Acoustics Presentation

1.Acoustics Seminar_Reading.docx, of 33.

The Sonic Arts Center @

The City College of New York

Acoustics Seminar Music 21700

© Paul Kozel

Edits and addendums, Jonathan Perl


The Sonic Arts Center @ The City College of New York

All Fig. numbers are examples from An Introduction to the Creation of Electroacoustic Music, by Samuel Pellman (out of print). What is Sound? Sound starts with the movement of an elastic medium (i.e. a string). The movement of this medium (sound propagation) disturbs the surrounding air molecules, which creates regular and/or irregular patterns of compression and rarefaction of these air molecules. Air molecules move toward each other during compression and away from each other during rarefaction. High air pressure is created during compression while low air pressure is created during rarefaction. These compressions and rarefactions are called vibrations. Vibrations travel to the ear where they are processed into perceptions of either noise or pitched sound or combinations of the two. Noise is created by irregular patterns of compression and rarefaction while pitched sound or frequency is created by regular patterns of compression and rarefaction. These vibrations radiate in various directional patterns at a constant speed of 1,130 feet per second (the speed of sound). The speed of sound is dependent on the medium and the temperature of the medium. Sound will travel through 72˚ Fahrenheit dry air at 1,130 feet-per-second.

(See last page of this handout for more information on noise.)

Irregular vibration = Noise

Regular vibration = Pitched sound/frequency


The Inner Ear: Pitch Perception and Frequency Masking

The Cochlea is the part of the inner ear containing several membranes, including the basilar membrane. Sounds excite the fluid within the inner ear via the action of the stapes which are in contact with the oval window. On the surface of the basilar membrane lies the Organ of Corti, which contains between 20,000 and 30,000 reedlike fibers/hairs, along its’ length, and these change properties: they are short and stiff at the start of the basilar membrane, and become increasingly longer and less stiff further down the membrane. Because of their variable stiffness and length, the hairs have different resonant frequencies. Nearer to the middle ear, the shorter stiffer hairs will be excited by higher-frequency components of the sound. Further on down the membrane, the longer looser hairs will be excited by lower-frequency components of the sound. The basilar membrane fires electrical impulses based on these excitations. These impulses travel via the cochlear nerve, which sends them on to the cerebral cortex of our brains. Our brains, in turn, have the amazingly complex job of interpreting and making sense of the impulses arriving from the basilar membrane. Among the most crucial interpretations is that of pitch: we can perceive specific identifiable pitches based on the location of firing of nerve impulses from the basilar membrane. Because most pitched sounds are not sine waves but instead contain many sine waves at different frequencies simultaneously, the concurrent firing from multiple locations can not only result in a perception of pitch, but in a perception of tone or timbre. If two fundamental tones are very close in pitch, they will excite areas that are very close to each other on the basilar membrane. In this case, our brains may be unable to parse the two distinct pitches. Whichever tone is louder and is causing the greater amplitude of vibration will be perceived as a distinct pitch, and the adjacent pitch will be masked, only contributing to an increase in perceived volume. Sounds with greater sound pressure amplitude will cause the hairs of the membrane to vibrate with a greater range of motion/displacement. The resulting nerve impulses are interpreted by the brain as being of sounds with a greater perceived loudness.


The Mechanics of a Vibrating System – The Damped String In order to demonstrate a vibrating system for pitched sound we will employ a plucked, vibrating string as demonstrated in Fig. 1.1 below. When a string is plucked it is displaced (Fig. 1.1a) or moved away from the point of rest/equilibrium. The point of equilibrium or rest is called the node. The point of maximum displacement from the node is called the antinode. The antinode also represents extremes in air pressure. When on the upper side of the node, antinodes represent maximum compression. While on the lower side of the node, antinodes represent maximum rarefaction. The antinode represents the amplitude of the sound. The further the antinode is displaced from the node the louder the sound (Fig. 1.6). When the string is released from the point of displacement, restoring force begins to act on the string, which attempts to bring the string back to the node or point of rest (Fig. 1.1b). However, momentum pushes the string past the node (Fig. 1.1c) to a point that is nearly the same distance from the resting point as the initial point of displacement. This point however, is on the opposite side of the node and represents another antinode. The string then begins the same journey, now in the opposite direction, moving from Fig. 1.1d back to Fig. 1.1f. During the string’s movement, friction from the surrounding air molecules creates drag on the string so that the string never returns to the point of the original displacement and never is able to reach the equivalent point on the opposite side of the node. The force of friction will eventually cause the string to stop vibrating altogether (damped string) unless more energy is supplied by displacing the string again in some manner. The gradual dampening of the motion of the string is called decay.

Upper Antinodes = Points of Maximum Compression


Lower Antinodes = Points of Maximum Rarefaction Pitched Sound/Frequency and Pitch The movement from Fig 1.1a through Fig. 1.1f is called a cycle. A cycle consists of one compression and one rarefaction (Fig. 1.4). Frequency is the number of cycles occurring during the time period of one second (440 cycles per second, etc.). Cycles per second are also called Hertz after the 19th-century German physicist Heinrich Hertz. Pitch is a letter name applied to certain frequencies (A - 440 Hertz etc.). The higher the frequency the more cycles take place in one second. The human ear has the ability to perceive the number of regular vibrations (cycles) that take place in one second. In short, it can distinguish between a regular vibration of 500 times a second, 200 times a second, etc. The range of frequencies most humans perceive is approximately 20-20,000 cycles per second. Frequencies above this range are called ultrasonic. Frequencies below this range are called infrasonic or subsonic.

Frequency of a String The frequency a string produces is dependent on three factors: string length, the tension of the string, and the mass of the string. Longer strings produce lower pitches. Increased tension increases frequency.


Increased mass decreases frequency. All strings of a guitar are of the same length. Differences in frequency between the strings of a guitar are not dependent on length but on tension and mass. The tuning pegs adjust the tension and the different string thickness combine to create six distinct frequencies. Note that some strings are made of a solid core (top strings) while others are solid core covered with over-wound strings (bottom strings). The over-wound string increases the mass and therefore lowers the pitch of the string. The piano uses a combination of differences in string length, tension, and mass to produce different frequencies.

The longer the string – The lower the frequency The more tension on a string – The higher the frequency

The more mass of a string – The lower the pitch


Physical Properties of a Wave Since each frequency radiates a specific number of cycles over the time of one second at the speed of sound, each cycle will have specific physical length called a wavelength. Wavelength is commonly designated with the Greek letter lambda (λ). Low frequencies have long wavelengths (since there are less cycles per second) while high frequencies have shorter wavelengths (since there are more cycles per second). The formulas below demonstrate the relationship between wavelength, frequency, and the speed of sound.

F (frequency in Hertz) x W (wavelength in feet) = S (speed of sound)

Therefore: 565 Hertz x 2 feet = 1,130 feet per second

Conversely:

S / W = F 1,130 feet per second / 2 feet = 565 Hertz

and

S / F = W 1,130 feet per second / 565 Hertz = 2 feet

Phase Angle and Phase Shift Phase Angle represents a specific point in time during the cycle of a wave. The length of time it takes to complete a single cycle is called a period. The passage of time during the period is divided into 360 equal parts, or degrees, and can be plotted about a circle (see drawing below). Phase Shift indicates the amount of time that exists between two identical waves. The time difference between these identical waves is classified by where the first wave is in its cycle when the second wave begins. For example, if the second wave begins when the first wave has already reached 90˚ of its cycle the waves are said to be phase shifted by 90˚ (see drawing below). If identical waves begin at the same time they are said to be “in phase” with each other. However, if these waves begin at different times they are “out of phase” with each other.


Phase Angles of a Single Wave

Phase Shift of Two Identical Waves

Dotted Waves are Phase Shifted from the Original Wave (Green)


Constructive Interference When two identical waves are in phase (starting at the same time) the amplitudes of the these waves are summed. This is called constructive interference or phase summation (Fig. 1.10).


Deconstructive Interference If two identical waves are 180º out of phase they will cancel each other out completely since they have opposite yet equal energy. No sound will be perceived by the listener. This is called destructive interference, or phase cancellation. (Fig. 1.11) If these two waves are 180˚ out of phase but have different amplitudes only partial phase cancellation will occur. Partial phase cancellation will reduce the amplitude of the two waves based on the differences in amplitude. The closer the two waves are in amplitude the greater the decrease in amplitude.

Polarity Polarity simply refers to energy relationships between two identical waves. It does not refer to time relationships. In the example below there is no time difference between the two identical waves. They both start at the exact same time. However, there is a polarity difference since these waves are of equal but opposite electrical energy. If these were acoustical waves they would be of equal but opposite air pressure. In the example below the waves are said to have opposite polarities or polarity reversed. Two identical sine waves that have opposite polarity or are phase shifted 180˚create the same result: zero output. However, they create this effect by different means. (180º phase-shifting of waveforms that are more complex then sine waves - or of actual audio tracks - does not result in total cancellation unless the waveforms are symmetrical - e.g., triangle waves and square waves: see below for the complete discussion of phase-shifting effects, and also the section on geometric waveforms.)


Polarity Reversal

Phase Shift and Frequency The amount of phase shift that results from a specific, fixed, time delay varies with frequency. In the recording studio a digital delay processor has the ability to delay any frequencies that are sent to it by a specific time value usually given in milliseconds. The digital delay processor then combines the original non-delayed signal with the delayed copy to create various effects such as echo, phasing, flanging, and chorusing. In the example below, if the four waveforms were delayed 5 milliseconds each signal would be phase shifted by different amounts due to the fact that their periods are of varying length. So when these delayed signals are combined with the non-delayed signals each frequency will have a different phase shift. In the example below you can see that a 5 millisecond delay applied to a 400 Hertz waveform will create a 720˚ phase shift (2 cycles or 360 x 2) while a 200 Hertz wave will be phase shifted 360˚ (1 cycle).

Diagram of a Digital Delay Processor

Static delay time

Dry/Unprocessed Signal

Delayed/Processed Signal

Input Signal

M ix/Output

(Delayed signal held in Shift Register)

(Determines the lenght of delay)Clock Input

A/D ConvertorRequired for processors with

analog inputs

(In a hardware processor this is an analog signal)

D/A ConvertorRequired for processors with

analog outputs

Set with software fields, Midi beat clock, switches, or LFO


Phase Shift and Perception Phase shift created by delay times over 60 milliseconds or so are not really considered phase shift since the time period between these two waves is discernable by the human ear. Once the ear can identify the time period between the original signal and the phased shifted/delayed copy it is considered a type of echo effect, not phase shift. If the copy of the original signal is delayed by less than 30-40 milliseconds and it’s mixed with the original signal you will not actually be able to hear the time difference between the two signals. The time differences will not be perceived as being distinct or individual. Instead they will be perceived as a collective whole much like the eye perceives a single image of a spinning coin consisting of both heads and tails. In both cases the eye and ear have passed their threshold to hear and see distinct reiterations. When a signal is delayed less then 30-40 milliseconds and is combined with the original signal the resultant shift in phase of the various frequencies will create patterns of constructive and deconstructive interference between the two waves. These patterns create what is known as the comb-filter response (see diagram below). These patterns will vary depending on the length of the delay. When the time differences


between two identical waves are less than 30 – 40 milliseconds listeners are actually detecting changes in the comb-filter patterns!

Comb Filter Response

Phase Shifting with Microphones and Speakers Phase shifting can be inadvertently created in the studio when several microphones pick up the same wave at different distances from the source of the wave (see Phase Problems with Microphones below). Each microphone will pick up the wave at a different place in the cycle since they are at two different distances from the sound source. These arrival time differences will create phase differences when the waves are combined together. Some of the differences in phase can create an audible reduction in the amplitude of some of the frequency components in the output of the two microphones when they are combined together (deconstructive interference). Changing the distance of one of the microphones from the sound source will change the phase relationship between the two microphones and may help to reduce the amount of deconstructive interference or comb-filter effect. Also many mixing consoles and outboard microphone preamplifiers will have a phase switch that can be engaged on one of the microphone inputs to change the phase by 180˚. Many times this can help restore amplitude lost due to phase cancellation. This “phase” switch is a real misnomer. It really should be called a polarity switch since it simply changes the electrical properties of the signal through polarity reversal, not through any time delay. The ear is extremely sensitive to phase shift. Even phase shifting of less than 1 ms are perceivable! In fact, the ear can perceive time differences that are as small as .030 microseconds (.03µs)!!!! Two speakers that are used to create a stereo image are very phase dependent. The sound from each speaker must reach the listeners ears in-phase in order for the listener to perceive the sounds accurately. Phase

Below 40 ms. 40 - 60 ms. 60 - 100 ms. Over 100 ms.Diferences in phase. Comb filter response.

Warbling/doubling effect.Blurred reiterations.

F l u t t e r .Very rapid reiterations.

Echo

Delay Times and Listening Responses


differences can affect the frequency response, spatial placement, and depth perception of the sounds emanating from a pair of speakers. Sitting in the “sweet spot” between two speakers is simply being equidistant from each of the speakers so that they will produce the most accurate picture of the audio.

Heterodyning (Amplitude Modulation) When two waves are played together that have the same amplitude and are close to each other in frequency (less than 1 to approximately 20 Hertz) the two frequencies, over a period of time, will sum their amplitudes when they are in phase and cancel their amplitudes when they are out of phase. The number of amplitude summations or "beats" that occur during one second can be calculated by subtracting the lowest frequency from the highest. For example, if you have a frequency of 500 hertz and 505 hertz you will hear 5 beats occurring each second (Fig. 1.12). Heterodyning is used to thicken up a sound to make it more interesting to listen to. For instance, the mid to upper range notes on a piano actually use three strings for each note. Each of the three strings is tuned slightly differently to create the Heterodyning effect. Once the difference in frequency reaches the 15-20 Hz range the listener at some point will begin to discern two distinct frequencies without amplitude fluctuations instead of one frequency with amplitude fluctuations.

30' From Source

25' From Source

110 Hz WaveSound Source

1 Wavelength1 0 '

1 Wavelength1 0 '

1 Wavelength1 0 '

MicrophoneA

MicrophoneB

25' From Source

30' From Source

MicrophoneB

MicrophoneA

55 Hz WaveSound Source

1 Wavelength2 0 '

1/2 Wavelength1 0 '

Phase Problems With Microphones

Mics pick up the same wave 180˚ out of phase. Note that this is one sigle frequency.

90˚

180˚

270˚

90˚

180˚

Mics pick up the same wave 90˚ out of phase. Note that this is one sigle frequency.


Modes Of Vibration Inversely Proportional Law- The frequency (number of cycles per second) of a vibrating string is inversely proportional to its length. In short, as frequency increases, the length of string decreases. Conversely, as the frequency decreases, the length of the string increases. If you play a single guitar string, you can produce progressively higher frequencies by shortening the vibrating length of the string (see diagram below).

The ratio that exists between any two frequencies is called an interval. The interval created when the string vibrates as a whole (open string) and when it vibrates at half its length has a ratio of 1:2. Therefore, if the open string is 440 Hertz then the string vibrating at half its length is 880 Hertz or 440 x 2. This interval is called the octave. Each doubling in frequency produces an octave interval. It represents a fixed ratio, not a specific number of Hertz between frequencies (for example all octaves being 100 Hertz apart). Therefore, it’s logical to deduce that the intervals express a logarithmic relationship.

Neck

Neck

Neck

Bridge

Bridge

Bridge

vibrating length

vibrating length

vibrating length

whole string vibrating

1/2 string vibrating



The graph below shows the relationship between the open string (vibrating as a whole) and other vibrating modes 1/2, 1/3, 1/4 etc. expressed in ratio, interval name, and frequency. These ratios and their resultant intervals are called pure intervals because they exist in natural vibrating systems like string and wind instruments.

The lower number of a ratio expresses the lower frequency while the higher number the higher frequency. The frequencies of 67 Hertz and 134 Hertz (2x67) create the ratio of 1:2. 134 Hertz (2x67) and 201 Hertz (3x67) create the ratio of 2:3. 201 Hertz (3x67) and 268 Hertz (4x67) create the ratio of 3:4, etc. The interval names listed above represent some of the consonant intervals. These intervals have simple ratios and are considered pleasant or neutral sounding. Dissonant intervals are expressed with more complex ratios and are considered more harsh sounding than consonant Intervals. The diminished fifth is a dissonant interval due to its sound and the ratio of 45:32. The formulas below demonstrate how to determine the second frequency of an interval when one is provided. Interval Ratio Up or Down Starting Pitch Formula Perfect 5 3:2 Up (2:3) A-440 440x3÷2=660 Hz Perfect 5 3:2 Down (3:2) A-440 440x2÷3=293.3 Hz Perfect 4 4:3 Up (3:4) A-440 440x4÷3=586.6 Hz Major 3rd 5:4 Down (5:4) A-440 440x4÷5=352 Hz Other Pure Interval/Ratios Minor 2nd 25/24 Major 2nd 9:8 Minor 3rd 6:5 Diminished 5th 45:32 Minor 6th 8/5 Major 6th 5:3 Minor 7th 9:5 Major 7th 15:8

Tuning Systems - Pure and Tempered Intervals/Ratios When pure or just ratios are used to create a tuning system for a musical instrument (how each note of a scale is tuned) each key is tuned to the tonic point of reference. To tune an instrument to the C Major scale for example, you would start with the tonic frequency (C) and then use the pure/just ratios to locate the frequencies of the remaining pitches. Since each scale has it’s own unique point of reference this type of tuning doesn’t work well for music that changes keys. While each scale may sound in tune with itself it’s not necessarily in tune with other keys since they use different points of reference.

frequency: 67 134 201 268 335 fx1 fx2 fx3 fx4 fx5

ratio: 1:2 2:3 3:4 4:5

i n t e r v a l : octave fifth fourth maj. 3rd

string length: whole 1/2 1/3 1/4 1/5


An equal tempered tuning system divides all notes of the chromatic scale into twelve equal parts. Intervals are not derived by pure or just ratios but by using a constant frequency multiple between notes of a chromatic scale (1.059463…). This makes all intervals and all scales equal. Since all half steps are equal in this system we further divide the equal half step into Cents. There are 100 cents in every half step so 50 cents would equal a quarter step. Overtone Series All strings have the ability to vibrate in many different modes simultaneously. Strings can vibrate as a whole (a below), in halves (b below), in thirds (c below), etc. all at the same time. The entire string is moving in each mode but various parts of the string may be moving in different directions as demonstrated below.

A string vibrating in two equal halves (a below) will produce the same frequency that half of the string produces (b below). A string vibrating in thirds will produce the same frequency that 1/3 of the string produces, etc.

String Length is related to frequency inversely: Therefore, F = Frequency, L = string Length.

F x 2 = L÷ 2 and L x 2 = F ÷ 2 etc.

Each whole # multiple of the fundamental frequency represents the next pitch, or mode of vibration, in the harmonic overtone series (see below)

Neck Bridgea .whole string vibrating

Neck Bridgeb .

vibrating in halves

Neck Bridgec .

vibrating in thirds

vibrating length

vibrating length

vibrating length

Neck Bridgea .

vibrating in halves(440 Hertz open string. 880 Hertz when string vibrates in two halves)

Neck Bridge

vibrating length


b .

vibrating length

(440 Hertz open string. 880 Hertz when only half the string vibrates)


The Overtone System While it is possible for a string to vibrate in just one of these modes at a time to create a simple vibration it usually creates a complex vibration resulting from the simultaneous vibration of many modes simultaneously. When a guitar string is plucked the string is vibrating as a whole, in halves, in thirds, fourths, etc. simultaneously! These different modes of vibration are called partials. The first partial or mode of vibration (called the fundamental) generally produces the loudest amplitude and therefore is the pitch we discern. However, even if the fundamental is not the loudest partial you will still perceive the fundamental pitch because of the consistent pattern of partials above the fundamental. In addition, when you calculate the difference between any two adjacent partials they will always give you the fundamental frequency! In short, since the ear recognizes the pattern of partials above the fundamental it perceives the fundamental. In fact it can perceive fundamentals that are completely missing! Partials above the fundamental are called overtones or harmonics since they exist over the fundamental. This simultaneous vibration of multiple modes creates the harmonic overtone series – or just overtone series – that exists naturally in nearly all periodic vibrating systems that deal with pitched sound. Although this system represents an unchanging set of intervallic relationships, not all the partials have to be present, and the amplitude relationships between the partials are infinitely variable. The inexhaustible variety of these two variables produces the many different timbres or colors of pitched sounds.

The Harmonic Overtone Series

Interval: OctaveFifth Fourth

Maj 3rd Min 3rd*Min 3rd? **Maj 2nd?

Partial #:

Frequency in Hertz:

Overtone/Harmonic:

67 134 201 268 335 402 *469 536

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7

Mode of Vibration: Whole Halves Thirds Fourths Fifths Sixths Sevenths Eights

1 yrd 1/2 yrd 1/3 yrd 1/4 yrd 1/5 yrd 1/6 yrd 1/7 yrd 1/8 yrdString Lengths:

Ratios: 1:2 2:3 3:4 4:5 5:6 6:7 7:8

*The interval between the 6th and 7th partial (created by the ratio 6:7) does not yeild a true B flat in the Equal Tempered System of tuning. A true tempered B flat would be created by a ratio of 5:6 which would yield a frequency of 482.4 hz. The interval created by the ratio of 6:7 exists as a pure interval of the overtone series but does not exit in the tempered tuning system we employ.**Since the pure ratio of 6:7 does not create a tempered B flat than the interval from the 7th partial to the 8th is not an interval of a major 2nd in the tempered system. The interval created by the ratio of 7:8 exists only as a pure interval of the overtone series. A tempered major second has a ratio of 8:9.

Fund fx2 fx3 fx4 fx5 fx6 fx7 fx8


Geometric Waveforms The different combinations of present partials and their associated amplitudes create various waveforms. The names of some of the standard geometric waveforms refer of the shapes they create when viewed with an oscilloscope. Geometric waveforms use static or fixed amplitudes for all of the partials. Sine Wave The sine wave has only a single partial. No overtones exist in a sine wave.

Sine Wave

Triangle Wave

A = Amplitude n = partial number


Square Wave

Sawtooth Wave

http://www.audiocheck.net/audiofrequencysignalgenerator_sinetone.php


Harmonic/Inharmonic Spectra If all of the partials in a given waveform are whole-number multiples of the fundamental, the wave is said to have harmonic spectra. (Fig. 7.2) Conversely, if some of the partials have a fractional relationship to the fundamental the wave has inharmonic spectra (Fig. 7.3).


Fourier Theorem, Analysis, and Synthesis In the early 18th century Joseph Sauveau discovered the overtone series. In the early 19th century the French mathematician Fourier developed a theory based on Sauveau's discovery. It stated that any complex periodic vibration could be reduced to a summation of simple sine waves with specific amplitudes. This theory is known as Fourier's Theorem. Analysis done on complex waves that break them down into simple sine waves with intensity/amplitude envelopes is called Fourier Analysis or Fast Fourier Transfer Analysis (see below).


Creating complex waves by adding together different individual sine waves is call Fourier synthesis (Fig. 7.5).


Formant Regions Sympathetic vibration or resonance occurs when a vibrating wave comes in contact with an elastic medium and sets this medium in motion. Some vibrating spaces, due to the material of construction, size or shape, will emphasize (add volume to) one or more frequencies due to sympathetic vibration. When these frequencies are played the vibrating space will begin to sympathetically vibrate, causing an increase in amplitude of those frequencies. These emphasized frequencies are called formant regions. (Fig. 7.7)

Formant regions can also occur due to the absorption coefficient (ability to absorb acoustical energy) of materials in a vibrating space. All materials employed in a vibrating space (whether inside a violin or a concert hall) will have a unique absorption coefficient rating. All ratings are done by frequency range (amount of absorption in a particular frequency band). Some materials will absorb high frequencies while others are best suited for low frequency absorption. The combination of all of the materials in a vibrating space and their unique absorption capabilities will create an overall frequency response with characteristic formant regions. Another cause of formant regions is the overall dimensions and layout of a vibrating space that creates favorable or unfavorable interactions between vibrations in the space. Waves emanating from one part of the vibrating space (incident wave) can reach a boundary and reflect back (reflected wave) in the direction of the incident wave. In some instances the interaction of the incident and reflected waves will create varying degrees of constructive interference that in turn creates a formant region. The interaction of the incident and reflected waves


resulting in constructive interference is called a standing wave. Standing waves are created by parallel surfaces. Standing waves can be reduced or eliminated by avoiding parallel surfaces. Measuring Amplitude - Objective, Empirical Measurement - Power and Intensity The amplitude of a sound is the direct result of the amount of energy used to create the sound. One empirical/objective measurement of amplitude can be accomplished by simply determining the amount of energy/work spent over the period of one second to create the sound. This is a measurement of power.

W = E/T W = Power E = Energy (measured in joules) T = Time (in seconds)

Another way of creating an empirical measurement of amplitude is to determine the amount of energy/work that is dissipated in a vibrating space over the period of one second. This is a measurement of intensity.

I = W/S I = Intensity W = Power (measured in watts) S = surface area (in square meters)

An analogy to these types of measurements can be seen by looking at the inherent power of a light bulb (determined by its construction and the electrical energy applied to it) and its resultant intensity in a room. This intensity relies on the power of the light bulb but also the color of the walls, the size of the space, etc. Therefore, in order for two rooms to have the same intensity of light they must both be of the exact same construction and use the exact same power source. A piano will have inherent power characteristics (determined by it’s construction and the mechanical energy applied by the performer) but the perceived intensity will be determined by the distance from the piano and the construction of the room that the piano is in.

Measuring Amplitude – Human Subjective Perception of Amplitude The subjective perception of amplitude change by the ear is accomplished through the comparative analysis of amplitude A vs. amplitude B. Simply stated: amplitude A is louder than amplitude B. Or, that amplitude is softer than this amplitude. The ear does not perceive amplitude as an empirical or absolute value as it does with pitch/frequency (A-440 Hertz, etc.). But like frequency the ear perceives changes in amplitude in a logarithmic fashion. The ear quantifies amplitude as ratios of power. For instance, a doubling of power is perceived as the same change in amplitude regardless if the sounds are soft or loud: the ear perceives the same amount of amplitude change no matter if the intensity ratio is, 1:2, 2:4, or 5,000:10,000 because each ratio represents a doubling of power or a 1:2 power ratio. A 1:10 power ratio creates the perception of one sound being twice as loud as another. The subjective perception of amplitude is called loudness.

Power Source

Intensity determined by room characteristics

Room Boundaries


The graph below demonstrates how incremental doubling of intensity/power is perceived as a progressive increase in loudness in equal increments.

Amplitude measurements for all audio signals (electrical or acoustic) reflect the subjective way in which we perceive changes in amplitude. This system employs logarithmic power/intensity ratios to provide linear changes in loudness. The increments of amplitude change are called bels named after Alexander Graham Bell. The bel represents a 10:1 power/intensity ratio. A 10:1 power ratio results in a doubling of perceived loudness (this figure is disputed in the literature: loudness perception is affected by many factors and is quite variable and subjective - see below section on the equal loudness curves). Each bel can be subdivided into 10 decibels or dB. The decibel represents the unit of measurement that corresponds roughly to the amount a sound must be raised in level to be heard as louder. Each incremental increase in dB is perceived as an equal change in loudness that is accomplished with a logarithmic change in intensity.

Logarithmic Change in IntensityLinear Loudness Perception

1 : 2

2 : 4

4 : 8

8 :16

16 :32

32 :64

1

2

3

4

5

6The listener perceives a progressive

increase in loudness in equal i n c rements

Equal increments of

loudness

Incremental doubling of

i n tens i ty

The listener perceives a progressive increase in loudness in equal increments

Equal increments of loudness

Incremental doubling of

intensity

Logarithmic Change in Intensity | Linear Loudness Perception


The diagram above shows the logarithmic relationship from one bel to the next. Note that all measurements are using the threshold of hearing as the point of reference. The point of reference is an absolute, empirical measurement (in power or intensity values) that represents the energy required to create the softest sound an average person can hear. This is the value that all other amplitudes are compared to. The diagram below demonstrates the logarithmic ratios and their linear equivalents in the decibel. Note that a doubling of intensity/power (2:1 ratio) will always result in an increase of 3 dB. dB scales and the levels in your DAW: • We know that an increase of 3 dB results in a doubling of Power or Intensity. • However, it requires a 6 dB increase to double Voltages, or to double Sound Pressure Levels. You can observe this when working with metering in Logic or Pro Tools, which typically reference a voltage ratio, not a power or intensity ratio. If you duplicate a solo’d track, the combined output of both the original with the duplicate will register as 6 dB greater on the summing meter, not 3 dB greater. • If you find this confusing, take comfort, you are not alone.

Point of Reference

OBJECTIVE REALITY = INTENSITY/POWERLogrithmic intensity/power is required to

produce linear (equal) loudness relationships

SUBJECTIVE PERCEPTION/LOUDNESSLinear (equal) loudness changes are

perceived between one adjacent bel or decibel and another

10 times the intensity of the point of reference10:1 power ratio

100 times the intensity of the point of reference100:1 power ratio

1,000 times the intensity of the point of reference1,000:1 power ratio

10,000 times the intensity of the point of reference10,000: 1 power ratio

100,000 times the intensity of the point of reference100,000:1 power ratio5 bels

4 bels

3 bels

2 bels

1 bel

dB (decibels)

Each bel is 10x the intensity/power or twice the loudness of the previous bel

10:1 power ratio

10:1 power ratio

10:1 power ratio

10:1 power ratio

10:1 power ratio

(Threshold of hearing)


The ear can perceive an incredibly wide range of amplitude change from the very threshold of hearing up to approximately the intensity ratio of 10,000,000,000,000:1 that represents the threshold of pain. The graph below demonstrates some typical sounds sources, their power/intensity ratio compared to the threshold of hearing, log, and the equivalent bel/decibel. Sound Power/Intensity Ratio Log Bel/dB Threshold of pain 10,000,000,000,000:1 10/13 13/130 Jet taking off from 500 ft. away 1,000,000,000,000:1 10/12 12/120 Loud band in a club 100,000,000,000:1 10/11 11/110 Power saw 10,000,000,000:1 10/10 10/100 Subway 1,000,000,000:1 10/9 9/90 Heavy highway traffic 100,000,000:1 10/8 8/80 Busy street traffic 10,000,000:1 10/7 7/70 Close normal conversation 1,000,000:1 10/6 6/60 Office background noise 100,000:1 10/5 5/50 Quiet Conversation 10,000:1 10/4 4/40 Waiting room in doctor’s office 1,000:1 10/3 3/30 Recording studio ambient level 100:1 10/2 2/20 Breathing 10:1 10/1 1/10 Threshold of hearing 0:0 10/0 0/0 Below are a few power ratio and dB equivalents.

1:10 ratio = 10 dB or 1 Bel = Twice the perceived loudness 1:2 ratio = 3 dB increase

2:1 (half power) ratio = 3 dB decrease

What is a decibel? (link)

Logarithmic Ratios Decibels (db)1

2

3

4

5

6

7

8

9

1 0

1 : 2 (3.01)

1 : 3 (4.77)

1 : 4 (6.02)

1 : 5 (6.98)

(7.78)1 : 6

1 : 7 (8.45)

1 : 8 (9.03)(9.54)1 : 9(10.00)1 : 10

B e l s

1

(Doubling Intensity = 3 db increase )


Psychoacoustics - Equal Loudness Curves In the early 1930s Harvey Fletcher and Wilden Munson discovered that the perception of loudness is related to both the amplitude and the frequency of the sound. The chart below, called an Equal Loudness or Fletcher Munson Curve, represents the actual intensity of various frequencies required for listeners to hear all frequencies with the same loudness (using a 1 k Hertz signal as the point of reference). It’s clear from these loudness curves that we do not perceive the amplitude of all frequencies equally. Low and high range frequencies need to be boosted in level in order for them to be heard as loud as the frequencies around 1 k Hertz. It’s also clear from the curves that as overall intensity increases our perceptions of amplitude/frequency flatten out. Phons, as seen in the curves, represent a subjective loudness level similar to dB. For instance, compare the 20 phons curve to the 100 phons curve. In the 20 phons curve an increase of 30 dB is required at 100 Hertz in order to hear it equally as loud as a 1kHz tone. However, in the 100 phones curve only a 1 dB increase is required at 100 Hertz. Most people listen to recorded music on speakers around 80-85 dB so the curve in this area is an important reference level for all mixing and mastering engineers. If a mix/mastering engineer listens most of the time in the 110 dB range they will tend produce tracks that are light in bass and high frequencies when playback is at the 85 dB standard. Conversely, if a mix/mastering engineer listens most of the time in the 20 dB range they will tend to produce tracks that are heavy in bass and high frequencies when playback is at the 85 dB standard. In short, our perception of bass and high frequencies varies greatly with the overall level of playback. It should also be self evident that low frequency sounds, like basses and kick drums, will represent a major portion of the overall intensity/power of most recordings since they require so much more intensity to compete with mid-range frequencies.

Psychoacoustics – Loudness and Overall Listening Level When overall listening levels are low a greater change in level will be required for us to perceive a Just Noticeable Change or JNC in amplitude as would be perceived at a higher overall listening level. For instance, a 1 dB change in volume may represent the JNC at 80 phons . However, at 20 phons it might require a 4 dB change in volume to represent the JNC.


Psychoacoustics – Loudness and Duration The duration of a sound will have a significant impact on our ability to perceive the loudness of a sound. Sounds that last less than a few tenths of a second (generally under three tenths/30 ms) will sound less loud than the same sound (same intensity) with a longer duration. Psychoacoustics – Loudness and Frequency Changing the intensity of a frequency may also create subjective perceptions by the ear particularly if the frequency in question is high. A high frequency sound played with a lot of intensity will seem higher in pitch to most listeners when compared to the same frequency played at a lower intensity. Conversely, a low sound played with a lot of intensity will seem lower in pitch to most listeners when compared to the same frequency at a lower intensity. This phenomenon is particularly apparent with pure tones. Amplitude/Intensity Envelopes The shape of a sounds amplitude in time is called an amplitude or intensity envelope.

As the diagram above indicates, these shapes when plotted on a graph consist of time values (rate) and amplitude values (level). R1 represents how long it takes the sound to go from no sound to the first amplitude peak (L1). R2 represents the time it takes to go from the first amplitude peak to (L1) to the second amplitude peak (L2). Finally R3 represents the time it takes to go from L2 to no sound at all. In order to create envelope shapes on a synthesizer the rate and level values are used to instruct an amplifier how and when to open and close. If you want the sound above to continue to sustain as long as the note is held down you need to tell the synthesizer that L2 is the sustain level. Now when you depress a key the synthesizer runs through R1, L1, R2, and holds on L2 as long as your finger is down. When you finger is released from the keyboard the synthesizer then runs through R3 and then the sound stops (see Fig. 1.8 on next page for various amplitude/intensity envelopes).

Amplitude

Finger DownFinger Up

Time

R1

L1R2

L2R3

R = RateL = Level


Noise Irregular or random vibrations create noise. Noise can be created on a synthesizer with a noise generator. The noise generator produces all frequencies in the audible spectrum and assigns random amplitudes to these frequencies. Noise generators are defined by color, the color representing the part of the frequency spectrum (as with light) that is emphasized. White noise is defined as having equal energy per unit frequency. This means that there is the same amount of energy between 500 Hz and 501 Hz as there is between 1500 Hz and 1501 Hz. Equal energy per unit frequency means that the noise is spectrally flat. Its' response is the same for all frequencies. Pink noise contains equal energy per octave. For example, there is the same amount of energy between 50 and 100 Hz as is between 7,500 and 15,000 Hz. Therefore, there is more energy in the lower part of the frequency spectrum than the upper part. Pink noise derives its name from the color pink that is at the low end of the light spectrum.

Acoustics Presentation

Documents

Transcript of Acoustics Presentation