The ear-brain system can be easily fooled in many ways giving rise to what are known collectively as phantom tones. The missing fundamental is one example of a phantom tone. As another example, under the right circumstances listeners will perceive a frequency which is the difference between two frequencies: f = f1 – f2, instead of the two individual frequencies. A similar effect can sometimes occur when the sum of two frequencies is heard instead of the two individual frequencies: f = f1 + f2. Since most musical instruments have harmonics and overtones, these combination tones (sometimes called Tartini tones) can have an effect on the timbre of a musical note, both for an individual instrument and for two or more instruments playing together.
Most musical instruments produce a combination of frequencies that change in amplitude as the note is played. These transitory pitches are called attack frequencies and they sometimes enable us to tell the difference between different instruments. When music is played backwards, particularly on stringed instruments it often sounds very strange because the attack frequencies occur in the wrong place. Our perception of timbre is determined not only by harmonics, intensity and duration as we have discussed previously, but also by the attack frequencies. Here is a YouTube video of a piano played backwards.
A similar effect occurs when we listen to a conversation in a noisy room. Although there is a wide range of frequencies present, because of our prior experiences we are able to separate out and recognize the words of the person we are talking to. An interesting test of this is to play a conversation recorded in a noisy environment backwards. Although the frequencies are the same the words of the conversation become much more difficult to detect.
The phenomena of beats, discussed previously, occurs when two notes are close together in frequencies and we perceive one note which varies in loudness. A guitar string can be tuned by comparing a note with a known pitch and tuning the string until the beats disappear. What happens if the two frequencies get further and further apart? If the notes are a little further apart we still hear a single note but it sounds rough or wavering. Eventually we hear two separate notes instead of beats. The range of frequencies where the two frequencies are close enough to cause us to hear a single sound is called a critical band. Notice this is not the same thing as just noticeable difference in frequency. We may still be able to tell one note from another if they played one after the other but when played at the same time our hearing mechanism combines them into a single note.
The following diagram shows the phenomena of critical bands. For a given frequency f1 a second frequency is played at the same time but its frequency is changed to be first below and then above the frequency f1.
The critical band width is different at different chosen frequencies, f1. It is about 100 Hz for frequencies below 1000 Hz and then steadily increases to about 3000 Hz at 10,000 Hz. If you start with a frequency, f1, of 10,000 Hz you will hear beats if a second frequency is introduced that is between 7,000 Hz and 13,000 Hz. But if you start with a frequency, f1, at 800 Hz you will only hear beats if the second frequency is between 700 Hz and 900 Hz.
The critical band also determines how well a tone is masked by noise (a random spread of frequencies). If there is one (or several) background frequencies (noise) but the frequency you are trying to hear is outside the critical band surrounding the noise then it is less difficult to hear the signal. If the signal frequency falls in the critical band, however, it is more difficult to perceive. Effectively this is because the ear mechanisms are already responding to that range of frequencies and the signal doesn’t cause additional changes to the hearing system.
Dissonance is the musical term that is used to describe musical notes that have harmonics close enough to fall in the roughness regime but not close enough to form beats. When two notes are played on a piano an octave apart (the second note has a fundamental exactly twice the fundamental of the first or a 1 to 2 ratio) all of the harmonics line up and fall on top of each other and there is no dissonance. (For example a 200 Hz fundamental has harmonics at 400 Hz, 600 Hz, 800 Hz, etc. and the note an octave above has frequencies at 400 Hz, 600 Hz, 800 Hz, etc.). However, when a perfect fifth is played (the fundamentals have a 2 to 3 ratio) the harmonics do not exactly line up. The match of harmonics is worse for harmonics of a minor third (ratio of 4 to 5) and progressively worse for a perfect fourth (3 to 4 ratio) and minor seventh (9 to 16 ratio). If the harmonics fall close enough together there may be beating or roughness which changes the timbre of the combination. This is the primary reason a minor seventh chord does not sound as pleasant to the ears as an octave chord or a perfect fifth; the harmonics produce more roughness in the minor seventh case. The effect is stronger when the harmonics of the instrument are stronger and weaker for instruments with fewer or weaker harmonics.
A more subtle from of beats, called waveform beats, can occur for a slight difference between two frequencies about an octave apart (one frequency is twice the other). So for example, a 201 Hz sound and a 400 Hz sound together will have a 2 Hz oscillation in volume. This is because the octave above 201 Hz would have a 202 Hz frequency which is 2 Hz from the 400 Hz signal. Interestingly, a 200 Hz plus a 401 Hz signal is heard with a beat frequency of 1 Hz because the octave above 200 Hz is 400 Hz which is only 1 Hz difference from 401 Hz. The difference between the two cases has to do with the actual shape of the waveform and indicates that our ear-brain system does detect some information about the shape of the wave being heard, in addition to the frequencies (and is best explained by the place theory of hearing). This phenomena can have a slight effect on the timbre of a musical note.
Simulation exercise 10A (turn in answers on a separate sheet of paper): Beats and Critical Bands.
The following is a list of other examples where our ears are fooled into perceiving something that isn’t actually present. We will listen to these examples in class. Examples of several of them can be found at Wikipedia or Richard Waren’s site. [NOTE: Many of these effects can only be heard using headphones.]
- Shepard’s Illusion. The illusion of a continuously rising (or falling) scale of notes. The continuous version of Shepard’s Illusion is called the Risset rhythm.
- The Octave Illusion. Two notes an octave apart are played alternating back and forth between each ear (eg. 400 Hz to left ear, 800 Hz to right ear, then switch). Most listeners perceive a low tone in one ear and a high frequency tone in the other (instead of back and forth). Which ear depends on handedness (right handed people hear the high note in the right ear). Wikipedia (with sound sample).
- Deutsch’s Scale Illusion. If a two different patterns of notes which alternate up and down in pitch are played simultaneously to each ear the perception of a rising scale is heard.
- Glissando illusion. An instrument or voice can be made to appear to change from one side of a stereo output to the other by simultaneously playing a sine wave that is shifting in pitch.
- Continuity of Tones. If you interrupt a continuous rising tone with a brief burst of noise your perception will be that the tone did not stop during the break.
- McGurk effect. Your vision affects what you think you hear. For example if you see a set of lips forming the sound “ga” while you are hearing the sound “ba” you will perceive the sound “da”.
- Context effects. The sound immediately preceding a sound may affect the how the second sound is perceived. For example “al” plus “da” sounds like “al ga” but “ar” plus “da” sounds like “ar da”. Similar to this is prior information affects what you hear: YouTube example.
- Chromatic Auditory Illusion. Alternating pitches may be perceived as two scales going in opposite directions.
- Melody Illusion. Keeping the same notes but changing their octave can make a melody unrecognizable; YouTube example.