Music of all cultures is based on the use of instruments (including the human voice) which produce notes of different pitches. The particular set of pitches used by any culture may be unique but the psychoacoustic basis on which pitch is perceived is basic to all human listeners. This chapter explores the acoustics of musical notes which are perceived as having a pitch and the psychoacoustics of pitch perception. It then considers the acoustics and psychoacoustics of different tuning systems that have been used in Western music.
The representation of musical pitch can be confusing because a number of different notation systems are in use. In this book the system which uses A4 to represent the A above middle C has been adopted. The number changes between the notes B and C, and capital letters are always used for the names of the notes. Thus middle C is C4, the B immediately below it is B3, etc. The bottom note on an 88-note piano keyboard is therefore AO since it is the fourth A below middle C, and the top note on an 88 note piano keyboard is C8. This notation system is shown for reference against a keyboard in Figure 3.21.
Fig 3.1 Acoustic pressure waveform of A440 (A4) played on a violin trumpet flute and oboe
When we listen to a note played on a musical instrument and we perceive it as having a clear unambiguous musical pitch, this is because that instrument produces an acoustic pressure wave which repeats regularly. For example, consider the acoustic pressure waveforms recorded by means of a microphone and shown in Figure 3.1 for A4 played on four orchestral instruments: violin, trumpet, flute and oboe. Notice that in each case, the waveshape repeats regularly, or the waveform is 'periodic' (see Chapter 1). Each section that repeats is known as a 'cycle' and the time for which each cycle lasts is known as the 'fundamental period' or 'period' of the waveform. The number of cycles which occur in one second gives the fundamental frequency of the note in Hertz or Hz. The fundamental frequency is often notated as 'f0', pronounced 'F zero', a practice which will be used throughout the rest of this book. Thus the f0 any waveform can be found from its period as:
f0 in Hz = 1/(period in seconds)
and the period from a known f0 as:
(period in seconds) = 1/ f0 in Hz
Example 3.1 Find the period of the note G5, and the note an instrument is playing if its measured period is 5.41 ms.
Figure 3.21 gives the f0 of G5 as 784.0 Hz, therefore its period rom Equation 3.2 is:
Period of G5 in seconds = 1/784.0 = 1.276 x 10-3 or 1.276 ms
The f0 of a note whose measured period is 5.405 ms can be found using Equation 3.1 as:
f0 in Hz = 1/ 5.41 x 10-3 = 184.8 Hz
The note whose f0 is nearest to 184.8 Hz (from Figure 3.21) is F#3.
For the violin note shown in Figure 3.1, the f0 equivalent to any cycle can be found by measuring the period of that cycle from the waveform plot from which the f0 can be calculated. The period is measured from any point in one cycle to the point in the next (or last) cycle where it repeats, for example, a positive peak, a negative peak or a point where it crosses the zero amplitude line. The distance marked 'T violin' in the figure shows where the period could be measured between negative peaks, and this measurement was made in the laboratory to give the period as 2.28 ms. Using Equation 3.1:
f0 = 1/2.27ms = 1/2.27 x10-3 s = 440.5 Hz
This is close to 440 Hz, which is the tuning reference to for A4 (see Figure 3.21). Variation in tuning accuracy, intonation or, for example vibrato if the note were not played on an open string, will mean that the f0 measured for any particular individual cycle is not likely to be exactly equivalent to one of the reference to values in Figure 3.21. An average f0 measurement over a number of individual periods might be taken in practice.
Fig 3.2 Spectra of waveforms shown in 3.1
Figure 3.1 also shows the acoustic pressure waveforms produced by other instruments when A4 is played. Whilst the periods and therefore the f0 values of these notes is similar, their waveform shapes are very different. The perceived pitch of each of these notes will be A4 and the distinctive sound of each of these instruments is related to the differences in the detailed shape of their acoustic pressure waveforms, which is how listeners recognise the difference between, for example, a violin, a clarinet and an oboe. This is because it is acoustic pressure variations produced by a musical instrument that impinge on the listener's tympanic membrane, resulting in the pattern of vibration set up on the basilar membrane of that ear which is then analysed in terms of the frequency components of which they are comprised (see Chapter 2). If the pattern of vibration on the basilar membrane varies when comparing different sounds, for example, a violin and a clarinet, then the sounds are perceived as having a different 'timbre' (see Chapter 5) whether or not they have the same pitch.
Every instrument therefore has an underlying set of partials in its spectrum (see Chapter 1) from which we are able to recognise it from other instruments. These can be thought of as the frequency component 'recipe' underlying the particular sound of that instrument. Figure 3.1 shows the acoustic pressure waveform for different notes played on four orchestral instruments and Figure 3.2 shows the amplitude-frequency spectrum for each. Notice that the shape of the waveform for each of the notes is different and so is the recipe of frequency components. Each of these notes would be perceived as being the note A4 but as having different timbres. The frequency components of notes produced by any pitched instrument, such as a violin, oboe,clarinet, trumpet, etc., are harmonics, or integer mulltiples of f0, (see Chapter 1), Thus the only possible frequency components for the acoustic pressure waveform of the violin note shown in Fig 3,1 whose f0 is 440.5 Hz are: 440.5 Hz (1X440,5 Hz); 881.0 Hz (2 x 440.5 Hz); 1321.5 Hz (3 x 4405 Hz); 1762 Hz (4 x 440.5 Hz); 2202.5 Hz (5 x 4405 Hz), etc, Figure 3.2 shows that these are the only frequencies at which peaks appear in each spectrum (see Chapter 1), These harnonics are generally referred to by their 'harmonic number', which is the integer by which f0, is multiplied to calculate the fequency of the particular component of interest.
Table 3.1. The relationship between overtone series, harmonic series and fundamental frequency for the first ten components of a periodic waveform
An earlier term still used by many authors for referring to the components of a periodic waveform is 'overtones', The first overtone refers to the first frequency component that is 'over' or above f0, which is the second harmonic. The second overtone is third harmonic and so on, Table 3.1 summarises the relationship between f0 overtones and harmonics for integer multipliers from one to ten
Example 3,2 Find the fourth harmonic of a note whose f0 is 101 Hz, and the sixth overtone of a note whose f0 is 120 Hz,
The fourth harmonic has a frequency which is (4 f0), which is (4 x 101) Hz = 404 Hz.
Thee sixth overtone has a frequency which is (7 f0) which is (7 x 120) Hz = 840 Hz.
There is no theoretical upper limit to the number of harmonics which could be present in the ouput from any particular instrument, although for many instruments there are acoustic limits imposed by the structure of the instrument itself. An upper limit can be set though, in terms of the number of harmonics which could be present based on the upper frequency limit of the hearing system, for which a practical limit might be 16000 Hz (see Chapter 2). Thus an instrument playing the A above middle C, which has an to of 440 Hz, could theoretically contain 36 (=16 000/440) harmonics within the human hearing range. If this instrument played a note an octave higher, 1«) is doubled to 880 Hz, and the output could now theoretically contain 18 (=16 000/880) harmonics. This is an increasingly important consideration since although there is often an upper frequency limit to an acoustic instrument which is well within the practical upper frequency range of human hearing, it is quite possible with electronic synthesisers to produce sounds with harmonics which extend beyond this upper frequency limit.
Acoustically, a note perceived to have a distinct pitch contains frequency components that are integer multiples of to usually known as harmonics. Each harmonic is a sine wave and since the hearing system analyses sounds in terms of their frequency components it turns out to be highly instructive in terms of understanding how to analyse and synthesise periodic sounds, as well as being central to the development of Western musical harmony to consider the musical relationship between the individual harmonics themselves. The frequency ratios of the harmonic series are known (see Table 3.1) and their equivalent musical intervals, frequency ratios and staff notation in the key of C are shown in Figure 3.3 for the first ten harmonics. The musical intervals (apart from the octave) are only approximated on a modern keyboard due to the tuning system used.
Fig 3.3 Frequency ratios and common musical intervals between the first ten harmonics of the natural harmionic series of C3 against a musical stave and keyboard.
The musical intervals of adjacent harmonics in the natural harmonic series starting with the fundamental or first harmonic, illustrated on a musical stave and as notes on a keyboard in Figure 3.3, are:
The frequency ratios for intervals between nonadjacent harmonics in the series can also be inferred from the figure. For example, the musical interval between the fourth harmonic and the fundamental is two octaves and the frequency ratio is 4:1, equivalent to a doubling for each octave. Similarly the frequency ratio for three octaves is 8:1, and for a twelfth (octave and a fifth) is 3:1.
Intervals for other commonly used musical intervals can be found from these (musical intervals which occur within an octave are illustrated in Figure 3.15). To demonstrate this for a known result, the frequency ratio for a perfect fourth (4:3) can be found from that for a perfect fifth (3:2) since together they make one octave (2:1): C to G (perfect fifth) and G to C (perfect fourth). The perfect fifth has a frequency ratio 3:2 and the octave a ratio of 2:1. Bearing in mind that musical intervals are ratios in terms of their frequency relationships and that any mathematical manipulation must therefore be carried out by means of division and multiplication, the ratio for a perfect fourth is that for an octave divided by that for a perfect fifth, or up one octave and down a fifth:
frequency ratIo for a perfect fourth = (2/1) / (3/2) = (2/1)X(2/3) = 4/3
Two other common intervals are the major sixth and minor sixth and their frequency ratios can be found from those for the minor third and major third respectively since in each case, they combine to make one octave.
Example 3.3 Find the frequency ratio for a major and a minor sixth given the frequency ratios for an octave (2:1), a minor third (6:5) and a major third (5:4).
A major sixth and a minor third together span one octave. Therefore:
Frequency ratio for a major sixth = (2/1) / (6/5) = (2/1) x (5/6) = 10/6 = 5/3
A minor sixth and a major third together span one octave. Therefore:
Frequency ratio for a minor sixth =(2/1) / (5/4) = (2/1) x (4/5) = 8/5
These ratios can also be inferred from knowledge of the musical intervals and the harmonic series. Figure 3.3 shows that the major sixth is the interval between the fifth and third harmonics, in this example these are G4 and E5; therefore their frequency ratio is 5:3. Similarly the interval of a minor sixth is the interval between the fifth and eighth harmonics, in this case E5 and C6; therefore the frequency ratio for the minor sixth is 8:5. Knowledge of the notes of the harmonic series is both musically and acoustically useful and is something that all brass players and organists who understand mutation stops are particularly aware of.
Fig 3.4 The positions of the first ten harmonics of A3 (f0 = 220Hz) E4(f0=330Hz)
and A4 (f0=440) on linear (upper) and logarithmic (lower) plots
Figure 3.4 shows the positions of the first ten harmonics of A3 (f0 = 220.0 Hz), plotted on a linear and a logarithmic axis. Notice that the distance between the harmonics is equal on the linear plot and that it becomes progressively closer together as frequency increases on the logarithmic axis. Whilst the logarithmic plot might appear more complex than the linear plot at first sight in terms of the distribution of the harmonics themselves, particularly given that nature often appears to make use of the
Fig 3.5 Octaves, perfect fifths, perfect fourths, major sixths and
minor thirds plotted on a logarithmic scale relative to A1 (f0 = 55Hz).
most efficient process, notice that when different notes are plotted, in this case, E4 (fo = 329.6 Hz) and A4 (fo = 440.0 Hz), the patterning of the harmonics remains constant on the logarithmic scale but they are spaced differently on the linear scale. This is an important aspect of timbre perception which will be explored further in Chapter 5. Bearing in mind that the hearing system carries out a frequency analysis due to the place analysis which is based on a logarithmic distribution of position with frequency on the basilar membrane, the logarithmic plot most closely represents the perceptual weighting given to the harmonics of a note played on a pitched instrument. The use of a logarithmic representation of frequency in perception has the effect of giving equal weight to the frequencies of components analysed by the hearing system that are in the same ratio. Figure 3.5 shows a number of musical intervals plotted on a logarithmic scale and in each case they continue to around the upper useful frequency limit of the hearing system. In this case they are all related to Al (fo = 55 Hz) for convenience. Such a plot could be produced relative to an 10 value for any note and it is important to notice that the intervals themselves would remain a constant distance on a given logarithmic scale. This can be readily verified with a ruler, for example by measuring the distance equivalent to an octave from 100 Hz (i.e. between 100 Hz and 200 Hz, 200 Hz and 400 Hz, 400 Hz and 800 Hz, etc.) on the x axis of Figure 3.5 and comparing this with the distance between any of the points on the octave plot. The distance anywhere on a given logarithmic axis that is equivalent to a particular ratio such as 2:1, 3:2, 4:3, etc., will be the same no matter where on the axis it is measured. A musical interval ruler could be made which is calibrated in musical intervals to enable the frequencies of notes separated by particular intervals to be readily found on a logarithmic axis. Such a calibration must, however, be carried out with respect to the length of the ratios of interest: octave (2:1), perfect fifth (3:2), major sixth (5:3), etc. H the distance equivalent to a perfect fifth is added to the distance equivalent to a perfect fourth, the distance for one octave will be obtained since a fifth plus a fourth equals one octave. Similarly, if the distance equivalent to a major sixth is added to that for a minor third, the distance for one octave will again be obtained since a major sixth plus a minor third equals one octave (see Example 3.3).
A doubling (or halving) of a value anywhere on a logarithmic frequency scale is equivalent perceptually to a raising (or lowering) by a musical interval of one octave, and multiplying by 3/2 (or by 2/3) is equivalent perceptually to a raising (or lowering) by a musical interval of a perfect fifth, and so on. We perceive a given musical interval (octave, perfect fifth, perfect fourth, major third, etc.) as being similar no matter where in the frequency range it occurs. For example, a two-note chord a major sixth apart whether played on two double basses or two flutes give a similar perception of the musical interval. In this way, the logarithmic nature of the place analysis mechanism provides a basis for understanding the nature of our perception of musical intervals and of musical pitch.
Fig 3.6 Acoustic pressure waveform (upper) and spectrum (lower) for a brushed snare drum
By way of contrast and to complete the story, sounds which have no definite musical pitch (but a pitch, nevertheless-see below) associated with them such as the 'ss' in sea have an acoustic pressure waveform that does not repeat regularly, and is often random in its variation with time and is therefore not periodic. Such a waveform is referred to as being 'aperiodic' (see Chapter 1). The spectrum of such sounds contains frequency onents that are not related as integer multiples of some frequency and there are no harmonic components, and it will contain all frequencies, in which case it is known as a continuous spectrum. An example of an acoustic pressure waveform and spectrum for a non-periodic sound is illustrated Figure 3.6 for a snare drum being brushed.
Repetition Pitch(Audio Demo)