Auditory Theory: Acoustics

Lecture 016 Instruments III

Reading Assignment for Lecture 017

Before next lecture please read Sections

4.5 The speaking and singing voice 198

pages 198 to 208 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.

Brain Bullets

Brass Instruments

The sound source in all brass instruments is the vibrating lips of the player in the mouthpiece. They form a double soft reed, but the player has the possibility of adjusting the physical properties of the double reed by lip tension and shape. The lips act as a pressure-controlled valve in the manner described in relation to the woodwind reed sound source, and therefore the mouthpiece end of the instrument acts acoustically as a stopped end.
If the wall material at the point of constriction is elastic and the force exerted by the Bernoulli effect is sufficient to move their mass (such as the brass players lips) from its rest (equilibrium) position, then the walls are sucked together a little (compare the right- and left-hand illustrations in the figure). Now the kinetic energy (airflow velocity) becomes greater because the constriction is narrower, thus the potential energy (pressure) must reduce some more to compensate (compare the graphs in the Figure), and the walls of the tube are sucked together with greater force. Therefore the walls are accelerated together as the constriction narrows until they smack together, cutting off the airflow. The air pressure in the tube tends to push the constriction apart, as does the natural tendency of the walls to return to their equilibrium position.
All brass instruments consist of four sections (see Figure 4.26): mouthpiece, a tapered mouthpipe, a main pipe fitted with slide or valves which is cylindrical (e.g. trumpet, French horn, trombone) or conical (e.g. cornet, fluegelhorn, baritone horn, tuba), and a flared bell (Benade, 1976; Hall, 1991). If a brass instrument consisted only of a conical main pipe, all modes would be supported (see discussion on woodwind reed instruments above),
but if it were cylindrical, it acts as a stopped pipe due to the pressure-controlled action of the lip reed and therefore only odd numbered modes would be supported (see Figure 4.18). However, instruments in the brass family support almost all modes which are essentially harmonically related due to the acoustic action of the addition of the mouthpiece and bell.
The bell modifies as a function of frequency the manner in which the open end of the pipe acts as a reflector of sound waves arriving there from within the pipe. A detailed discussion is provided by Benade (1976) from which a summary is given here. Lower frequency components are reflected back into the instrument from the narrower part of the bell whilst higher frequency components are reflected from the wider regions of the bell.
The frequency relationship between the modes of the stopped cylindrical pipe (odd-numbered modes only: If, 3f, Sf, 7f, etc.) will therefore be altered such that they are brought closer together in frequency. This effect is greater for the first few modes of the series.
The addition of a mouthpiece at the other end of the main bore also affects the frequency of some of the modes. The mouthpiece consists of a cup-shaped cavity which communicates via a small aperture with a short conical pipe. The mouthpiece has a resonant frequency associated with it, which is generally in the region of 850 Hz for a trumpet, which is otherwise known as the popping frequency since it can be heard by slapping its lip contact end on the flattened palm of one hand (Benade, 1976). The addition of a mouthpiece effectively extends the overall pipe length by an increasing amount. Benade notes that this effect 'is a steady increase nearly to the top of the instruments playing range', and that a mouthpiece with a 'lower popping frequencey will show a greater total change in effective length as one goes up in frequency' (Benade, 1976, p.416). This pipe length extension caused by adding a mouthpiece therefore has a greater downwards frequency shifting effect on the higher compared with the lower modes.
In a complete brass instrument, it is possible through the use of an appropriately shaped bell, mouthpiece and mouthpipe to construct an instrument whose modes are frequency shifted from the odd only modes of a stopped cylindrical pipe to being very close to a complete harmonic series. In practice, the result is a harmonic series where all modes are within a few per cent of being integer multiples of a common lower-frequency value except for the first mode itself, which is well below that lower frequency value common to the higher modes, and therefore it is not harmonically related to them.
The second mode is therefore the lowest musically usable mode available in a brass instrument (note that the lowest mode does not correspond with 1F). Overblowing from the second mode to the third mode results in a pitch jump of a perfect fifth, or seven semitones.

Percussion instruments

Sound source in percussion instruments

The sound source in percussion instruments usually involves some kind of striking. This is most often by means of a stick, but not, for example, in a cymbal crash. Such a sound source is known as an 'impulse'. The spectrum of a single impulse is continuous since it is non-periodic (Le. it never repeats), and all frequency components are present.
Any instrument which is struck is excited by an acoustic sound source of short duration in which all frequencies are present. All modes that the instrument can support will be excited, and each will respond in the same way that the plucked reed vibrates. The narrower the frequency band of the mode, the longer it will 'ring' for.

Sound modifiers in percussion instruments

Percussion instruments are characterised acoustically by the modes of vibration that they are able to support, and the position of the strike point with respect to the node and antinode points of each mode (e.g. see the discussion on plucked and struck strings earlier in this chapter). Percussion instruments can be considered in three classes: those that make use of bars (e.g. xylophone, glockenspiel, celeste, triangle); membranes (e.g. drums) or plates (e.g. cymbals).
In each case, the natural mode frequencies are not harmonically related, with the exception of longitudinal modes excited in a bar which is stimulated by stroking with a cloth or glove coated with rosin whose mode frequencies are given by Equation 1.20 if the bar is free to move (unfixed) at both ends, and 1.21 is it is supported at one end and free at the other. Transverse modes are excited in bars that are struck, as for example when playing a xylophone or triangle, and these are not harmonically related.
Halving the length of a bar will raise its transverse mode frequencies by a factor of four, or two octaves, whereas the longitudinal modes will be raised by a factor of two, or one octave. The transverse mode frequencies vary as the square of the mode number, apart from the second mode of the clamped bar whose factor (2.988) is very close to (3)
In order that notes can be played which have a clearly perceived pitch on percussion instruments such as the xylophone, marimba, and vibraphone, the bars are shaped with an arch on their undersides to tune the modes to be close to harmonics of the first mode. In the marimba and vibraphone the second mode is tuned to two octaves above the first mode, and in the xylophone it is tuned to a twelfth above the first mode. These instruments have resonators, which consist of a tube closed at one end, mounted under each bar. The first mode of these resonators is tuned to the f0, of the bar to enhance its loudness, and therefore the length of the resonator is a quarter of the wavelength of f0
In percussion instruments which make use of membranes and plates, the modal patterns which can be adopted by the membrane or plate themselves govern the frequencies of the modes that are supported. The membrane in a drum and the plate of a cymbal are circular, and the first five mode patterns which they can adopt in terms of where displacement nodes and antinodes can occur are shown in Figure 4.30. Displacement nodes occur in circles and/or diametrically across and these are shown in the figure. They are identified by the numbers given in brackets as follows: (number of diametric modes, number of circular modes). The drum membrane always has at least one circular mode where there is a displacement node, which is the clamped edge.