# Physics Of Scale Length Effect on Tone/Timbre

#### tedtan

##### SS.org Regular
Anyone have any links that describe the effect of scale length on tone/timbre (preferably scientific in nature)?

##### SS.org Regular
Anyone have any links that describe the effect of scale length on tone/timbre (preferably scientific in nature)?

Not exactly, but regarding timbre, this is a pretty interesting one I learned back in college.

A Fourier series breaks down a periodic signal into an infinite sum of sines and cosines with different amplitude coefficients, and this is what gives different instruments their timbre. So for example an F on a piano and an F on a clarinet will break down into a very similar infinite sum of sines and cosines with different amplitude and wavelength coefficients.

• tedtan

#### tedtan

##### SS.org Regular
Thanks!

I figure @bostjan will probably be able to come up with something physics-y.

#### ixlramp

##### SS.org Regular
the effect of scale length on tone/timbre
I know this is probably what you mean, but to be precise ... what actually matters is the 'vibrating length' of the string, determined by whether and where it is fretted.

The consideration here is actually the effect of string stiffness on tone. (By 'stiffness' i do not mean 'tension', these are 2 different things, i am using the scientific definitions.)
The vibrating length and the string gauge/construction together determine the effective stiffness of the string. The stiffness then affects the tone.

If you keep the string the same gauge/construction and reduce the vibrating length, it effectively becomes stiffer, because its width relative to its length increases, becoming more like a short, fat metal rod.
Increased stiffness results in a darker tone, less sustain and more inharmonicity (harmonics out of tune with the fundamental).

So alternatively, you could search for information on the affect of string stiffness on tone.

• bostjan, odibrom and tedtan

#### bostjan

##### MicroMetal
Contributor
I know this is probably what you mean, but to be precise ... what actually matters is the 'vibrating length' of the string, determined by whether and where it is fretted.

The consideration here is actually the effect of string stiffness on tone. (By 'stiffness' i do not mean 'tension', these are 2 different things, i am using the scientific definitions.)
The vibrating length and the string gauge/construction together determine the effective stiffness of the string. The stiffness then affects the tone.

If you keep the string the same gauge/construction and reduce the vibrating length, it effectively becomes stiffer, because its width relative to its length increases, becoming more like a short, fat metal rod.
Increased stiffness results in a darker tone, less sustain and more inharmonicity (harmonics out of tune with the fundamental).

So alternatively, you could search for information on the affect of string stiffness on tone.
Very well said.

The equation for inharmonicity due to stiffness:

fn = n f0 ( 1 + ( n² π² d^4 Y ) / ( 128 L² F ) )

fn is the frequency of harmonic with overtone number n, so f0 is the fundamental, f1 is the first harmonic, etc. d is the diameter of the string (in SI units), so d^4 is the fourth power of that. Y is Young's modulus of elasticity, which depends on the construction and material of the string (actually, it basically is constant if one calculates carefully based off of the core and considers the length of the string rather than just the scale length...). L is the scale length, so L² is the square of that, and F is the force of tension.

The closer the harmonics fit the harmonic series, the brighter the tone will sound, as the various modes of vibration can more efficiently excite each other. The more inharmonicity, the duller the tone will be as lower frequency harmonics will have less ability to excite higher frequency modes.

There's quite a bit more to it than that. The material of the string matters a lot, as less elastic material will tend to sound duller for other reasons as well. Yielding support of the strings can result in a duller tone, as higher frequency vibrations will bleed off mechanically from exciting that motion. Non-uniform strings will sound duller, as will strings with contaminants stuck to them, as the thicker portions of the string contribute damping to the vibrations.

@tedtan Are you interested in the inharmonicity formulae for those as well?

• tedtan

#### tedtan

##### SS.org Regular
Thanks bostjan and ixlramp.

At this point, I’m more interested in the general overview, so I think you guys have covered it.