Part 1.

1. 'What are the critical bands?'

The notion of critical bands, introduced by Fletcher (1940), explains the masking of a narrow band (sinusoidal) signal by a wideband noise source. Consider figure 1.

In this diagram, the noise signal is distant, in frequency terms, from the sinusoidal signal. Under these circumstances, the noise signal does not affect the threshold of hearing of the sinusoidal signal. Figure 2 shows the noise signal centred on the sinusoidal signal.

With the noise centred on the frequency of the sine tone the threshold of hearing of the latter is increased. The noise signal is therefore masking the sine tone at amplitude levels between the old threshold and the new. As the noise bandwidth, dF, is increased, so the threshold of the sine tone will increase. However, there will come a point when an increase in dF gives no increase in the threshold. At this point one can say;

One can think of a critical band as a frequency selective 'channel' of psychoacoustic processing; only noise falling within the critical bandwidth can contribute to the masking of a narrow band signal. The mammalian auditory system consists of a whole series of critical bands, each filtering out a specific portion of the audio spectrum. Figure 3 shows a graph of the responses of several critical bands.

The frequency response of each critical band is, roughly, of Gaussian shape. The overlap in the bands ensures that the combined response is approximately flat.

2. 'The role of critical bands in the perception of pitch.'

The American National Standards Institute gives the following definition; 'Pitch is that attribute of auditory sensation in terms of which sounds may be ordered from low to high.'.

Pitch is entirely subjective, and cannot be measured in an analytical way. Perceptual experiments are required. The intensity, frequency, waveform shape, and duration of a sound all contribute to how one perceives the pitch of a sound. However, the scientific analogy of pitch can usually be taken as the fundamental frequency of the sound.

Periodic and Non-Periodic Sounds.

Figure 4 shows both a periodic and a non-periodic waveform in both the time and frequency domains. The non-periodic waveform repeats itself exactly after a little less than 10 ms. This period (T) of repetition gives the fundamental frequency of the waveform;

This gives a fundamental of just over 100Hz, represented by the first line on the spectrum. Notice that the spectrum consists of discrete lines (harmonics) situated at integer multiples of the fundamental frequency.

The non-periodic waveform does not repeat itself, and gives a continuous spectrum. The importance of a pitched sound's line spectrum will become apparent later.

Frequency Discrimination.

Figure 5 shows the 'Just noticeable difference' in frequency, that an average listener can discern, over the range 50Hz to 10kHz. Also shown are the values for the critical bandwidth.

One can see that the critical bandwidth is about 30 times the 'Just noticeable difference' at all frequencies. There is therefore another mechanism at work, fine tuning the ear's ability to discern frequency. However, the ear can only make sense of one signal per critical band. For example, if two sine tones of 100Hz and 110Hz are played, two distinct tones are not heard. Instead, one hears a beating or 'harsh' and unresolved sound. This is because the critical bandwidth at 100Hz is about 80Hz, and the two tones would be within the same critical band; one will only hear two distinct tones when the two excitations exist in separate critical bands.

The Place Theory of Pitch Perception.

Figure 6 shows a schematic representation of the Basilar Membrane. This organ is within the cochlea of the inner ear, and performs transduction from the acoustic to the neural domain. A sound consisting of a series of harmonics, i.e. a pitched sound, will excite locations on the Basilar membrane as shown in the diagram. (A specific place on the Basilar membrane will react to a specific frequency.) Neural connections running from these sites will provide the cortex with an immediate Fourier Transform of the sound.

It is suggested that the brain can recognise this 'pattern' of harmonics, allowing it to decide which is the fundamental and therefore ascribe a pitch. This may explain why, if one removes the fundamental from a harmonic tone, the pitch is unchanged. The brain can extrapolate where the fundamental should be, and thus ascribe a pitch.

Although the critical bands do not perform pitch determination themselves, they allow it to occur. For the frequency of a particular harmonic to be determined it must be alone within a critical band. If two harmonics occupy the same critical band, then neither of them will be resolved. In general, harmonics above about the sixth are not resolved and will therefore not contribute to pitch perception of this form. This poses a problem, as sounds comprising of only harmonics of higher order than 6 are still perceived as pitched.

The Temporal Theory of Pitch Perception.

Consider figure 7;

In the diagram, the first eight harmonics of the spectrum of some complex signal are shown. The critical bands centred at these harmonic frequencies are also shown. If one considers this as a network of bandpass filters, it can be seen that the first five bands only allow a component of one harmonic to pass. However, the bands centred at the 6th and 7th harmonics allow a component of the harmonics either side of the centre frequency to pass.

In the band centred at the 6th harmonic one has a strong component at 1.2kHz, and two weak components at 1kHz and 1.4kHz. This will cause a beat signal at 200Hz, the frequency of the fundamental. In the band centred at the 7th harmonic one has the component at the centre frequency, 1.4kHz, and components at 1.2kHz and 1.6kHz. Here the beat at the fundamental frequency will be more pronounced than in the previous band, due to reduced attenuation at 1.2kHz and 1.6kHz. If one were to consider critical bands centred at higher frequencies, one would find that the beat at the fundamental continues to increase.

It is known that the firing of the auditory nerve is phase locked to the stimulus waveform up to about 5kHz. Therefore, for beat frequencies in the critical bands up to 5kHz the brain is fed with a signal that is 'locked-in' to the fundamental frequency of the sound, thus providing a mechanism for pitch perception. Notice how this theory relies entirely on the notion of non-resolved harmonics within the critical bands.

3. Consonance and Dissonance.

It has been shown that two pure tones, not in unison, coexisting within one critical bandwidth will cause a harsh sound to be perceived. One can therefore calculate the expected consonance of a given musical interval, formed by two complex tones, by looking for critical bands that contain a harmonic of both tones.

By way of an example, the perfect fourth (5 semi-tones) and the augmented fourth (6 semi-tones) will be considered. The root note is taken as 200Hz. To put this in a contemporary musical context, the western tempered scale will be used. The formula for this is;

f0 is the frequency of the root note.

f1 is the frequency of a note n semi-tones above the root note.

In calculating the critical band, the harmonics of the perfect and augmented fourth (tritone) were taken. These will give a slightly wider critical band than the harmonics of the root note, thus giving the 'worst case' results. The formula for calculating the critical bandwidths is;

df is the critical bandwidth centred at fc.

These equations give the following results;

The criteria for consonance, of two pure tones, are defined as follows;

1. When the two notes coincide they are judged to be perfectly consonant.

2. When their frequencies fall outside the critical bandwidth the interval is said to be consonant.

3. When their frequencies differ by 5% to 50% of the critical bandwidth the interval is said to be dissonant.

Applying these criteria to the complex tone case, on a harmonic-by-harmonic basis, will give an indication of the consonance of the interval.

The Perfect Fourth.

This interval is therefore very consonant. None of the harmonics clash below 1.4kHz, and any clash above is ignored as 1.2kHz is the sixth harmonic of the root. ( Above the sixth harmonic more than one partial will be present in each critical band, even if only one complex tone is played.) Also, the harmonics at 800Hz and 800.91Hz appear in unison (perfectly consonant on the pure tone model) since the 'just noticeable difference' in frequency at 800Hz is 4Hz.

The Augmented Fourth (Tritone).

Here, one has three clashes below 1.4kHz. Clearly the augmented fourth is more dissonant interval than the perfect fourth.

Part 2.

Psychoacoustic Demonstrations; Theme and Variations.

The following theme was composed;

Recording of First Variation (to show repetition pitch).

Note: Repitition pitch is a binaural effect- use headphones when listening to this recording.

Here, the notes played are exactly as notated for the theme. However, the instrument used now is white noise. In the first piece on the tape a white noise signal is played into the left ear. Then, after a very short delay, the signal is played to the right ear. It is important to note that the two signals are IDENTICAL to each other. It is thought that the brain can correlate the signal from one ear with the signal from the other ear. This would occur where the two VIII nerves meet- the Cochlea Olives. The time delay between the two signals gives the period of the fundamental note perceived. Therefore;

( This recording was generated using CSOUND. Orchestra file. Score file. )

Often one hears a pitch from repeated sounds in day-to-day life. (e.g. the reflection of footsteps from a wall.) In these cases the sound is not presented to the listener in a binaural way. To investigate this I prepared the second recording. Here, both the original and delayed signals are presented to both ears, i.e. this is a monaural effect. ( Again CSOUND was used. Orchestra file. Score file. )

The effect of the first demonstration will vary from one listener to another. (I could only hear a slight timbral change in the fist variation.) However, everyone will detect a pitch sound in the second demonstration. The reason for this is that the first piece relies on a psychoacoustic effect and the second on a physical effect. If, as in the second case, the two signals are mixed, they will interfere. The result is a comb filter, changing the white noise into a harmonic series of amplitude peaks. See appendix C for Fourier transforms of two such signals.

Second Variation (to show auditory streaming).

Here, the following two parts were played simultaneously;

In the third recorded example the above is played slowly (30 b.p.m.). The result is that first part and the second, highly syncopated, part merge into one sequential piece. In the fourth recording the tempo is increased (165 b.p.m.) and the two parts seem to become separated from each other. One can hear two distinct patterns.

The fifth and final recording is the theme. (The theme and second variation were produced using an Atari ST running Cubase v2 and a Yamaha MT32 sound module.)