Intro To Music Technology Part I Fundamentals of Sound

Transcription

1 Intro To Music Technology Part I Fundamentals of Sound Outline ****Whenever you see an unfinished sentence that ends with periods (..) take notes**** ex. Sound is. Brief description of Sound: What is Sound? Sound is a form of energy, just like electricity and light. Sound is made when air molecules vibrate and move in a pattern called waves, or sound waves. Think of when you clap your hands, or when you slam the car door shut. That action produces soundwaves, which travel to your ears and then to your brain, which says, "I recognize that sound." Simple Vibrating Systems Simple Harmonic Motion: In an effort to explain what sound is, we will do a quick demonstration of the building blocks of sound. Here is a link to a graphic of simple harmonic motion. Also known as the mass-ona-spring demo. WWW: Check out this web site for a graphic of mass-on-a-spring The spring is the type that can be compressed or stretched. When the mass is still, it would be right in the middle of the the two extremes (A+

2 2 A-). It is in a state of equilibrium. If I push, or force the mass all the way up (Force) and load up the spring, at some point the force of my hand is going to equal the force (Restoring Force) of the compressed spring. This is A+ (maximum amplitude) At this point, the mass is again in a state of equilibrium. When I let go, it will start the motion. 1. The Restoring force of the spring.. 2. As the mass passes the equilibrium point, it s moving at it s fastest rate.. 3. The mass then starts to decelerate.. 4. Now the mass starts to accelrate back towards the Equilibrium point again. This system is called a simple harmonic oscillator and it demonstrates Simple Harmonic Motion. The distance from the equilibrium position to either point of maximum displacement A+ or A- is called the.. One Cycle of the oscillation is one complete excursion from a given point, through both extremities, back to the given point and moving in the original direction. Components of Simple Harmonic Motion Period: Frequency: Frequency (F) is the reciprocal of the Period (T) F = 1/T

3 3 Hertz: Graphs It will be useful in our discussion of acoustics to make diagrams of how some particular physical quantity works with some other quantity such as time. This type of diagram is called a graph. It is usual to plot values of quantity on the vertical y axis with positive values above and negative values below the horizontal axis. The other quantity, usually time, is plotted on the horizontal x axis, with positive values to the right. To draw this curve, we have to take measurements of the mass displacement at regular time intervals. Question - What is the Frequency? Figure 1 To plot this accurately, we would need to make many more measurements within one cycle. We would then find that the mass s acceleration decreases as it reaches its maximum amplitude (displacement) and the resulting graph would resemble the curve in fig. 2.

4 4 Figure 2 This curve is so important that it is given a special name. Mathematicians call them sinusoidal curves. We electronic musicians call it a.. Any system that is vibrating with simple harmonic motion may be said to undergo sinusoidal motion. All matter is made up of tiny particles called molecules and are constantly in motion whether it s in a liquid, solid or gas state. If we think of the mass-on-a-spring as a single molecule of a larger mass, we start to get the basic idea. When the matter is disturbed in a periodic fashion, the SHM of the individual molecules create a Wave. If this wave is in the audible frequencies and has enough energy to be detected by our ears, we will hear it as sound! WWW: Check out this web site for a graphic: Think of the blue dot as the mass-on-a-spring. The line it creates is a smooth curve. Acoustic Compression Waves WWW: waves Check out this web pages for a graphic of sound

5 5 In a compression wave the particle displacement is parallel to the direction of wave propagation. The animation shows a compression wave propagating down a tube. The individual particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions in SHM. Sound waves or Acoustic waves What we hear as sound is a class of physical kinetic energy called acoustical energy. Acoustical energy consists of fluctuation waves of pressure in a physical medium usually air. A single complete cycle of an acoustical pressure wave consists of one half cycle of compression (higher pressure) of the air molecules, followed by on half cycle of rarefaction (lower pressure) of the molecules. Sounds of higher amplitude (louder) compress and rarefy the air molecules to a greater extent than do lower amplitude (softer) sounds. If the frequency of a SHM oscillator is in the audible range, it will create a Sine Wave. A Sine Wave is. Frequency of Sound In order to be classified as sound, these waves must fluctuate at a rate between. Frequency corresponds to the musical attribute of pitch.

6 6 Pitch Pitch is one of the three major auditory attributes of musical sounds along with.. Pitch is Logarithmic. We perceive the pitch of A4 (440Hz) as sounding an octave higher than A3 (220 Hz). Doubling a given frequency is perceived as raising the pitch. Speed of Sound Sound waves travel through air on a standard temperature day ( 71 degrees Fahrenheit or 22 degrees Celsius) at the speed of approximately.. The physical distance covered by one complete cycle of a given frequency of sound as it passes through air is called the.. Here is a link to a Speed of Sound calculator for air with varying temperatures Sound waves travel through other mediums at different speeds. Other gasses have different densities from that of air, so the speed of sound waves in them will be different. Hydrogen, for example, is a very light gas, and sound waves in it travel four times as fast as they do in air. In liquids and solids the speed of sound is quite large; this is because the elasticity s of these materials are very much greater than that of air and more than compensate for the fact that their densities are also greater. Table 1 gives values of the speed of sound in various substances.

7 7 1. Wavelength Table 1 Speed of sound in various substances Substance Temp Speed Speed Celsius m/sec ft/sec Air Air Hydrogen Carbon Dioxide Water Steel ,400 Helium Water Vapor In our dealings with sound we will only be interested in the speed of sound waves in air at 22 degrees Celsius ( 71 F). Wavelength is expressed by the equation.. Where W = wavelength, S = speed of sound, F = frequency Example: The fundemental frequency of the pitch A, above middle C is 440Hz. Its wavelength in air is 344m/s divided by 440Hz =.78 meters long. Or 1130ft/s divided by 440Hz = 2.57 ft long. This is the actual length of one cycle of sound at that frequency in air. Using Exponents and Prefixes Let s stop here for a moment and discuss the use of Exponents and Prefixes. Since forces in nature are Logarithmic, we need a way to represent very large and very small numbers more easily. We can do this using Scientific Notation. Exponents: Base 10 Q. To what power must 10 be raised to make 1000? A.

8 8 Decimal Exponent (+n) 10(-n) 101 = 100 = We are still dealing with positive numbers. Negative exponents are smaller than 1, but larger than 0. More work with Exponents 1,000,000. = 2,600,000,000. = = 0.000,000,002,6 = 10,900,000,000,000. = Prefixes The use of prefixes to indicate the factors is quite useful. Example, the prefix centi means 1/100 or 10-2 of whatever it is attached to. ex. One centimeter = 0.01 of a meter, or 1 centigram is 0.01g. Some useful prefixes we need to know = 1,000,000,000,000 = 1 Trillion 10 9 = 1,000,000,000 = 1 Billion 10 6 = 1,000,000 = 1 Million 10 3 = 1,000 = 1 Thousand 10 0 = 1 = = 0.01 = 1 Hundredth 10-3 = = 1 Thousandth 10-6 = = 1 Millionth 10-9 = = 1 Billionth = = 1 Trillionth Giga, Mega, Whata? Where did the names Kilo, Mega, Giga and all those other prefixes come from? They have entered our language. Everyone uses them. The

9 9 terms, particularly with "byte", are almost commonplace. Kilobyte, Megabyte and Gigabyte are part of our lexicon. The familiar ones are Greek, Latin or Spanish. Tera from Greek teras meaning Monster Giga from Greek gigas meaning Giant Mega from Greek megas meaning Great Kilo from Greek khilioi meaning thousand Milli from Latin mille meaning thousand Micro from Greek mikros meaning small Nano from Greek nannos meaning dwarf Pico from Spanish pico meaning small quantity Electrical Representation of Sound Audio Signal An audio signal is an electrical representation of a sound in the form of a fluctuating voltage or current. The amplitude of an audio signal is called the.. Many different operating levels exist in audio systems. Level (acoustic or electrical) is specified in decibels.

10 10 WWW: Check out this web pages for a graphic of a speaker converting voltage into sound: Phase The time relationship of a sound wave (or an audio signal) to a known reference is called the phase of the signal. Phase is expressed in degrees. One complete cycle of a sine wave equals 360 degrees. Adding Sine Waves Phase relationships are very important to Audio Engineers. The main reason phase must be controlled is that it affects how sounds add together. When audio signals are mixed, or when sound waves mix in the air, they add algebraically. Figure 5 shows the effect of phase on the addition of two sine waves of equal level and frequency. In figure 6a the sine waves are in phase; they add to form a sine wave of twice the level of either one. In figure 6b the sine waves are 90 degrees out of phase. They add to form a sine wave that is times higher in level than either one. In figure 6c the sine waves are 180 degrees out of phase; they totally cancel one another.

11 11 WWW: Check out this web pages for a graphic of a superposition of two wave forms: Figure 5 Here are some practical demos of phase realationships in the studio! Bass track phase cancellation demo Drum mix phase cancellation demo

12 12 Complex Waves Partials and Harmonics Almost every musical sound that we hear is a composite of sine waves at different frequencies and amplitudes. These sine waves combine to form the sound and their frequency and amplitude relationships determine the quality, or timbre, of the sound. The individual simple tones making up a complex tone are called partial tones or simply partials. When a musical note with a complex waveform (partials) has a distinct pitch (as opposed to noise), that waveform can ultimately be created by combining a set of precisely related sine waves. These sine waves are called harmonics and their frequencies are related in simple integer multiples. The sine wave frequency at the pitch that we hear as the note is the. Harmonics are partials that have mathematical relationship to the fundamental. All Harmonics are Partials. Not all Partials are Harmonics. Complex sounds that are more noise like have many partials sounding. Some are multiples of the fundamental, most are not, like a Snare drum or cymbal. Complex sounds that have a definite pitch have the partials that are multiples of the fundamental sounding. We call these Harmonics. Harmonic Series Above the fundamental there are additional sine wave components whose frequencies are multiples of the fundamental frequency. If the fundamental is at 500 Hz, for example, the harmonics will occur at 1000 Hz, 1500 Hz, 2000 Hz, and so on. The fundamental is called the... The 2 nd harmonic has twice the frequency as the fundamental. The 3 rd harmonic has three times the frequency of the fundamental and so on. As the frequency multiplier increases, the amplitude of the harmonic usually decreases so the upper harmonics are generally much lower in level than the fundamental.

13 13 Harmonics: Modes, nodes and anti-nodes If we were to analyze how string and wind instruments vibrate, we would find that a string and open-air columns vibrate in modes whose frequencies are integral multiples of the fundamental. The sound of a string will contain all the harmonics. Figure 7_A Figure 7_A shows the first 7 harmonics of the series that are present when the string is plucked or struck. The term overtone has been used in reference to complex tones, such a tone described as consisting of a fundamental and its overtones. This causes some confusion. In the sound produced by a vibrating string, the first overtone is the second harmonic. Similarly, in the sound produced by the closed tube, the second harmonic is missing, so the first overtone is the 3 rd harmonic. So we won t get confused, we will not use the term overtone. The second harmonic will always refer to the partial that is twice the frequency of the fundamental.

14 14 Figure 7. Harmonic Series first 12 notes The first 12 harmonics are given in figure 7. Note that the 7 th and 11 th are in black. That is because they are very much out of tune with the note of our present equal-tempered scale. Some of the others are a little out of tune also, but not important for our discussion. 12 TET tuning: (Twelve Tone Equal Temperament) The accepted standard for tuning fixed-pitched instruments (such as the piano) since the time of J.S. Bach Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and was a better approximation to just intonation than the nearby alternative equal temperaments. It permitted total harmonic freedom at the expense of just a little purity in every interval. This allowed greater expression through modulation, which became extremely important in the 19th century music of composers such as Chopin, Schumann, Liszt, and others. Adjacent notes (1/2 step) are called. There are 12 individual notes per octave Each half-step (semitone) is 1/12 of an octave.

15 15 Semitones are divided into.. There are cents per octave. Name Just Cents in Decimal value Differen Cents intonation just in 12-TET ce interval intonation Unison (C) = Minor second (C ) = Major second (D) = Minor third (D ) = Major third (E) = Perfect fourth (F) = Diminished fifth (F ) = Perfect fifth (G) = Minor sixth (G ) = Major sixth (A) = Minor seventh (A ) = Major seventh (B) = Octave (C) =

16 16 Noise If the sine wave components, or partials, of a sound are not related in simple integer multiples, the sense of pitch is lost and the sound quality approaches noise. Drum sounds, for example, have very complex sets of components with non-integral frequency relationships. White Noise: Pink Noise: Since there are more frequencies packed into each upper octave than each lower octave, pink noise rolls off at higher frequencies. This means that there is less energy at any single high frequency than at any single low frequency so that when all noise in an octave is added up, it equals all the noise in any other octave. Complex Tone Structure (figure 8) To see how a complex tone is built up, let us assume that we have present in the air two single tones, one having twice the frequency of the other and, three quarters the amplitude. The individual waves would be as shown by the two solid curves in figure 8a. The superposition of the two waves will give a resultant sum of displacements shown in figure 8b, the sum of the first two harmonics. Adding the 3 rd harmonic, as shown by the dashed curve in 8a, the sum of all three is shown in curve 8c. Adding harmonics of higher frequencies would produce a waveform of still greater complexity which, however, will still repeat at the fundamental frequency.

17 17 Figure 8. Building a Complex Wave Realative Phase In figure 8 the individual harmonics were assumed to have their phases such that their magnitudes were at zero at the same instant. This does not always have to be the case. The phase of the second harmonic may have any arbitrary value at the time the fundamental is zero. We may then speak of the relative phase of the 2 nd harmonic with respect to the fundamental. Figure 9 shows the fundamental and second harmonic with identical amplitudes but with different relative phases of the second harmonic. The light lines are the harmonics and the heavy line is the resulting complex wave. The waveform of the complex wave obviously depends on the relative phase of the second harmonic as well as its amplitude. Similarly, the waveforms of more complex waves will be quite different for different relative phases of their harmonics.

18 18 Figure 9 Relative Phase Synthesis In electronic music, the process of adding harmonics to produce an arbitrarily complex waveform is termed synthesis. The converse of this is analysis of a wave. Any periodic waveform may be analyzed into the sum of harmonics of definite amplitude and phase, and there is only one set of such harmonics that will add up to the given waveform. With training, the ear can to some extent hear the individual harmonics of a complex sound wave. In electronics, complex waveforms can be built up, or synthesized in various ways. One important complex wave is made up of harmonics 1, 2, 3, 4 and so on of relative amplitudes 1, 1/2, 1/3, 1/4 and so on, all in the same phase. It is shown in figure 10a and is called for obvious reasons. Another important waveform in synthesized music is obtained by using the odd harmonics 1, 3, 5, 7 and so on, with their relative amplitude 1, 1/3, 1/5, 1/7 and so on, again all in the same phase and is shown in figure 10b. It is called the..

19 19 Figure 10 Synthesized Waveforms These two waves are important in electronic production of musical sounds. The dashed curves show the result of adding the first four harmonics for each waveform. As more harmonics are added, the sum approximates the waveform more closely. A waveform in the shape of a triangle can likewise be shown to be built up out of harmonics whose amplitudes and phases will depend on the shape of the triangle. All of the above examples of adding harmonics to produce complex waves are illustrations of the. This Theorem simply states: Any complex vibration can be reproduced by adding (with proper amplitudes and phase) the harmonics of the fundamental vibration. What s important, Phase or Amplitude relationship? Since the detailed waveform of a complex wave depends on the relative phases of the harmonics, it might be expected that the ear would ascribe a different quality to different waveforms produced by changing the phase but not the amplitudes of the harmonics. For example, we might expect the three different waveforms of figure 9 to sound different. Investigation of this question over a century ago by German physicist Ohm led him to the conclusion, that the perception of a complex sound depends upon its analysis into simple tones and is entirely independent of the phase relationship of these components. This Law

20 20 has been accepted without question for a long time. It is now firmly imbedded in most books on acoustics. It is known as.. In recent years the validity of this Law has been questioned, further research done to check it. As a result, evidence has now accumulated that demonstrates conclusively that under some conditions the ear can hear changes in the phases of the harmonics of a complex tone. Just what these conditions are is not clear, it seems to depend on how many sharp dips or peaks the waveform has. In most cases, it is not a pronounced effect; the three waveforms in figure 9 for example, as reproduced by a loudspeaker in an ordinary room, will all sound the same. For musical purposes, we may assume for now that the quality (timbre) of a tone does not depend significantly on the relative phases of its harmonics, only on the relative amplitudes of the harmonics. Spectrum Analysis Since the quality of a periodic complex tone depends mainly on the relative amplitudes of the harmonics and very little on their relative phases, we may characterize a tone by a diagram of its harmonics. An analogy to the optical case in which light is resolved into its constituent colors, the representation of the analysis of a tone into its constituent harmonics is called its spectrum. As an example, the spectra of the sawtooth and square waves are shown in figure 11a and 11b. Figure 11a and 11b Spectrum of Sawtooth and Square Waves For a Graphic display of Fourier composition of synthetic waveforms spectrum visit Dan Kettering s web site.

21 21 Recap Complex Tones Since the tones of musical instruments have such distinctive qualities, it is natural to look for a somewhat representative sound spectrum for each instrument. One would hope that a certain spectrum would characterize the trumpet, violin, and so on. The analysis of a large number of tones from many musical instruments, made over a considerable period of time, has demonstrated that there is no characteristic spectrum, somewhat the same from note to note, associated with a given musical instrument. An oboe, for example, does not have a certain pattern of harmonic amplitudes that characterizes its sound over its playing range. A few generalizations about tone quality can be made. The low tones of a clarinet are deficient in even harmonics, but not all tones deficient in even harmonics sound like a clarinet; however, such tones are sometimes said to have hollow sound. Tones with many highfrequency harmonics tend to sound brighter, and those with fewer tend to sound darker. This is about all that can be said; it is not possible to relate a tone of given harmonic structure to a particular instrument. Actually, it is not to be expected that the details of the harmonic structure of a tone should be important. For a particular instrument, the structure of a given single tone will depend on a number of factors. It changes with loudness, for example; soft tones will generally have few harmonics while loud tones will have many more harmonics covering a greater frequency range. Also, the spectrum of a tone will also depend on how the player produces it. It will also depend on the characteristics of the room in which the instrument is played.