Fundamentals of Acoustics The Nature of a Sound

Fundamentals of Acoustics

The Nature of a Sound Event n n n Sound consists of vibrations of air molecules Air molecules are analogous to tiny superballs Sound occurs when air molecules are disturbed and made to ricochet off of each other

The Nature of a Sound Event n The ricochets cause the density of the air molecules to oscillate Rarefied Normal Compressed

The Nature of a Sound Event n The ricochets cause the density of the air molecules to oscillate back and forth

Wave Types Sound consists of longitudinal waves propagation oscillation The wave’s oscillation is in the same direction as its propagation Water waves are transverse waves propagation oscillation The wave’s oscillation is perpendicular to the direction of its propagation

Sound Propagation Sound waves propagate in a sphere from the sound source (try to imagine a spherical slinky). Note that the molecules themselves are not travelling. What spreads is the energy of the wave.

Sound Perception n Speed of sound (in air): 1128 ft. /sec (344 m/sec) n When sound waves reach the eardrum, they are transduced into mechanical energy in the middle ear n The mechanical motion is transduced into electrical current in the inner ear. The auditory nerves interpret the current as sound

Sound Wave Plots n Sound waves are typically represented with molecular density as a function of time compressed normal time rarefied molecular density

Music vs. Noise Musical sounds are typically periodic – the wave repeats regularly repeats Sine wave Though they don’t exist in nature, sine waves are often useful for demonstrating properties of sounds Noise is aperiodic – there is no repeating pattern Noise

Properties of a Musical Event A musical event can be described by four properties. Each can be described subjectively, or objectively (in terms of measured properties) Subjective Pitch Objective Frequency Volume Amplitude/Power/Intensity Timbre Overtone content Duration in beats Duration in time

Frequency/Pitch Frequency is measured in cycles per second, or Hertz (Hz) one second f = 2 Hz Wavelength (l), the distance between corresponding points on the wave, is the inverse of frequency. l = c f = 1000 ft. /sec. 2 cyc. /sec. l = 500 ft. /cyc.

Frequency/Pitch Middle A = 440 Hz l = 2. 3 ft. 20 Hz l = 50 ft. < frequencies audible to humans < 20, 000 Hz (20 k. Hz) l = 0. 05 ft. Sound wavelengths are significantly larger than light wavelengths

Waves reflect from a surface if its height/width is larger than the wavelength

Waves refract around surface if the surface dimensions are smaller than the wavelength This explains why we hearsmall sound around corners, Light wavelengths are can far too to from refract around any but cannot see around corners: visible surface

Our Pitch Perception is Logarithmic Equivalent pitch intervals are perceived according to an equivalent change in exponent, not in absolute frequency For example, we hear an equivalent pitch class with every doubling of frequency (the interval of an octave) Frequencies of successive octaves of concert A 55 110 220 440 880 1760 3520 55 x 2 1 55 x 2 2 55 x 2 3 55 x 2 4 55 x 2 5 55 x 2 6

Our Pitch System is Based on Equal Division of the Octave 12 Tone Equal Temperament – the octave is divided into twelve equal increments We can describe an octave by: • choosing a starting frequency n/12 • multiply it by 2 for n = 0 to 11 A 220 0 x 212 220 A# 220 1 x 212 233 B 220 2 x 212 247 C 220 3 x 212 261. 6 C# 220 4 x 212 277 D 220 5 x 212 293. 6 D# 220 6 x 212 311 E 220 7 x 212 329. 6 F 220 8 x 212 349. 2 F# 220 9 x 212 370 G 220 10 x 212 392 Higher octaves may be created by doubling each frequency Lower octaves may be created by halving each frequency G# 220 11 x 212 415. 3

Phase = “the position of a wave at a certain time” If two waveforms at the same frequency do not have simultaneous zero-crossings, we say they are “out of phase” Wave 1 + Wave 2 Wave 1 Wave 2 Two waves at the same frequency but different phase In terms of sound perception, phase can be critical or imperceptible, as we’ll see. . .

Loudness is related to three measurements: • Power • Pressure • Intensity All three are related to changes in sound pressure level (molecular density)

Molecular Motion is Stationary n n As sound travels, molecules are not traveling with the sound wave What is traveling is an expanding sphere of energy that displaces molecules as it passes over them How strong is the force behind this energy wave? The more force is contained in a sound wave, the greater its perceived loudness.

Power = the amount of time it takes to do work (exert force, move something) Power is measured in watts, W There are two difficulties in measuring sound power levels. The range of human hearing encompasses many millions of watts. Sound power level is also relative, not absolute. Air molecules are never completely motionless. Given these two difficulties, sound power levels are measured on a scale that is comparative and logarithmic, the decibel scale.

Logarithmic Scale Logarithm = exponent (an exponent is typically an integer, a logarithm not necessarily) 102 = 100 103 = 1000 log 10100 = 2 log 101000 = 3 102. 698 = 500 log 10500 = 2. 698 102. 875 = 750 log 10750 = 2. 875 Logarithms allow us to use a small range of numbers to describe a large range of numbers

The Decibel Scale n The decibel scale is a comparison of a sound’s power level with a threshold level (the lowest audible power level of a sine tone at 1 k. Hz). Threshold (W 0): W 0 = 10 -12 watts Power level of a given sound in watts, LW(d. B): L W (d. B) = 10*log 10 (W/W 0 )

Decibels Typical power levels: Soft rustling leaves Normal conversation Construction site Threshold of pain 10 d. B 60 d. B 110 d. B 125 d. B Halving or doubling sound power level results in a change of 3 d. B. For example, a doubling of the threshold level may be calculated: LW(d. B) = 3. 01 d. B Thus, a power level of 13 d. B is twice that of 10 d. B. A power level of 60 d. B is half that of 63 d. B, and so on.

Pressure changes The degree of fluctuation present in a vibrating object Peak pressure level: Maximum change in sound pressure level (more generally: in a vibrating system, the maximum displacement from equilibrium position) The amplitude level fluctuates with the wave’s oscillation. Thus, power is the cause, pressure change is the result

Pressure changes Also may be described as changes in sound pressure level (molecular density). Pressure level is measured in Newtons per square meter (N/m 2 ) -5 Threshold: 2 x 10 N/m 2 (p 0 ) There is a direct relationship between pressure and power levels: For any propagating wave (mechanical, electric, acoustic, etc. ) the energy contained in the wave is proportional to the square of its pressure change. Pressure changes are also expressed in decibels, but in a way that describes an equivalent change in power level: L W (d. B) = 10*log 10(W/W 0) = 10*log 10(p/p 0)2 = 20*log 10(p/p 0) logmn = nlogm This is how pressure is measured

Pressure changes In audio parlance, “amplitude” (the degree of pressure change) is often equated with “loudness. ” The reason is that modifications to volume are made by adjusting the amplitude of electrical current sent to an amplifier. But perceived loudness is actually based on power level plus the distance of the listener from the source.

Intensity Power corresponds to the sphere of energy expanding outward from the sound source The power remains constant, spread evenly over the surface of the sphere Perceived loudness depends primarily on the sound power level and the distance from the sound event Power combined with distance is intensity, I, measured in watts per square meter (W/m 2 ). Intensity is also measured in decibels: L I (d. B) = 10*log 10(I/I 0 ) -12 I 0 = 10 W/m 2

Timbre The perceived difference in sound quality when two different instruments play at the same pitch and loudness Sine waves are useful as demonstrations because they are a wave with one frequency only, thus they are often termed pure tones Natural sounds are composed of multiple frequencies To understand how a wave can be composed of multiple frequencies, we can consider the behavior of a wave in a bounded medium, such as a string secured at both ends (or air vibrating within a pipe)

Timbre When we pluck a string, we initiate wave motion The wavelength is twice the length of the string The perceived pitch is the fundamental, the speed of sound divided by the wavelength

Timbre This curved shape represents the string’s maximum deviation It’s more accurate to think of it as a series of suspended masses (kind of like popcorn strung together to hang on a Christmas tree).

Timbre Each suspended mass can vibrate independently. Thus, many simultaneous vibrations/frequencies occur along a string. When a string is first plucked, it produces a potentially infinite number of frequencies.

Timbre Eventually, the bounded nature of the string confines wave propagation and the frequencies it can support Only frequencies that remain in phase after one propagation back and forth can be maintained; all other frequencies are cancelled out Only frequencies based on integer subdivisions of the string’s length, corresponding to integer multiples of the fundamental, can continue to propagate

Timbre These frequencies are called harmonics NOTE: These frequencies are equally spaced Therefore, they do not all produce the same pitch as the fundamental Therefore, other frequencies are introduced …etc.

Timbre n Harmonics are well known to many instrumentalists – – Strings Brass

Timbre n The first six harmonics are often the strongest: 220 440 660 880 1100 1320 Fundamental Octave Perfect fifth Octave Major third Perfect fifth n People can learn to “hear out” harmonics

Timbre Instruments and natural sounds usually contain many frequencies above the fundamental n These additional frequencies, as part of the total sound, are termed partials n The first partial is the fundamental n

Timbre The first partial is the fundamental n Other terms are also used n Overtones are partials above the fundamental (the first overtone is the second partial) n Harmonics are partials that are integer multiples of the fundamental n

The Spectrum n n n Jean Baptiste Fourier (1768 -1830) discovered a fundamental tenet of wave theory All periodic waves are composed of a series of sinusoidal waves These waves are harmonics of the fundamental Each harmonic has its own amplitude and phase The decomposition of a complex wave into its harmonic components, its spectrum, is known as a Fourier analysis

The Spectrum It is often more useful to represent complex waveforms with a spectral plot as opposed to a time domain plot = time domain spectral domain amplitude as a function of time amplitude as a function of frequency

Sound in Time n Our perception of sound and music events is determined by the behavior of frequency and loudness over time

Sound in Time n All instruments can be characterized by changes in amplitude over time (the envelope) loudness trumpet bowed violin harp Changes in amplitude often correspond with changes in frequency content. . . time

Sound in Time n n Most instrument’s sound begins with an initial transient, or attack, portion The transient is characterized by many high frequencies and noise Example: the scraping of a bow or the chiff of breath An instrument’s distinctiveness is determined primarily by the transient portion of its sound

Sound in Time Following the transient, instruments usually produce a steady-state, or sustained, sound n The steady state is characterized by n – – Periodicity Harmonic spectrum

The Spectrogram Most natural sounds (and musical instruments) do not have a stable spectrum. Rather, their frequency content changes with time. The spectrogram is a three-dimensional plot: Vibraphone note at 293 Hz (middle D) 2) frequency 3) power of a given frequency (darkness level) 1) time The instrument’s sound is characterized by the fundamental at 293 Hz and the fourth harmonic at 1173 Hz. The attack also contains noise below 2 k. Hz, the tenth harmonic at 2933 Hz and the seventeenth harmonic at 4986 Hz. Once the steady state portion sets in, the highest harmonic fades first, followed by a fading of the fundamental.

Localization n The auditory system localizes events through interaural time delay – the sound wave reaches the nearer ear a few milliseconds before it reaches the farther ear For stereo systems, using delay for localization is impractical because it requires people to listen from a “sweet spot” Localization effects are simulated through differences in loudness

Localization n In a multi-speaker system, a sound emanating from one speaker will be localized at that speaker A sound produced at equal volume from two speakers will be perceived as a “phantom image” placed in space between them Changing the volume balance between two speakers will cause the phantom image to “drift” towards the louder speaker

Measurement and Perception Our perception of auditory events is based on all these measurements in combination n And more n An auditory event may be more than the sum of its parts n

Measurement and Perception Phase n Changing the phase of components in a steadystate tone produces no perceptible change in sound, although the shape of the wave may change noticeably

Measurement and Perception Phase n n The behavior of components in the attack segment is likely to be far more complex than in the steady state segment Changing the phase of attack components can change the character of the attack Solo performance sounds different from group performance because no two players can ever sound at exactly the same time; thus the attack is blurred Since an instrument’s characteristics are defined primarily by the attack, the phase of attack components is critical

Measurement and Perception Timbre n n n We have discussed timbre as the result of overtone content It is also judged by the sound’s envelope Research in sound synthesis has shown the envelope shape to be more definitive than an exact match of overtone content The attack portion is critical—a faster attack can be confused with “brightness” (more high frequency overtones) Considerable research has gone into the creation of “timbre space, ” a multi-dimensional plot in which timbres are classified according to overtone content, envelope and attack time

Measurement and Perception Loudness While intensity is the measurement most closely correlated to loudness, the perception of volume is based on a number of factors, not all of them entirely measurable.

Measurement and perception Loudness Perceived loudness is frequency-dependent Perceived equal loudness of sine tones This is why many receivers have a Loudness knob Equal loudness curves (Fletcher, Munson, 1930 s).

Measurement and perception Loudness Perceived loudness is frequency-dependent Within close frequency ranges, perceived loudness is proportional to the cube root of intensity Two violins playing the same pitch will generate twice the intensity of one violin, but will not sound twice as loud To achieve twice the volume, eight violins are required

Measurement and perception Loudness Perceived loudness is bandwidth-dependent Increasing the bandwidth (component frequency content) of a sound makes it sound louder, even if the intensity remains constant Despite many efforts, no one has suceeded in creating a definitive perceptual scaling system for loudness

Measurement and Perception Loudness Some have argued that estimation of loudness is not automatic (measurable), but depends on a number of higher-level estimations of distance, import, context, etc. Hermann Helmholtz, On the Sensations of Tone (1885): …we are exceedingly well trained in finding out by our sensations the objective nature of the objects around us, but we are completely unskilled in observing these sensations per se; and the practice of associating them with things outside of us actually prevents us from being distinctly conscious of the pure sensations.

Measurement and Perception Conclusion Objective measurements can tell us more about sound events By the same token, they give us insight into what we don’t know This course will examine music in technical terms This examination will give us some new insights It will also give us an idea of where music crosses the barrier from the objective (acoustics) to the subjective (magic? )