# String gauges and scales - The physics behind it



## Varjo (Nov 25, 2011)

So, since Ive been studying some physics, I thought Id take to pratice what Ive learned. And what a better way to apply my new found skills by applying it to what we love most,

Guitars.

So what Ive written here down is a practical-yet-theoretical real-life physics-based summary (DAYMN that sounded fine) how actual physics work behind our ringing strings  especially how string gauge, tension and guitar scale work together. Every now and then someone who hasnt yet had the experience comes wondering how does a longer scale affect playing or how do gauges differ. Sometimes more experienced members, right as they might be, give more or less mixed answers. Sometimes were just interested how things work. So thats why I wrote this marathon text.

A string (in a guitar) is physically known as a harmonic vibrator. This means that it vibrates (in our context, simplified: goes up and down) harmonically (in a predetermined, standardized manner, such as a stable tuning). When we say tune, we mean frequency, which has a unit of a Hertz  Hz. It tells us how many times the harmonic vibrator, or in our case, the string, moves up and down in a second. For example, the standard A moves 440 times up and down in a second, creating the note A at 440 Hz. When we say how strong we hit the string with our pick, we mean amplitude. This tells us how large is the movement that the string makes while it vibrates, which results in the heard volume or attack. 

FREQUENCY and AMPLITUDE do NOT affect each other. Doesnt matter how hard or soft you pick the open A string, itll always be an A.

Now that weve got those cleared, lets go deeper. How does one define the actual tune of a string?

The formula is rather simple or not:

f = n / 2L * Sqrt( T / y )

f = frequency (A, 440 Hz)
n = harmonic multiple (basically determined by the fret you might hold)
L = Wavelength of the fundamental harmonic (In our interest, the length of the string)
T = Tension
y = linear density of the string (basically qualities of the string, including gauge. Linear density is a unit of calculating the density of a string per length)

Which basically tells us that the frequency (f) we achieve, when we play a certain note (n), is affected by the scale length (L, as the vibration is limited to a known length aka. the scale) and the tension and the density of the string, where the tension and the density of the string have a combined effect.

Another formula to think about is

F = ma = kx

F = Force
m = mass
a = acceleration
k = spring standard
x = spring movement

Which tells us that the force (f) applied gives an object with an mass (m, the string here) an acceleration (a, the speed the string begins to vibrate, or the loudness of our picking), which in turn affects the string movement (x, how large is the movement of the string) which is affected by the string standard (which constitutes of the qualities of the string, gauge, winding type, metals used and such).

Or, if we want to simplify it a bit

f = T / Sm

f = frequency (for example open A)
Sm = String mass, determined by multiplying the string gauge with the lenght of the scale
T = Tension (tuning)

Please note  the formula above is NOT an actual formula of anything. Its just a simplified mind tool. The actual formula is higher above with all the letters, multipliers and square roots.

Also, if someone is unclear about it  the only length of string that matters tune-wise is the part between the nut and the bridge. Its the only part that resonates, so its all that matters. Having 4 or 40 turns around the tuning post shouldnt make a difference in sound.

Naturally you dont have to understand the real physics behind this, but youve read so far. The point is  theres a very certain, simple and clear physical law how strings work. The law in itself takes everything we can affect into account, making it somewhat useful as a tool of understanding. But, taking these formulas into effect, lets work out some dilemmas.

First: lets remind ourselves how formulas work. Personally I know that the shining majority of people here are smart and cool. What I dont know is how far is the mathemathical / physical education of said folk. But everyone knows how to multiply and divide, so the important part about formulas: both sides of the equation must have the same value. For example, our laymans formula of

f = T / Sm

means that if you increase, say, tension, the frequency will rise also. To balance out the equation. Or if you increase the string mass (heavier gauge strings) and want to keep the same tuning, youll have more tension.

Its the same thing really when youre closing down on intimate relations. Making love to a woman also follows that formula. How much she enjoys is totally dependant on the size of your manhood (compare to the string mass) and the feeling you put into it (the tension on a string). Ridiculously big junk but lack of feeling  yes, she moans. A guitar with a power cable as a string will require very little tension, but is that really what we want? Likewise, the moans of your concubine will not be a testament to your skill. Its all about the balance.

So, aspects on the qualities of strings we can more or less affect:

STRING GAUGE
String gauge affects directly the string mass. After all, when we increase the diameter of an object, it grows in size. Im sure that everyone can come up with an example of growth in size related to increase in diameter or length. Anyway, as the mass grows, so grows the force needed do make it move  or vibrate, as a string. As the force required grows, so does the required tension to achieve the required frequency we want.

In the laymans formula, increasing the string mass will lower your tuning, unless you tighten the tension.

So  Heavier strings give you tighter strings on lower tunings

For the advanced class, looking at the actual formula, you propably noticed the density. This means that ANY change in the density of the string will affect the frequency, if not counterbalanced by a change in, say, tension. And this indeed is the reason why some strings feel tighter than others even if theyre the same gauge. Elixir has a whole other way of making their strings, starting from the soil they acquire their minerals from, than DR. The changes arent big, but when were talking about 440 vibrations per second and amplitudes of a few millimeters, small changes are enough. The change might measure as negligible, but our fingers feel it. So yes, you are correct  those damn dAddarios really are sloppier than DRs. Not much, but enough to feel it.

How much that matters is up to you, of course.

SCALE LENGTH

Scale length, also, affects directly the string mass. True, the gauge will not change, but the length will  which increases the mass. Which, again, looking at either one of our formulas affect the outcome, or the frequency. Increasing the scale without decreasing string gauge will result in a tighter tension, perhaps too tight even.

So  longer scale gives you tighter strings on lower tunings.

TUNING

Obviously, tuning defines the tension we wound into our strings.

EFFECT ON PLAYING

Thinking how changes in scale, gauge or tuning affect the individual string, we can try to think of the big picture. Playing in A standard without any fret buzz or too much flabbiness will require some tension on the strings. If youre used to playing with a 9-42 set in E standard, to pull the same off with the same feel in A standard will require some changes. You could increase the scale. You could increase the string gauge. You could do both, having less effect on both.

Works other way around too. Really really really love a light touch? Try playing E standard with a 26,5 or even a 27 scale. Theoretically this would give you an even lighter touch with the same 9-42 set, while producing the same sound.

With our formulas we can see why its such a challenging thought to play A standard with a 24,75 scale guitar. Its gonna need some serious string gauges, high action or toleration of fret buzz. But, since In Flames pulls it off, sure enough it can be done.

EFFECT ON SOUND

Increasing the scale or string mass results in more string vibrating over the pickups. This, in turn, results in a clearer output, since the pickup has a clearer reading of the movements of the string in its magnetic field (theres an physical formula for this one, too). End result: clearer, crispier, more attack, more defined. Less warmth, less round sound. Harsher sound. Cleaner, less natural.

Decreasing the scale or string mass results in the opposite, giving more roundiness and warmth and life as some say. More of a nature, perhaps. Of course, the shorter/the lighter you go, the closer you are getting towards an undefineable chaotic noise (in extreme conditions, anyway, but you get the idea).

So which is better, hard to say.

SO WHAT SCALE/STRING

Really, its quite impossible to say. The two key questions are: How much tension do I like and How clear and articulate do I want my strings to be. String gauge and scale length wont make you a great player, they wont make your playing super tight and aggressive, but they will play their part. Part of Claptons warm sounds were his well-crafted strat, his personal playing technique, his single coil pickups, but also the tension on the strings and scale length. Part of Tosin Abasis clarity in his playing is due to his extended scale guitars and heavier strings, but I think we all can agree that hed shred the fuck out shit with a cigar box guitar tuned to G with a scale length of three inches. Knowing how the parts work together might help you have some idea. For example, if you enjoy playing in D standard and want to experiment in C standard, but dont want to lose that feel, try an increase in scale (for example, Ibanez 25,5 vs. Schechter 26.5 scale) or try heavier strings. Learn to use the string tension calculator: String Guage and Tension Calculator - Version 0.1.4 - 26 apr 1998

For example, for a 25,5 scale guitar, playing with a .046 in D would have approximately the same tension as a .052 in C. Or a .050 in C with a 26,5 scale. How about that.

So, to determine the best string gauge for you:
1)	Learn what you like or dislike in your current strings
2)	Find out the approximate tension via calculator above
3)	Change the tuning, change the string gauges  compare to ready sets by different manufacturers
4)	When an approximate match has been found, try it.

After that, learn your new strings. If youre not happy, continue experimenting. Try different brands. As said before, gauge isnt everything, the minerals in the steel wires affect also  they have a direct impact on the density.

OH AND ONE LAST THING

Most of you propably knew all this, but I just wanted to state some facts. Also to see if I've understood something wrong. Also, have fun playing.


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## ixlramp (Nov 25, 2011)

Varjo said:


> f = n / 2L * Sqrt( T / y )
> 
> f = frequency (A, 440 Hz)
> n = harmonic multiple (basically determined by the fret you might hold)
> ...



Please excuse me, just a little correction to this 

n is the harmonic number as in n=1 for fundamental / 1st harmonic, n=2 for 2nd harmonic / 1st overtone, n=3 for 3rd harmonic / 2nd overtone. With guitars we are usually talking about the frequency of the fundamental so n is usually set to 1.
L is the vibrating length of the string, from bridge to fret / nut. L=scale length for open notes only. The wavelength of the fundamental is actually 2L.

With n=1 the equation can be re-arranged to:
T = y * (2Lf)squared
Which is the tension equation found in the absolutely essential D'Addario tension guide: http://www.daddario.com/upload/tension_chart_13934.pdf Anyone interested in string tension or designing custom sets should study this.


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## skeels (Nov 25, 2011)

I love science!
But there are subtleties in the art of of playing that elude a "perfect" formula. For instance - if I have a big ol' low string and I absolutely bash it, the tension can increase due to the pull of the sheer vibration of the string (minus the friction of the nut, bridge saddles, air and so forth) to the degree where the actual note produced will be sharp and fall ever so slightly in pitch as the motion and subsequently tension subside. This produces a rather pleasant BWOOOOOOOOOAAAAAAOOOOOONNNNNGGGGG sound. 
Also, it has been observed that while string lengths beyond the nut and behind the bridge shouldn't affect the audible sounds reproduced by the pickups, this is not the case. Hans Reichel built guitars capitalizing on this phenomenon - give his stuff a listen. Is it a matter of sympathetic resonance or harmonic? 
Fripp used to say "One note is clear, two notes create overtones based on their intervallic relationship .. Three notes? Now you've got a mess." 
Okay, so im paraphrasing him, but ..
What am I saying? 
I love science!


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## skeels (Nov 25, 2011)

Whoa - double post! Phenomenon or fat fingers? You decide!


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