SUBurban Bassic Physics

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Bassic Physics for String Instruments

For a string instrument, it is rather obvious, that the fundamental object is the string, or strings. They have to be there, and they have to vibrate in a desired way, to provide any sound.

To make the string "be there", we need something that keeps it in place. In a piano or a harp, it is a big, strong and heavy frame. In portable instrument, we usually call it a neck, but usually the neck is not going from one string end to the other. The neck vary often holds the string in one end, and is connected to a "body", in which the other end of the string is attached. It is therefore more accurate to use the word "backbone", to describe the function "keep the strings in place" on a portable string instrument – such as a bass or a guitar, or a violin.

Then, we know that a string vibrates rather quietly, so we need some sound reinforcement. On the portable instrument, we introduce a body. This can be a big hollow box as on a double bass, or a tiny solid block as on a Steinberg electric guitar. What it does is resonate with some of the string frequencies, thus reinforcing them to audible volumes. And, due to the size, shape and material, it will reinforce some frequencies more or less, which is what we call tone shaping.

In the last century, tone reinforcement and shaping has been added to, with the use of electronic pickups and amplification. This is not included in this little ramble.

The ideal approach

The ideal string has a certain length, L [m], a certain weight per unit length, r [kg/m], and is made from a monocrystal with a minimal diameter of 0.000…01 [m]. It is attached between two anchorages and under a certain tension, T [N]. The anchorages are such, that it allows rotation around all axii, but prevents any translation in any direction.

PICT 1 An ideal(!) string (green)

The string is not affected by gravity, magnetic fields, air resistance or any other disturbance.

By adapting infinitesimal calculus, we finally end up with the string equation:

f = (1/2L)*(T/r)^½ [Hz]

How close can we get?

We all know that ideal is far from real. Knowing that, we search for a compromise, that is sufficiently satisfactory.

The closest we have got this far is the grand piano, with its rigid frame and constant string length and tension. However, the character of the sound is not always wanted, due to the hammer exciting. The finger excited variant, the harp, is also as stationary as the piano, while we like to carry the instrument with us. We end up searching again. And we find the guitar, viola, mandolin, banjo and similar instrument families.

String

We can come pretty close to the ideal string, by making a thin metal or nylon wire. But when we want bass tones, i.e. high weight (r), it gets too stiff. By using a tiny centre core, and winding more wires around the core, we distribute more weight without adding as much stiffness. We increase r more than the stiffness and the tension, thus lowering the frequency.

Tampering with different materials and gauges will give us more variants of T/r. And of course, we increase the string length to the limit of the human hand.

The price will be deviations from the (physically) ideal tone.

Anchorage

For a portable string instrument, anchoring the string is a crucial dilemma:

How can I produce the distinct transitional lock, and yet allow rotation?
How can I produce the precise and rigid positioning of these points?

The answer to the latter is a matter of backbone, i.e. the physical connection between head and bottom string anchorages. This is the neck, and – in case of glued in or bolted on bodies – the body joint and the centre of the body. Violas and acoustic guitars (et al) count in the neck and the body rim and back, sometimes also the body top.

The importance of this backbone is so great, that I refer to the neck as the base feature of the instrument, and the body as secondary addition. Hence, it’s "body joint" and bolt-on bodies, not neck joint and bolt-on necks.

The backbone should obviously be as rigid as ever possible, to come close to the ideal. It should also be as lightweight as possible, to be portable and playable.

A compromise is necessary, but what can be compromised? Well, as long as we keep the resonance frequencies of the backbone well above the frequencies that the strings (and pickups) will give, we will be in business. Providing that the backbone isn’t bent by the total string tension! OK, again: how close can we get?

Both prerequisites are functions of stiffness, which in turn is a function of material and design. It is usually formulated as EI, where E is the material Modulus of Elasticity (Young's modulus), and I is the bending stiffness of the cross-section.

For a rectangular cross-section, I=w*h³/12. NOTE that the height of the section, h, is to the power of 3!

PICT 2 A rectangular crossection, just to define w and h.

We have the stiffness EI=E* w*h³/12, for a rectangular cross-section.

Then we worry about mass, or density (r). The heavier material, the lower resonance frequencies we will get.
The resonance frequency is proportional to r/EI

Now, some E and r for popular materials:

	E [GPa]	r [kg/dm³]
Alder	10.0	0.50	Alnus Incana (swedish 'grey alder')
Maple	11.2	0.73	Acer Nigrum (black/hard/rock maple)
Oak	13.0	0.69	Quercus Robur (swedish oak)
Ebony	15.6	1.25	Dalbergia Melanoxylon (african black ebony)
Birch	16.3	0.65	Betula Pubecens (swedish 'glass birch')
Aluminum	70.0	2.70
Steel	210.0	7.80
Graphite reinforced epoxy	330.0	1.95
D Species latin names added due to response from Peter Puleo, who pointed out that other variants of alder and maple have different values. This proves again, that the properties of the individual plank is of more importance than the general properties of the species.

This table indicates, that a high graphite backbone would be optimal, but we have to make it hollow and decrease the height to make it playable. It will, however, be rather too expensive… Which is why we usually end up with a wooden backbone, often reinforced with a couple of graphite stringers, and a truss rod to counteract the string tension.

Experiments have also shown, that the bending stiffness of wood is higher when loaded parallel to the annual rings (a.k.a. grain). As much as 9% higher for spruce!!! Which corresponds with the h³ above. But, just to fuzz things up a bit: experiments show, that a horizontal laminate of equal boards are stiffer than a vertical! This is due to the "sandwich effect", that also makes it possible to make a three-piece laminate of a weak centre and two stiff outer boards stiffer, than a single piece of the stiff material, all other properties kept the same.

OK, what do we finally do, to make a suitable backbone?

We take a material with lowest plausible r/E, make a horizontal laminate of three pieces, reinforce with a material with even lower quotient, and make it as thick as possible while still playable. To help control the string tension, we use a truss rod.

PICT 3 Three-laminate neck with truss rod and dual reinforcement bars, and fingerboard

At times, people discuss the impact of bridge and tuners on tone. They have an impact, because they contain the connection between the backbone and the string, i.e. the string anchorage. What is important here is rigidity. The joint must be rigid, meaning rigid tuner pegs, distinct gears and everything well attached to the backbone. And low weight, because high weight will have an effect on the resonance frequencies, especially heavy tuners in the head.

Tone shaping

Having spent the last minutes trying to explain what we do to not have any resonance, it is time to change the subject. But only slightly. Towards how to control the desired resonance! A.k.a. "tone".

This is a task for the body, with some minor addition from the fingerboard, when it comes to attack and note change response. But the body is the primary factor holder, namely the following factors:

Neck-body joint: the less stiff joint, the less sustain and more "thump". The sustain of the strings fundamental frequency is most affected.
Surface hardness and porosity: harder surface makes for harder attack and brighter tone. However, size and shape of the surface pores adds effects that are very unpredictable!
Materials: stiffness, weight and hardness have the same effect as on necks.
Shape, i.e. mass and stiffness distribution: as on necks.
Type: solid, hollow, acoustic

This is such a huge area, I’d rather let it be! Just two guidelines:

Bigger, lighter and softer makes for lower tone emphasis.
Smaller, heavier and stiffer makes for brighter tone emphasis.

Blend according to taste…. And do remember:
When you use wood, the properties of the individual piece has more impact on the sound than the wood species!

Finally

Good luck with designing – and building – your own instrument!

Oh, and by the way, if you find any mistakes, please report them to SUBurban™!

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