# Physics of strings

Discussion in 'Strings [BG]' started by Geej, May 30, 2005.

1. ### Geej

Aug 3, 2004
Brisbane, Australia
OK, this is for a university assignment....

I want to know how the thickness, tension and length of a string relates to the pitch... and for that matter what makes the pitch change...

EG... Why is the E-string the largest and the lowest sounding? Does the extra thickness allow for a lower tension and therefore lower pitch?
Similarly, how does the thickness (or lack of) of the G-String make it the highest?

Why does fretting a string higher up (decrease in distance) result in a higher pitch? What change in distance relates to what pitch change? (numerical values, ratios etc)

Are there any formulas that this relates too?

What frequency relates to what value of string thickness/ tension?

I'm looking for answers with values, and specific words... not subjective words like "tone" and "feel"

And a second question, relating to tuning strings using harmonics... i.e. <7> on the D-string sounds the same as <5> on the E-string

The beats in the sound increase as the strings are tuned further apart, specific values? What speed of beats relates to what frequency/phase difference?

This is for an assesment at uni, my group has to perform some experiments, and I chose sound waves as one of them... the whole asking you guys questions is part of the background research... and any help is greatly appreciated... But again, not to be rude or anything, but I only want relevant answers, involving a scientific approach to the problem. Not a "play it by ear" approach.

Thanks alot to who ever answers

2. ### CJK84

Jan 22, 2004
Maria Stein, OH
3. ### 7flat5

Nov 28, 2003
Upstate NY
Geej, this .pdf contains the tension/mass/pitch math.

Note that "thickness" means nothing, but unit mass is what you are after. All your other questions, except the harmonics question, can be answered by this. I don't have a good link for harmonics, but any good first-year physics text I have seen addresses that, too. Basically, if you play a "harmonic" at the 12th fret, you are dividing the string in half, and the note resulting is one octave up from the open string. Play a 5th fret harmonic and you divide the string in 1/4, or 2 octaves up. At the 7th fret, 1/3...etc.

Hope this helps.

4. ### Jazzin'...Bluesin' and Funkin'

Yeah, thickness does not affect the pitch, it is only the mass and the length as well as tension.

5. ### i_got_a_mohawk

the greater the mass and the lower the tension result in a lower frequency, i think

6. ### bill spangler

Mar 4, 2001
Albany GA
This is going to involve a little math.

1st The speed of a wave on an ideal string is

speed = sqr root (tension/linear density).

The linear density isnt too hard, its just the mass of the vibrating portion of the string divided by the mass of the vibrating portion. The tension is how much force the string is stretching the string. (Note, the speed found here is NOT the speed of sound.)

Since youre an Aussie, the tension should be in Newtons, the mass in kgs and the length of the string in meters. The wave speed will be in meters/second then.

The wave speed found from the above formula is also

speed = wavelength x frequency

The wavelength of the fundamental frequency will be TWICE the length of the portion of the string that is vibrating. When you pluck an open A string on most electric basses, the frequency will be 55 Hz and the wavelength 68 inches (about 1.73 meters) so the wave speed on the A string will be about 95 m/s. When you shorten the string by fretting it, the wave speed will still be 95 m/s but the frequency (& ergo the pitch) will be higher because the wavelength will be smaller.

Now back to the first formula: The wave speed for A string is 95 m/s. There are only two parameters to adjust, the tension and the linear density. The only ways to adjust mass part of the linear density is 1) to use different alloys for each string OR 2) as the string manufacturers do, make the lower pitched strings thicker. The lower pitched strings will support waves of lower wavespeed (the E string on a bass will be about 71 m/s). Look at the first formula; notice that as the linear density goes up (e.g. the strings get thicker) the wavespeed goes down. When the wavespeed goes down, the frequency also goes down (pitch is lower), because the wavelength of the open E string is still about 1.73 meters. To make the strings of the bass feel somewhat consistent, the tensions of all the strings are kept pretty close to each other which means that the lower pitched strings tend to be thicker. I hope some of this theory helps. The DAddario tables mentioned previously use the theory that I have attempted to expound here.

7. ### Geej

Aug 3, 2004
Brisbane, Australia
Yea, thanks alot guys, that exactly the kind of stuff i was looking for