SESEIBLE EXPERIENCE AND PHYSICAL PARAMETERS

by Simone Bianchi, Tangerinetech Engineering http://www.tangerinetech.net

1. Introduction

An old joke tells about a sod which wanted to gamble on horses, who asked his physicist friend about a sure method to win. The physicist thought a bit about the question and said:
 

'Let's suppose horses are spherical and move with straight and uniform motion...'

Now I'll be using Paolo's courtesy once again to try and resume the limits of my explainations and thoughts written in my last article on this blog.
 
When we stimulate a material object (please knock on that wooden table), it will respond to the vibrations we deliver with a characteristical sound. It's obvious that if you knock on the wall (go light, pleeease) it will respond with a definitely different noise.
 
Technically speaking the representation of a great deal of this behaviour is defined frequency response. The older among us certainly remember those graphical equalizers with their cursors which dominated stereo rigs, the positions of the (usually 10) cursors represented the attenuation or the boost applied by the device on the various frequency ranges so to modify the sound of the stereo setup.

Every object has a similar response. It attenuates or boosts a part of the frequencies contained into the vibrations which reach it, and gives them back to the environment as its characteristical noise. A table has a different behaviour than a wall.
 

Unless your table is made of brick (or your wall is wooden).
 
That's it. Material counts, a great deal of the sound behaviour of an object is due to the material composing it. Brick, wood or else.

The rest is due to the physical dimensions of the object in question. Take a wooden plank and knock on it. Then saw it in two and knock on one of its halves. You'll have another sound.

So let's reexamine one of my phrases from last week:
 

"The frequency response of a musical instrument is never linear. That is to say stimuli impressed by the musician's bow and fingers won't be amplified by the instrument in the same way"

So if we impress to an instrument a stimulus (pick the strings of a bass, play a violin with the bow, crush a Fender on stage) it will respond with a sound which is the consequence of:

a) the stimulus impressed

b) the instrument's characteristics.

(By the way DO NOT DAMAGE innocent guitars for the pleasure of theatre.)

2. Simplified Model of the Bow system.

When we speak about musical instruments, bowed instruments are an integrated system. The bow is an integral part of the instrument. A violin played with the bow is different from a violin played with fingers or a violin played with a file. The fact is evident in itself.
 
The bow itself has a frequency response (see 'Il Signor Bruschino' by G. Rossini)...

"...among the “particularities” which made rumors when it was first performed there's the bold experimentation in the research for new effects which can be found in the symphony, in which the composer prescribes that the second violins hit the lecterns with their bows."

http://www.festivalditorrechiara.it/bruschino.htm

Since this blog speaks about bowmaking, we focused on bow vibrations and its 'resonant frequency'.

Resonant frequency is a very important parameter in elastic systems, which tend to boost a particular frequency range in comparison to all others. When this frequency range changes, the elastic system changes its sound characteristics.
 
This behaviour is present in instruments' bows, since their construction qualifies them as proper elastic systems.

Bow mass and its elasticity define the vibrations which will be privileged by the bow. Elaborating on the base parameters of the bow seen as an elastic system, last week we could isolate two kinds of significant vibrations. They do not cover all the possible vibrating behaviours of the bow, however they can be very representative.
 

1) Vibrations which propagate in the stick, characterized by a resonant pulsation depending on the geometry and wood sound velocity.

w = v(sound) * sqr (A/LV)

Let's rewrite isolating the frequency f1, bound to pulsation w and collecting all the other factors (mainly geometric ones) into the constant C1:

f1 = v(sound) * C1

2) Transverse vibrations of the stick when kept steady at an extremity and moved at the other.

They also depend on the sound velocity and on the system geometry; let's rewrite the expression found last week, here also collecting various factors (mainly geometric) into the constant C2.

f2 = v(sound) * C2

Both vibration kinds depend on material geometry and on sound velocity in the material itself, and this is good news, because we can characterize the vibrating behaviour of a wood with a non-destructive measurement. Not bad.
 

3. Conclusions

We also have our spherical horses...models we used are simple and reductive, suffice to say we didn't take into consideration all the complex movement bow and instrument undergo during a performance, or the vibrations of the bow hair-string interface!

Wood properties measurement provides very useful informations about quality of base materials, but don't believe a hi-quality wood log with hair attached can be at all compared with a real instrument bow.

It's always like this: a mathematical model can be very rigorous, but when a project exits physics and enters engineering, horses get back to horses and leave horseshoe traces, not a continuous rolling strip.
 

Reality is complex and seems to show a perverse will to escape models and simplifications, equal temperament is not all the music can be and the elasticity of a double bass is not the same of a spring.
 
Our ears have a dynamic range greater than 100 decibels and a bandpass of ten octaves. More important they've a brain in between which is able to make 'measurements' of dynamics and bandpass with a synthesis ability difficult to imagine. To satisfy their needs a good mathematical model is not enough, we need art, technique, skill, in a word we need a great deal of talent.

Greetings to all readers who patiently followed me up to this point and thanks to Paolo who allowed me to speak to a very special audience.

Simone Bianchi