut and bridge saddles also affects ease of fretting and bending.
A Strat without string trees feels looser and sounds "looser" - with less attack - than one with string trees pulling the string down sharply past the nut, just as a Les Paul with the tailpiece set high feels looser than the same guitar with the tailpiece screwed right down.
It's all a matter of how firmly the string is pulled over the nut and saddle and how much length of string there is past these points. (The Strat and Les Paul have been used here as convenient examples. There are many variations applicable to other guitars)

Figure 3. Effect on "Bending Feel" of the distance between machine and nut, and bridge and tailpiece. 3a is easier to deflect than 3b.

For Experimental guitarists

To get a better feel for the forces involved in string movement, get a friend and a towrope or similar and hold one end firmly whilst the friend jiggles the rope to stimulate the modes of vibration shown in Fig 1. You'll soon get an impression of how violent the forces at the end of the string are, and how they change according to the mode you're simulating. (Try holding the rope against your chest and singing a note whilst performing the above experiment.)

Strength of Strings

By clamping a string at each end in a machine called and extensiometer and subjecting it to increasing tension until it broke, while measuring how much is stretched, I was able to plot a graph for a Dean Markley .026" wound string (see Fig 4).
The main point of interest here is that guitar strings are bloody strong!! Also it has become quite clear that guitar strings break during play due to mechanical damage (i.e. wear at the nut, a particularly vicious stroke of the pick or whatever, rather than due to over-tensioning).

Figure 4. Tension vs extension of a guitar string. A: Normal tension in regular gauge electric strings; B: Normal tension in medium gauge acoustic strings; C: Tension in regular gauge electric string when held by interval of a fourth (five fret bend); D: Limit of elasticity (yield point); E: Breaking point (ultimate Strength).

Summing up and Settling in
Although all of the above facts and figures are basically true, there are numerous deviations from theoretical behaviour in practical situations, and numerous assumptions involved in doing practical tests. For example, when a string is stretched, it must decrease in diameter but it is assumed that the diameter remains constant.

Temperature and moisture also have their parts to play. One significant deviation from the ideal elastic behavior is the settling in process a string goes through when new. Even when a string is pulled up to normal tension and pitch and fixed firmly at each end point, it will stretch and detune a little when pulled, until, after a number of pulls, it settles in and remains fairly faithful in tune despite further plucking and pulling. This phenomenon is known as 'work hardening' and is experienced by every guitarist 'stretching in' a new string or set of strings.

Also, as the bridge and nut are not frictionless carriages for the string, tension can get 'stored up' in the lengths of string between nut and machine and bridge and tailpiece and, with playing, this stored tension in the vibrating length and thus the pitch. That's another reason why 'playing in' new strings is important.
Well, now you'll be thoroughly clued up on what your Dean Markleys are doing when you produce your next Hendrix-inspired wail, and you'll understand why your guitar feels a little odd when someone strings it with the top E where the bottom E should be. Ain't physics wonderful!