Physics of Hearing II PDF

Physics of hearing II Sound decibel level-obsi-cobg/fo) - np2 Lever sours pressure - obse-zolog/d Intensity is perceived in a peculiar way, what we call sound level, as we perceive it in a logarithmic way. Measured in decibels. The 3rd characteristic of sound: timbre. Fourier expansion = tool to ana...

Physics of hearing II Sound decibel level-obsi-cobg/fo) - np2 Lever sours pressure - obse-zolog/d Intensity is perceived in a peculiar way, what we call sound level, as we perceive it in a logarithmic way. Measured in decibels. The 3rd characteristic of sound: timbre. Fourier expansion = tool to analyse the presence of different frequencies. We will also discuss what are the consequences of timbre EXAMPLES - same pitches, di erent timbre - Guitar ALL INSTRUMENTS COMMON NOTE HAS FREQUENCY WHICH - Piano EACH timba distinguish instruments from - Voice These 3 sounds represent the same pitch (same frequency); however, our ears can differentiate the different sources of the sound (person singing, guitar,...) A we IS TO their Where is this difference coming from? We have to take a close look at the behaviour of the cochlea. It’s quite peculiar. How to di erentiate pitches The different frequencies (guitar, voice,...) will each excite different responses on the cochlea (key point). For each specific frequency, the basilar membrane will trigger an electric response in a specific place corresponding to said frequency - High frequencies = produce vibration of the membrane close to the oval window - Low frequencies = close to the apex of this system In other words, thanks to this behaviour, our cochlea is able to differentiate different frequencies / pitches. KEY POINT. As we can see in the drawing below, our guitar and singing person both emit the same pitch, so they’re frequencies will induce a vibration in the same region of the basilar membrane, in this case quite close to the oval window. How to di erentiate source. Timbre HOWEVER, even at a same given pitch, if you look at the behaviour of the cochlea when you perceive a person speaking and playing guitar, we can see that the excitation is also different (at same pitches) (1 sound → more than one response on the basilar membrane. Many frequencies perceived, making different parts of the membrane vibrate) Guitar = there's stimulation in one initial region, corresponding to the frequency of the pitch, but also in another 2 regions. There will be a set of frequencies that are triggered at the same time Singing person = once again, same situation, but for a singing person we will perceive an even more complex set of frequencies. 4, up till 5 excitations of the membrane at different points (following that initial excitation that corresponds to the pitch) There’s a complex behaviour in the cochlea that cannot be described using a single frequency. We will explain why we need this behaviour to distinguish between different sources of sound. ORIGIN OF THIS BEHAVIOUR Let’s take the example of a guitar. The sounds in this instrument are produced by oscillating strings. The strings are fixed between two points, at a given tension. Blue diagram = shows maximum (top) and minimum positions of the string. Max and min amplitudes it can reach. In this first situation, the wavelength of this expanding wave is twice the length between the fixed points. The shorter the wavelength, the higher the frequency = higher pitch (explains why we play the guitar as we do. Approximate the finger to the fixed point to lower the length of the string = lower the wavelength, higher pitch sounds) Velocity of sound = frequency x wavelength However, we will see that for a specific situation, where there are 2 fixed positions, we can excite a different set of oscillations, which will each have different frequencies (f2, f3,f4,...) but all derived from the initial f1. F1 represents the most simple situation = the one above. But there are other possibilities, like the drawings below show (a part going up, the other down, oscillating between 2 different positions, behaviour which can be multiplied several times) We can excite even more complex behaviours. There is no limit to the complexity of oscillations we can induce on the system. The fixed points are referred to as nodes. If we do the numbers, f1 being the original frequency, we’ll recover that all the other frequencies that arise in more complex oscillations of said string are multiples of the original frequency. At the same time, we are decreasing the wavelength (a medida que dividimos la cuerda en distintas ondas). IN ANY OSCILLATING SYSTEM, WE CAN TRIGGER A WHOLE SET OF OSCILLATIONS WITH A FREQUENCY THAT ARE MULTIPLES OF THE MOST SIMPLE ONE (f1). Depending on how we play the string, we can induce any of the different behaviours. Once the first, most simple frequency is fixed, all the others are also fixed as they are multiples of f1. It’s important to remember that many solutions are possible for that string, not only the simplest one. Additionally, the frequencies derived from more complex situations are related, multiples of the most simple one. f1 = FUNDAMENTAL FREQUENCY Why is this whole idea important? It’s important because we have to remember that we have an instrument and we would like to play a specific sound, a pitch of 440 Hz. Instead of a single frequency, however, what playing a string triggers here is a whole set of frequencies in our cochlea. When playing the guitar, it’s impossible to generate only one of the solutions. ALL OF THEM APPEAR AT THE SAME TIME The only instrument where we can trigger a single frequency, is a flute. All the other instruments (guitar, person singing,...) have a lot of solutions. Sequence of events: Whenever playing a guitar, singing,... we are exciting a whole set of frequencies which derive from the initial one (f1) that determines the pitch. When they reach our ear, they will excite different positions in the cochlea, corresponding to each set of frequencies. All the sounds we hear in our ear are generated by vibrating objects, which will trigger a whole set of frequencies. How can we describe this behaviour? → FOURIER EXPANSION. FOURIER EXPANSION Harmonic behaviour In order to explain this phenomenon, 1st we need to characterize a pure oscillating sinusoidal behaviour, which is also denominated as a harmonic behaviour. Harmoning waves behave in the following way: Our magnitude / variable P(t) oscillates in time and can be described in a sinusoidal, periodic manner ⇒ Harmonic wave. We just need 2 parameters to characterize its behaviour: Amplitude, Po Frequency (w1) P(t) is our variable, the amplitude of the wave in a function of time corresponding to different frequencies. This is the situation for harmonic waves. Non-harmonic behaviour (Sound) What would happen when we have a non-harmonic behaviour? For example, a pressure wave such as sound. NOT sinusoidal. We’ll see that even in these cases, we can detect periodicity! “Whenever we find a wave is non harmonic but periodic, we can propose that this is created as the superposition of a set of harmonic behaviours”. Coming back to a guitar:according to our proposal, the sound being played by this guitar is the combination of all the solutions (f1, f2, f3,...) which are played at the same time, emitting a specific sound. This took 90 years to be demonstrated! Omega 1 (w1) = fundamental frequency, corresponds to 2pi /T (period of the wave) All the other frequencies are called higher harmonics (w2, w3, w4,...): 2nd, 3rd, 4th harmonic,... which are multiples of the first w1. Because of that, we usually refer to each of these waves as: - w1 = FUNDAMENTAL WAVE or FREQUENCY - wn = HIGHER HARMONICS or MODES Whenever an amplitude is different from 0, we will say the harmonic waves or modes are being triggered on the system. This whole theory is called the FOURIER EXPANSION. Usually written in a compact way, as shown (p(t)): Once again, remember a fundamental point: ONCE THE PERIOD OF THE WAVE IS FIXED, ALL THE DIFFERENT FREQUENCIES WILL BE DERIVED FROM THAT FUNDAMENTAL WAVE. The only thing we can change from that wave is the Amplitude. 2 sounds can have the same basic pitch sound (with same fundamental frequency), but different relative amplitudes of sound. P(t) will be different for each. CONSEQUENCES OF THESE SUPERPOSITION EVENTS Diagram = on one axis, the sound level; on the other, the frequencies corresponding to each note (each note has a given pitch). The diagram shows the amplitudes which we can find on each system. Different instruments playing the same notes. - We see that there are 10 different harmonics at least for the flute. - But in the bottom diagram situation, we have an even larger set of frequencies, at least 12 harmonics. The point is that in the top and bottom situation, the number of harmonics are different. WHAT IS THE TIMBRE when we refer to different sounds? TIMBRE = REFERS TO THE NUMBER OF HARMONICS WE HAVE TRIGGERED AT EACH SPECIFIC SOUND. These two sounds above, therefore, have a different timbre. We have 2 sounds with the same pitch. They share the same fundamental frequency. On sound n1, we will have the 1st, 2nd, 3rd frequencies,.... With a specific amplitude On sound n2, we will have the exactly the same set of possible frequencies, BUT the difference is on the amplitudes of all of them (p = AMPLITUDE) The ratio between the amplitudes in sound 1 and sound 2 are different Diagram II: sound levels on one axis, frequencies on the other - Green = person with increased hearing threshold (hears less, problems for large frequencies) - Blue = normal threshold Let’s propose a situation in relation to this diagram. We are talking with someone else, and the voice of that person has a fundamental frequency of 1000 Hz. When this person is talking or singing, you will perceive 1000, 2000, 3000, 4000 Hz. Because the behaviour is exactly the same as in the instruments. When talking or singing, we produce a whole set of harmonics. Each of the harmonics will have a specific amplitude, so they will correspond to different sound levels. Consequence? The person with an increased threshold cannot perceive the harmonics below his or her threshold. They will perceive a distorted sound. This has dramatic consequences as consonants are particularly sensitive to high pitches. PA-DA-TA turns into DA-DA-DA (what the person perceives). There is a strong deformation of the original sound. If you speak louder, the distortion still remains. Speaking louder doesn’t help. Additionally, when we try to speak louder, we speak with higher frequencies, so it makes it worse. Alternative way to represent, where the frequencies are represented on one axis and the amplitudes on the other, rather than This representation is known as the FOURIER SPECTRUM. The spectrum will be different for each different sound. We see that amplitude decreases as frequency increases. FINAL CONCLUSION WHAT CHANGES BETWEEN A GUITAR AND MY VOICE? The ratio between amplitude P1 and P2. The ratio between all the different amplitudes produced by the vibration of my vocal cords as I speak, is different from the ratio between the amplitudes produced by a guitar when we play a string. They’ll produce different TIMBRE. Each amplitude is different for the different frequencies (w1,w2,...) If we want to shout, we increase all the amplitudes simultaneously, but the ratio between them remains the same, thus maintaining the timbre and, if the fundamental frequency remains unchanged, my pitch.