INDEX

College of Santa Fe Auditory Theory

Lecture 016 Instrument III

INSTRUCTOR CHARLES FEILDING

  1. Brass Instruments
  2. Percussion Instruments
  3. Sound source in percussion instruments
  4. Sound modifiers in percussion instruments
  5. Brain Bullets

4.3.7 Brass Instruments 189

The brass instrument family has an interesting history from early instruments derived from natural tube structures such as the horns of animals, seashells and plant stems, through a variety of wooden and metal instruments to today's metal brass orchestral family (e.g. Fletcher and Rossing, 1999; Campbell and Greated, 1998). 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 (pressure anti node and displacement node-see Figure 4.18). The double reed action of the lips can be illustrated if the lips are held slightly apart, and air is blown between them. For slow airflow rates, nothing is heard, but as the airflow is increased, acoustic noise is heard as the airflow becomes turbulent. If the flow is increased further, the lips will vibrate together as a double reed. This vibration is sustained by the physical vibrational properties of the lips themselves, and an effect known as the 'Bernoulli effect'. As air flows past a constriction, in this case the lips, its velocity increases. The Bernoulli effect is based on the fact that at all points, the sum of the energy of motion, or 'kinetic' energy, plus the pressure energy, or 'potential' energy, must be constant at all points along the tube. Figure 4.25 illustrates this effect in a tube with a flexible constriction. Airflow direction is represented

Fig 4.25 An illustration of the Bernoulli effect (potential energy + kinetic energy = a constant) in a tube with a constriction. Note lines with arrows represent airflow directionand the distance between them is proportional to airflow velocity. PE = potential energy KE = kinetic energy.

by the lines with arrows, and the velocity of airflow is represented by the distance between these lines. Since airflow increases as it flows through the constriction, the kinetic energy increases. In order to satisfy the Bernoulli principle that the total energy remains constant, the potential energy or the pressure at the point of constriction must therefore reduce. This means that the force on the tube walls is lower at the point of constriction.

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. Like two displaced pendulums, the walls move past their equilibrium position, stop and return towards each other and the Bernoulli effect accelerates them together again. The oscillation of the walls will be sustained by the airflow, and the vibration will be regular if the two walls at the point of constriction have similar masses and tensions, such as the lips.

The lip reed vibration is supported by the resonator of the brass instrument formed by a length of tubing attached to a mouthpiece. Some mechanism is provided to enable the player to vary the length of the tube, originally, for example, in the horn family by adding different lengths of tubing or 'crooks' by hand. Nowadays this is accomplished by means of a sliding section as in the trombone or by adding extra lengths of tubing by means of valves. The tube profile in the region of the trombone slide or tuneable valve mechanism has to be cylindrical in order for slides to function.

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. Frequencies higher than a cut-off frequency determined by the diameter of the outer edge of the bell (approximately 1500 Hz for a trumpet) are not reflected appreciably by the bell. Adding a bell to the main bore of the instrument has the effect of making the effective pipe length longer with increasing frequency. 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 effects of the addition of the bell and mouthpiece/mouth pipe on the individual lowest six modes are broadly as summarised in Figure 4.27. Here the odd-numbered modal frequencies of the stopped cylindrical pipe are denoted as integer multiples of frequency 'f, and the resulting brass instrument modal frequencies are shown as multiples of another frequency 'F'.

Fig 4.27 Brass instrument mode frequency modification to stopped cylindrical pipe by the addition of mouthpiece/mouthpipe and bell


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. The addition of three valves to brass instruments (except the trombone), each of which adds a different length of tubing when it is depressed, enables six semitones to be played, sufficient to progress from the first to the second mode. Assuming this is from the written notes C4 to G4, the six required semitones are: C#4, D4, D#4, E4, F4, and F#4. Figure 4.28 shows how this is achieved. The centre (or second) valve lowers the pitch by one

Fig 4.28 The basic valve combinations used on brass instruments to enable seven semitones to be fingered. Note Black circle = valve depressed, white circle = valve not depressed. On a trumpet the first valve is nearest the mouthpiecethe second in the middle and the third is nearest the bell.


semi tone, the first valve (nearest the mouthpiece) by two semitones, and the third valve by three semitones. Combinations of these valves therefore in principle enable the required six semitones to be played. It may at first sight seem odd that there are two valve fingerings for a lowering of three semitones (third valve alone or first and second valves together) as shown in the Figure. This relates to a significant problem in relation to the use of valves for this purpose which is described below.

Assuming equal-tempered tuning for the purposes of this section, it was shown in Chapter 3 that the frequency ratio for one semi tone (1/12 of one octave) is:

r = 12th root(2) = 1.0595

The decrease in frequency required to lower a note by one semi tone is therefore 5.95%, and this is also the factor by which a pipe should be lengthened by the second valve on a brass instrument. Depressing the first valve only should lower the !C, and hence lengthen the pipe by 12.25% since the frequency ratio for two semitones is the square of that for one semitone (1.05952 = 1.1225). Depressing the first and second valve together will lengthen the pipe by 18.2% (12.25% + 5.95%), which is not sufficient for three semitones since this requires the pipe to be lengthened by 18.9% 0.0595'1 = 1.1893). The player must lip notes using this valve combination down in pitch. The third valve is also set nominally to lower the fo by three semitones, but because of the requirement to add a larger length the further down that is progressed, it is set to operate with the first valve to produce an accurate lowering of five semi tones. Five semitones is equivalent to 33.51% 0.05955 = 1.3351), and subtracting the lowering produced by the first valve gives the extra pipe length required from the third valve as 21.26% (33.51 %-12.25%), which is rather more than both the 18.2% available from the combination of the first and second valves as well as the 18.9% required for an accurate three-semitone lowering. In practice on a trumpet, for example, the third valve is often fitted with a

Fig 4.29 Waveforms and Spectra for C3 played on a trombone and a tuba.

tuning slide so that the player can alter the added pipe length while playing. No such issues arise for the trombonist who can alter the slide position accurately to ensure the appropriate additional pipe lengths are added tor accurate tuning of the intervals. Figure 4.29 shows waveforms and spectra for the note C3 played on a trombone and a tuba. The harmonics in the spectrum of the trombone extend far higher in frequency than those of the tuba. This effect can be seen by comparing the shape of their waveforms where the trombone has many more oscillations during each cycle than the tuba. In these examples, the first three harmonics dominate the spectrum of the tuba in terms of amplihide and eight harmonics can be readily seen, whereas the fifth harmonic dominates the spectrum of the trombone, and harmonics up to about the 29th can be identified.

4.4 Percussion instruments 194

The percussion family is an important class of instruments which can also be described acoustically in terms of the 'black box' model. Humans have always struck objects, whether to draw attention or to imbue others and themselves with rhythm. Rhythm is basic to all forms of music in all cultures and members of the percussion family are often used to support it. Further reading in this area can be found in Benade, 1976; Rossing, 1990; Hall, 1991; and Fletcher and Rossing, 1998.

4.4.1 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. Therefore 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 as illustrated in Figure 4.21. The narrower the frequency band of the mode, the longer it will 'ring' for. (One useful analogy is the impulse provided if a parent pushes a child on a swing just once. The child will swing back and forth at the natural frequency of the swing and child, and the amplitude of the swinging will gradually diminish. A graph of swing position against time would be similar to the time response for the hard reed plotted in Figure 4.21.)

4.4.2 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. The following equations (adapted from Fletcher and Rossing, 1999) relate the frequencies of higher modes to that of the first mode:

For transverse modes in a bar resting on supports (e.g. glockenspiel, xylophone):

fn = 0.11030 (2n)+ 1)^2 f1

where

For transverse modes in a bar clamped at one end (e.g. celeste):

f2 = 0.70144 (2.988)^2 f1

fn = 0.70144 (2n + 1)^2 f1

where

The frequencies of the transverse modes in a bar are inversely proportional to the square of the length of the bar:

f transverse is proprtional to (1/L^2)

whereas those of the longitudinal modes are inversely proportional to the length

f longitudinal is proportiional to (1/L)

Therefore 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 (see Equation 4.13) whose factor (2.988) is very close to (3). Table 4.1 shows the frequencies of the first five modes relative to the frequency of the first mode as a ratio and in equal-tempered

Fig 4.1 Frequency rations and semitone spacings of the first five theoretical modes relative to the first mode for a bar clamped at one end and a bar resting on supports


semitones (Appendix 2 gives a frequency ratio to semitone conversion equation) for a bar resting on supports (Equation 4.12) and one clamped at one end (Equation 4.13). None of the higher modes are a whole number of equal-tempered semitones above the fundamental and none form an interval available within a musical scale. The intervals between the modes are very wide compared to hannonic spacing as they are essentially related by the square of odd integers (i.e. 32, 52, 72, 92, ... ). The relative excitation strength of each mode is in part governed by the point at which the bar is hit.

Benade (1976) notes that often the measured frequencies of the vibrating modes of instruments which use bars differs somewhat from the theoretical modes (in Table 4.1) due to the effect of 'mounting hole(s) drilled in the actual bar and the grinding away of the underside of the center of the bar which is done for tuning purposes'.

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 (see Equation 1.21).

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.

The frequencies of the modes can be calculated mathematically, but the result is rather more complicated than for the bars. Table 4.2 gives the frequencies of each mode relative to the first mode (Fletcher and Rossing, 1999) and the equivalent number of semitones (calculated using the equation given in Appendix 2). As with the bars, none of the modes are an exact number of equal tempered semitones apart or in an integer ratio and therefore they are not harmonically related. These frequencies are for an 'ideal' membrane since they will change when the membrane is mounted on a drum body. In the case of the tympani or kettle drum, Rossing (1989) notes that the air loading of the air enclosed in the drum body is 'mainly responsible for establishing the harmonic relationship of kettledrum modes'.

Fig 4.30 THe first five modes of an ideal drum membrane (upper) and a cymbal plate (lower). Notes: 1) the edge of the cymbal platge is shown dotted as it is unclamped (displacement antinode) 2) the mode numbers arew given in brackets as (number of diametric modes, number of circular modes)

Table 4.2 Modes and frequency ratios as well as semitone spacings of the first five theoretical modes relative to the first mode for an ideal circular membrane and plate.

Modes for circular membranes

Circle 0,1

Circle (1,1) Mode

The (2,1) Mode


The (0,2) Mode

The (1,2) Mode

The (0,3) Mode

 

 

 


Modes for rectangular soundboards

Mode 1,1

 

Mode 1,2

Mode 2.1

Mode 2.2

 

 

You should know

 

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.

 

4.4 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 as illustrated in Figure 4.21. 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 (see Equation 4.13) 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.