Bass Guitar and Structural Engineering
I thought I'd make a post about some things that I've found interesting over the last couple of years, combining my love for bass guitar and my area of study (structural engineering), which some of you may find interesting. Some of it may actually be really obvious, but hopefully it's still a little interesting to a few of you
One of the first things I noticed when I started playing bass was that the scale length of all the strings was generally different, not exactly 34 inches. And almost all of the basses that I'd played or seen others play had a similar thing in common, that the E string had a longer scale length than the A string, and so on, with the G string having the shortest scale length. Of course, this only differed by a small fraction of an inch but it seemed to be pretty consistent among all basses.
My explanation for this is actually found in a lot of structural engineering applications. The 'effective length' of the string is actually shorter than the scale length because of the small amount of flexural stiffness of the string near its ends, where it touches the bridge/nut. For very thin strings like the G string, this flexural stiffness is pretty negligible (it's very easy to bend a G string into a tight coil) but for thicker strings, the flexural stiffness becomes more noticeable (imagine bending an E or B string into the same radius coil as a G string). The fact that the ends of the string are restrained by the bridge/nut to some extent means that the 'effective length' of the string is shortened, between two points of contraflexure which occur somewhere close to (but not coincident with) the nut and the bridge. And of course it's the effective length of the string which determines its frequency (along with linear mass, etc). The same principle applies to columns in buildings: where a column is rigidly connected on either end to slabs, for example, the points of contraflexure of the columns occur somewhat closer to the centre of the column, and it's buckling behaviour depends on its effective length, rather than its actual length. The greater the degree of flexural stiffness and rigidity at the connection, the smaller the effective length.
In the image you can see that for the case of a relatively flexible string, the effective length would be close to 1.0 times its actual length, but somewhat less than that due to its end stiffness. If the string were exceptionally stiff and rigidly connected to the nut/bridge, you would have a case like the first. In reality it's somewhere in between.
Another thing I've found interesting is the way tones are formed by the superposition of different frequencies when we pluck a string. Any sort of structure has an infinite number of modes in which it can vibrate, and bass strings are no exception. We are all familiar with the first mode of vibration - this is where the string vibrates up and down along its entire length, much like looking at a skipping rope from the side. The first mode of vibration produces the fundamental frequency, essentially the note that you intended to play. However, it's practically impossible to pluck a string and not induce other modes of vibration, especially since we pluck the strings so close to their ends. We can also hear the second mode of vibration, where the string oscillates with a node at its middle, producing a frequency which is 2x that of the fundamental frequency, and of course this is the next octave up. The reason the second mode of vibration is not as audible is because modes with lower frequencies are generally easier to excite. Higher modes corresponding to different harmonic frequencies are also audible, but to a lesser extent.
The resultant tone that you hear is the contribution of infinitely many modes of vibration, each with a slightly lesser intensity as the frequency increases. And of course, the tone also depends on the position of the pick-ups which are different for different basses, as well as the position of your plucking fingers. This all has a lot of application in structural engineering as well; a building will vibrate according to its natural frequencies, much like a string. The resultant vibration in a building is a superposition of contributions of its infinitely many modes of vibration, however, much like a string, the lower-frequency vibrations are easier to excite and are the most noticeable.
The case of a vibrating string is similar to the simply supported beam on the right of the image, although the beam relies on flexural stiffness as opposed to the string's axial stiffness, however the same modes of vibration occur.
I'd be happy to elaborate on anything, or answer any sort of questions that relate to both bass and structural engineering :)
I understood about 25% of that but still found it very interesting.
Hi Batmilk. I am an engineer and bass lover as well!
Your post is very interesting!
I think that you are right. The flexural stifness may play a role in the required string length. Have you tried taper wound strings? For this type of string the point of contact between string and sadle has the same gauge for all strings, and I would expect less difference in string length (note that the nut still have different string gauges).
Another interesting point is that most string manufacturers sell strings by it's gauge, and the string pack is not designed to provide even tension between strings. This is why we may find some basses with a loose B string (poor bridge design may act in favor of a loose B string as well - e.g.: dampening effect). String gauge shall be a consequence of the required linear mass, for a particular string length, to provide even tension between strings.
Another aspect related to string tension is the definition of the scale (position of the frets). The positions are designed considering the theoretical lengths to represent the harmonics. However, whenever you press the string against the frets, you are aplying an additional force, which increase the string tension and thus slightly alter the note frequency. The result is that all fretted instruments have intonation issues. In my opinion, this is an unsolvable problem - some solutions, like the buzz feiten system, may improove intonation, but is not a perfect solution.
Thats it. Maybe some day I become a luthier, hehehhe.
I think that this is really very interesting. Some time ago I experimented a lot with B-strings, to gain experience on how their tone and response changed when played in the upper registers (i.e. at a shorter effective lenght). The flexural stiffness seems to be an important factor here, too, also because it influences fundamental and overtones in different ways (?). I was pointed at the properties of tapered strings - but still, the core wire of a B-string is bigger in diameter, therefore a relatively larger effect of increased flexural stiffness seems to be present, compared to lighter strings. And since I didn't have much use for the range below D, and since I did not like the behaviour of the B-string in the upper register, I went back to 4 strings.
Could you elaborate on the way the flexural stiffness influences different harmonics / overtones? I guess that would help us to understand the behaviour of heavier strings when played in upper registers a bit better.
holy crap, could anyone translate that to english?
That is why you measure scale length from the nut to the 12th fret and multiply by 2:D
Yes, since you can never get a fretted instrument to be exactly in tune through the length of the neck, intonation involves slightly changing the string length based on the specific design and gauge of the string (not necessarily purely driven by the overall thickness in general), to 'average out' the inherent flaws in neck design and make the fretted instrument the 'most in tune' you can. Depending on the instrument and string, often it is actually the D string that will need to have the saddle moved up the most (i.e., the length shortened) to obtain the best 'average intonation' up and down the neck.
The Buzz Feiten system also varies the string length at the nut to try to again, get a closer 'average in tune' across the entire length of the neck.
Even putting on strings from a different manufacturer of exactly the same gauge can result in needing to readjust the intonation (as does tapered strings, etc.), due to different tuning tension driven, I believe, but the mass of the string.
Complicated stuff. It is why guys with REALLY good ears find fretless instruments to be much more in tune than fretted ones, when played correctly.
Indeed. Piano manufacturers have been dealing with these issues for centuries. It's good to see some actual analysis as to why it happens.
It is English, just a more accurate version. Interesting to see the engineers viewpoint of the more common musical 'science' version. I'd not really made the link to stiffness and rigidity that engineers consider. I suspect musical 'technologists' consider strings as elastic in structure, not rigid.
I'd also not considered that if the initial deviation from rest state is quite big, then as the string vibrates, it's also stretching, and this presumably detunes it, which means ..... what? harder plucking changes the note?
I'm an electrical engineer. Trust me, mechanical engineering jargon is a lot closer to ordinary English than musical theory jargon!
Demonstrating modes is always a cool fun thing to see.
And of course there is the always super famous...
I think it's probably an unintended (but probably inevitable) consequence of needing such a thick string, both the core wire and the outer wire. It probably comes back to the flexural stiffness of the string. For a G string, it's essentially only stiff along its axis (you'd have a hard time trying to stretch it any noticeable length using your hands, but you can easily bend it into a tight coil). For a thick E or B string, the flexural stiffness is much more significant, and I think that when we are fretting higher up the neck, this flexural stiffness begins to influence the way the string vibrates, to a greater extent than for lower frets. For lower frets, since the vibrating length of the string is quite long, the flexural stiffness of the string doesn't contribute much to the string's displacement (the deflection of a flexural element is proportional to its length squared); the string relies on its axial stiffness. For very short vibrating lengths (e.g. if we played the 14th fret), suddenly the flexural stiffness of the string begins to contribute quite a bit towards the deflection. (When I say deflection I just mean the deformed shape of the string as it vibrates).
I think that this probably causes a lot of odd harmonic overtones to come through, since the string is acting less like a idealised string (purely axial stiffness), and somewhat more like a flexural element (combination of axial and flexural stiffness).
I'd love to hear some other opinions on this though, as I said this is just my best explanation for it, but I'm sure there's better ones out there
^As for tapered strings, I think that the issue of end stiffness would be alleviated somewhat, but now you have a string which doesn't have a uniform distribution of mass and which has different end stiffnesses (the string is still quite thick at the nut), and I would imagine this would cause similar strange harmonic overtones.
I think that the problem of murky sounding B strings is probably unavoidable at higher fret positions
I find the B string is better left alone past the 5th fret.......reverting at that point to the E string.
But that's just me and I'm a old-school thumper.
The tonal characteristics of B-strings is often explained as being caused by the qualities of the instrument itself, stating that certain basses provide even B-string sound even in high registers.
To me, while respecting these opinions, they were always hard to understand, because the differences regarding the mechanical behaviour of the witness points seem to be marginal, compared to the mechanical properties of the string itself - or, the differences between properties of e.g. E- and B-strings.
It should be pretty easy to construct an experimental setup, using a stiff beam with two string witness points whose distance can be varied. Some analysis should be possible regarding the mechanical behavior of the string when plucked.
Too bad I left academia and don't have access to a lab any more... - I am a biologist, was working in biomechanics during my former life.
in for later
I love enginerding. This answers a question I posted some time back about the Boomy on the E string, much of my suspected culprits being exactly as you explained.
I can say that I notice less of the Boomy on the same instrument using rounds or TIs (lower tension) and more using XLs (hi tension) as one would expect.
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