Auditory Theory: Acoustics

Lecture 014 Instruments I

Reading Assignment for Lecture 015

Before next lecture please read Sections

4.3 Wind instruments 166

pages 166 to 189 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.

Brain Bullets

Black Box Model

a sound source, and
sound modifiers.
If we are modelling the effect of an instrument being played in a room, then the output we require is the sound heard by the listener and not the output from the instrument itself. The environment itself acts as a sound modifier and therefore it too acts as a 'system' in terms of the input/ system/ output model.
An environment which has no acoustic effect is one where there are no reflections of sound-ideally this is known as 'free space'. In practice, free space is achieved in a laboratory in an anechoic ('no echo') room in which all sound reaching the walls, floor and ceiling is totally absorbed by large wedges of sound-absorbing material.

Stringed instruments

All stringed instruments consist of one or more strings stretched between two points, and the f₀ produced by the string is dependent on its mass per unit length, length and tension.
For any practical musical instrument, the mass per unit length of an individual string is constant and changes are made to the tension and/ or the length to enable different notes to be played.
A vibrating string on its own is extremely quiet because little energy is imparted to the surrounding air due to the small size of a string with respect to the air particle movement it can initiate. All practical stringed instruments have a body which is set in motion by the vibrations of the string(s) of the instrument, giving a large area from which vibration can be imparted to the surrounding air. The body of the instrument is the sound modifier.
There are three main methods by which energy is provided to a stringed instrument. The strings are either 'plucked', 'bowed' or 'struck'. Instruments which are usually plucked or bowed include those in the violin family, instruments which are generally plucked only include the guitar, lute, and harpsichord, and the piano is an instrument whose strings are struck

Sound source from a plucked string

When a string is plucked it is pulled a small distance away from its rest position and released. The nature of the sound source it provides to the body of the instrument depends in part on the position on the string at which is plucked.
if the string is plucked at the centre, as indicated by the central dashed vertical line in Figure 4.6, modes which have a node at the centre of the string (the 2nd, 4th, 6th, 8th, 10th, etc., or the even modes) are not excited, and those with an antinode at the centre (the 1st, 3rd, 5th, 7th, 9th, etc., or the odd modes) are maximally excited.
If the string is plucked at a quarter of its length from either end (as indicated by the other dashed vertical lines in the Figure), modes with a node at the plucking point (the 4th, 8th etc.) are not excited and other modes are excited to a greater or lesser degree.
In general, the modes that are not excited for a plucking point a distance (d) from the closest end of a string fixed at both ends, are those with a node at the plucking position.

Sound source from a struck string

Piano strings are very hard and they are under enormous tension compared with the string on plucked instruments. When a piano string is stuck, it behaves partly like a bar because it is not completely flexible since it has some stiffness. This results in a slight raising in frequency of all the component modes with the effect being greater for the higher modes, resulting in the modes no longer being exact integer multiples of the fundamental mode.
The effect would therefore be considerably greater for bass strings if they were simply made thicker to give them greater mass, and in many stringed instruments, including pianos, guitars and violins, the bass strings are wrapped with wire to increase their mass without increasing their stiffness.
The notes of a piano are usually tuned to equal temperament (see Chapter 3) and octaves are then tuned by minimising the beats between pairs of notes an octave apart. When tuning two notes an octave apart, the components which give rise to the strongest sensation of beats are the first harmonic of the upper note and the second harmonic of the lower note
Inharmonicity on a piano increases as the strings become shorter and therefore the octave stretching effect increases with note pitch. The stretching effect is usually related to middle C and it becomes greater the further away the note of interest is in pitch.

Sound source from a bowed string

The sound source that results from bowing a string is periodic and a continuous note can be produced while the bow travels in one direction.
A bow supports many strands of hair, traditionally horsehair. Hair tends to grip in one direction but not in the other. This can be demonstrated with your own hair. Support the end of one hair firmly with one hand, and then grip the hair in the middle with the thumb and index finger of the other hand and slide that hand up and down the hair. You should feel the way the hair grips in one direction but slides easily in the other.
The hairs of a bow are laid out such that approximately half grip when the bow is moved in one direction and half when it is moved in the other. Rosin is applied to the hairs of a bow to increase its gripping ability. As the bow is moved across a string in either direction, the string is gripped and moved away from its rest position until the string releases itself, moving past its rest position until the bow hairs grip it again to repeat the cycle.

Sound modifiers in stringed instruments

The sound source provided by a plucked or bowed string is coupled to the sound modifiers of the instrument via a bridge. The vibrational properties of all elements of the body of the instrument play a part in determining the sound modification that takes place. In the case of the violin family, the components which contribute most significantly are the top plate (the plate under the strings which the bridge stands on and which has the f holes in it), the back plate (the back of the instrument), and the air contained within the main body of the instrument.
Two acoustic resonances dominate the sound modification due to the body of instruments in the violin family at low frequencies: the resonance of the air contained within the body of the instrument or the 'air resonance', and the main resonance of the top plate or 'top resonance'. Hall (1991) summarises the important resonance features of a typical violin as follows:
- practically no response below the first resonance at approximately 273 Hz (air resonance)
- another prominent resonance at about 473 Hz (top resonance)
- rather uneven response up to about 900 Hz, with a signifcant dip around 60G-700 Hz
- better mode overlapping and more even response (with som exceptions) above 900 Hz
- gradual decrease in response toward high frequencies.
Sound from stringed instruments does not radiate out in all directions to an equal extent and this can make a considerable difference if, for example, one is critically listening to or making recordings of members of the family. The acoustic output from any stringed instrument will contain frequency components across a wide range, whether it is plucked, struck or bowed. In general, low frequencies are radiated in essentially all directions, with the pattern of radiation becoming more directionally focused as frequency increases from the mid to high range.
In the case of the violin, low frequencies in this context are those up to approximately 500 Hz, and high frequencies, which tend to radiate outwards from the top plate, are those above approximately 1000 Hz. The position of the listener's ear or a recording microphone is therefore an important factor in terms of the overall perceived sound of the instrument.