Sinusoidal modelling aims at representing sounds by means of sinusoids plus noise signal models, in order to analyse and synthesis sounds. The algorithms yielding a. Synthesis of Sinusoidal Signals. Question: What types of audio signals can we make up by combining sinusoids of various frequencies, phases, and durations? Harmonic additive synthesis is closely related to the concept of a Fourier series which is a way of expressing a periodic function as the sum of sinusoidal functions. Sine Wave Synthesis. SINEWAVE SYNTHESIS. and provides examples of sinewave synthesis for you to hear. sinusoidal signals preserve the dynamic properties of. Fourier Synthesis. A periodic signal can be described by a Fourier decomposition as a Fourier series, i. e. as a sum of sinusoidal and cosinusoidal oscillations. Signal Processing First Lab 04: Synthesis of Sinusoidal Signals—Music Synthesis Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of. Page 00000096 Sinusoidal Synthesis Optimization Georgios Marentakis and Kristoffer Jensen Department of Datalogy, University of Copenhagen email: babis, krist @diku. 1 Digital Speech processing in Noisy Environments Sinusoidal Analysis/Synthesis Source: T.Quatieri “Discrete-Time Speech Signal Processing”, chapter 9. Page 1. Sinusoidal Synthesis Optimization Georgios Marentakis and Kristoffer Jensen Department of Datalogy, University of Copenhagen email: babis, krist @diku.dk Sinewave synthesis, or sine wave speech, is a technique for synthesizing speech by replacing the formants (main bands of energy) with pure tone whistles.Categories
Harmonic additive synthesis is closely related to the concept of a Fourier series which is a way of expressing a periodic function as the sum of sinusoidal functions with frequencies equal to integer multiples of a common fundamental frequency. These sinusoids are called harmonics. overtones. or generally, partials. In general, a Fourier series contains an infinite number of sinusoidal components, with no upper limit to the frequency of the sinusoidal functions and includes a DC component (one with frequency of 0 Hz ). Frequencies outside of the human audible range can be omitted in additive synthesis. As a result only a finite number of sinusoidal terms with frequencies that lie within the audible range are modeled in additive synthesis.
A waveform or function is said to be periodic if
The Fourier series of a periodic function is mathematically expressed as:
The simplest harmonic additive synthesis can be mathematically expressed as:Inharmonic form
Additive synthesis can also produce inharmonic sounds (which are aperiodic waveforms) in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency. [ 3 ] [ 4 ] While many conventional musical instruments have harmonic partials (e.g. an oboe ), some have inharmonic partials (e.g. bells ). Inharmonic additive synthesis can be described as
Example of inharmonic additive synthesis in which both the amplitude and frequency of each partial are time-dependent.
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Problems listening to this file? See Media helpTime-dependent frequencies
In the general case, the instantaneous frequency of a sinusoid is the derivative (with respect to time) of the argument of the sine or cosine function. If this frequency is represented in hertz. rather than in angular frequency form, then this derivative is divided by . This is the case whether the partial is harmonic or inharmonic and whether its frequency is constant or time-varying.
In the most general form, the frequency of each non-harmonic partial is a non-negative function of time, , yielding
Additive synthesis more broadly may mean sound synthesis techniques that sum simple elements to create more complex timbres, even when the elements are not sine waves. [ 5 ] [ 6 ] For example, F. Richard Moore listed additive synthesis as one of the "four basic categories" of sound synthesis alongside subtractive synthesis. nonlinear synthesis, and physical modelling. [ 6 ] In this broad sense, pipe organs. which also have pipes producing non-sinusoidal waveforms, can be considered as additive synthesizers. Summation of principal components and Walsh functions have also been classified as additive synthesis. [ 7 ]Implementation methods
Modern-day implementations of additive synthesis are mainly digital. (See section Discrete-time equations for the underlying discrete-time theory)Oscillator bank synthesis
Additive synthesis can be implemented using a bank of sinusoidal oscillators, one for each partial. [ 1 ]Wavetable synthesis
In the case of harmonic, quasi-periodic musical tones, wavetable synthesis can be as general as time-varying additive synthesis, but requires less computation during synthesis. [ 8 ] As a result, an efficient implementation of time-varying additive synthesis of harmonic tones can be accomplished by use of wavetable synthesis .Group additive synthesis
Group additive synthesis [ 9 ] [ 10 ] [ 11 ] is a method to group partials into harmonic groups (of differing fundamental frequencies) and synthesize each group separately with wavetable synthesis before mixing the results.Inverse FFT synthesis
An inverse Fast Fourier transform can be used to efficiently synthesize frequencies that evenly divide the transform period or "frame". By careful consideration of the DFT frequency-domain representation it is also possible to efficiently synthesize sinusoids of arbitrary frequencies using a series of overlapping frames and the inverse Fast Fourier transform. [ 12 ]Additive analysis/resynthesis
Sinusoidal analysis/synthesis system for Sinusoidal Modeling (based on McAulay & Quatieri 1988. p. 161) [ 13 ]
It is possible to analyze the frequency components of a recorded sound giving a "sum of sinusoids" representation. This representation can be re-synthesized using additive synthesis. One method of decomposing a sound into time varying sinusoidal partials is Fourier Transform -based McAulay-Quatieri Analysis. [ 14 ] [ 15 ]
By modifying the sum of sinusoids representation, timbral alterations can be made prior to resynthesis. For example, a harmonic sound could be restructured to sound inharmonic, and vice versa. Sound hybridisation or "morphing" has been implemented by additive resynthesis. [ 16 ]
Additive analysis/resynthesis has been employed in a number of techniques including Sinusoidal Modelling, [ 17 ] Spectral Modelling Synthesis (SMS), [ 16 ] and the Reassigned Bandwidth-Enhanced Additive Sound Model. [ 18 ] Software that implements additive analysis/resynthesis includes: SPEAR, [ 19 ] LEMUR, LORIS, [ 20 ] SMSTools, [ 21 ] ARSS. [ 22 ]Products
Additive re-synthesis using timbre-frame concatenation:
Concatenation with crossfades (on Synclavier)
In Helmholtz's time, electronic amplification was unavailable. For synthesis of tones with harmonic partials, Helmholtz built an electrically excited array of tuning forks and acoustic resonance chambers that allowed adjustment of the amplitudes of the partials. [ 47 ] Built at least as early as in 1862, [ 47 ] these were in turn refined by Rudolph Koenig. who demonstrated his own setup in 1872. [ 47 ] For harmonic synthesis, Koenig also built a large apparatus based on his wave siren. It was pneumatic and utilized cut-out tonewheels. and was criticized for low purity of its partial tones. [ 41 ] Also tibia pipes of pipe organs have nearly sinusoidal waveforms and can be combined in the manner of additive synthesis. [ 41 ]
In 1938, with significant new supporting evidence, [ 48 ] it was reported on the pages of Popular Science Monthly that the human vocal cords function like a fire siren to produce a harmonic-rich tone, which is then filtered by the vocal tract to produce different vowel tones. [ 49 ] By the time, the additive Hammond organ was already on market. Most early electronic organ makers thought it too expensive to manufacture the plurality of oscillators required by additive organs, and began instead to built subtractive ones. [ 50 ] In a 1940 Institute of Radio Engineers meeting, the head field engineer of Hammond elaborated on the company's new Novachord as having a “subtractive system” in contrast to the original Hammond organ in which “the final tones were built up by combining sound waves”. [ 51 ] Alan Douglas used the qualifiers additive and subtractive to describe different types of electronic organs in a 1948 paper presented to the Royal Musical Association. [ 52 ] The contemporary wording additive synthesis and subtractive synthesis can be found in his 1957 book The electrical production of music. in which he categorically lists three methods of forming of musical tone-colours, in sections titled Additive synthesis. Subtractive synthesis. and Other forms of combinations. [ 53 ]
A typical modern additive synthesizer produces its output as an electrical. analog signal. or as digital audio. such as in the case of software synthesizers. which became popular around year 2000. [ 54 ]Timeline
The following is a timeline of historically and technologically notable analog and digital synthesizers and devices implementing additive synthesis.
Research implementation or publication
Company or institution
Synthesizer or synthesis device
New England Electric Music Company
The first polyphonic, touch-sensitive music synthesizer. [ 56 ] Implemented sinuosoidal additive synthesis using tonewheels and alternators. Invented by Thaddeus Cahill .
7. Introduction to Synthesis
Basic Tutorial In Sound Design
To c reate your own Synth Sounds, you will need a basic understanding of Sound Design or Synthesis.
There are a variety of types of Synthesis, including�
A detailed description of each form of Synthesis is outside the scope of the book � For further reading, Google it. � The fundamental areas of Synthesis are explained below.
Introduction to Additive Synthesis
In it�s simplest form, Additive Synthesis is the addition of sine waves at various frequencies to create more complex sounds. � In theory, any sound can be reproduced using up to an infinite number of sine waves. � Square Saves, Saw Waves and other more complex waves are generated by adding Sine waves of different harmonic frequencies together. � A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the fundamental frequency is f, the harmonics have frequencies f, 2f, 3f, 4f, etc.
Fig 1: One sine wave at 440 Hz.
Fig 2: Three addition sine waves are added at double 880 Hz, treble 1,320 Hz and Quadruple 1,760 Hz the original frequency.
Fig 3: More harmonic frequencies of the Sine Wave are added to create a Square Wave.
Introduction to Subtractive Synthesis
The most common form of Synthesis in most VSTs is Subtractive Syntheses. � Filters are used to subtract frequencies from different types of waves, e.g. Sine, Saw Tooth, Square to create a variety of sounds. � It is a 4 stage process to create an infinite array of any sound imaginable
Subtractive Synthesis = 1) VCO + 2) ADSR + 3) Filter + 4) DSP
1) VCO = Voltage Controlled Oscillator e.g. Sine, Saw or Square wave.
Le terme Additive synthesis est cité dans le Wikipedia de langue anglaise. Il est défini comme suit:
Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.
The timbre of musical instruments can be considered in the light of Fourier theory to consist of multiple harmonic or inharmonic partials or overtones. Each partial is a sine wave of different frequency and amplitude that swells and decays over time .
Ceci est un extrait de l'article Additive synthesis de l'encyclopédie libre Wikipedia. La liste des auteurs est disponible sur Wikipedia.
Synthèse sonore additive
Le terme Synthèse sonore additive du Wikipedia en langue allemande correspond au terme Additive synthesis issu du Wikipedia en langue anglaise. Il est défini comme suit :
La synthèse sonore additive consiste à créer un son en superposant des signaux sinusoïdaux harmoniques.
Ceci est un extrait de l'article Synthèse sonore additive de l'encyclopédie libre Wikipedia. La liste des auteurs est disponible sur Wikipedia.
Images pour Additive synthesis
Entrées blog pour le terme
Every pitched sound can be though of as a collection of individual sine waves at frequencies related to the fundamental. Gordon Reid introduces a powerful method of synthesis that works by manupulating these individual harmonics. • Read all parts
Fun with Additive synthesis and interfaces posted by me on 2013.10.01, under Supercollider 01: When I started learning SuperCollider, I was motivated more by algorithmic and generative music. I was (and still am) fascinated by bits of code that can go on forever, and create weird soundscapes.
A laboratory of media, art, and technology
If you have been around computer-based music production for any length of time, VirSyn will be a very familiar name as the company are well-known for their range of plugin virtual instruments and effects for desktop computers. They are also carving out a similar collection of iOS music apps and I have reviewed their Harmony […]Click here for the original post from Music App Blog.
Cube Synth review – Additive synthesis for iOS from VirSyn. Music App Blog - reviews, resources and news for the iOS musician.
In working a great deal with Additive synthesis. I have discovered why I believe that Additive synthesis is ineffective based on a flawed premise.First, let me explain that I have the mathematical background to understand the mathematical basis for Additive synthesis so this is coming as much from a technical standpoint as a musical one.The basis of Additive synthesis and what is often called the "frequency domain" of sound is based on a mathematical idea called the fourier transform:http://en.wikipedia.org/wiki/Fourier_transformThe basic idea of this transform is that an periodic waveform can be broken down into two infinite series of sin waves (or as they are called in acoustics) overtones each a multiple higher than the fundamental (the pitch that is heard). Each of the two series are seperated by 90 degrees.
This tutorial will show you how you can create an additive synthesizer using a combinator in Reason. Additive synthesis is one of the less common forms of synthesis in use today, but it can be used to create some great sounds. Furthermore, following this tutorial is a great way to improve your understanding of some of the fundamental principals of sound and synthesis in general. | Difficulty: Beginner; Length: Short; Tags: Audio Production, Reason
For this blog entry, I felt a good topic to cover would be Additive synthesis and its implementation in C. Based on Fourier theory, Additive synthesis concerns combining simple periodic waveforms (such as sine or cosine waves) together that result in more complex sounds. Put another way, any waveform can be expressed as a sum…
Computer Music: Musc 216
ADDITIVE SYNTHESIS (also called Fourier Synthesis ) is a type of synthesis which produces a new sound by adding together two or more audio signals. The "classical" form of additive synthesis, still used on some synthesizers, may be called "harmonic synthesis." Here the sources added together are simple waves (for example, sine waves) and are in the simple frequency ratios of the harmonic series. The resultant absolute amplitude is the sum of the amplitudes of the individual signals. The resultant frequency is the sum of the individual frequencies taking into account the effects of constructive and destructive interference . This is potentially a very powerful technique, as it was shown well over 200 years ago by the Frenchman, Francois Baptiste Joseph Fourier that any periodic sound (i.e. pitched sound) can be represented by the sum of simple sine waves. In practice, however, the approach can be extremely time-consuming since often hundreds of harmonics may be used to create a complex sound. Also see PARTIAL and OVERTONE .
If you have a MACINTOSH COMPUTER it might be helpful to download and run the application WAVEMAKER. This Additive Synthesis Emulator was developed by David Sebald at the Institute for Music Research . The University of Texas at San Antonio.
Visit this link to read a description of WAVEMAKER .
Click here to download the application (1.2 Mgbytes) (In Zipped format. You will need to Un-Zip it.)
Click here to download ZIPIT.
Check out these links:
Additive Synthesis (Computer and Music, Dartmouth University -- an E-Textbook on Computer Music)
An Introduction to Additive Synthesis (from Synth Secrets, a series of articles from Sound on Sound -- an Electronic Music Magazine_
I'm attempting to write a general additive synthesis c program that will generate a complex sinusoid created from a sequence of pure sine waves of arbitrary frequency following a single envelope. The input file will be something of this nature
I want my program to generate a WAV file up to the last breakpoint where all the sine waves will be generated at the given frequencies, scaled to the % contribution as listed, and added together to make the final sound
I've attempted to try some of the c programming out but I'm not naturally a C programming so this is what I've done so far:
However, I'm not really sure if I'm doing this right at all. Any ideas on how I can fix this and proceed?
asked Oct 22 '12 at 14:45
As I've started noticing more and more little details, the comments start getting annoying to read (but make sure you still answer my compiling Hello World question), so I've coalesced them into this answer and will update it as I see more:
answered Oct 22 '12 at 19:36
something was wrong with my gcc but that's fixed so i was able to run the hello world question. i've fixed up a few things but i'm still getting tons of errors – NuNu Oct 24 '12 at 4:43
is there an email where i can email you at because this comment thread might go back and forth for a while – NuNu Oct 24 '12 at 4:45
sure, send an email to this: <firstname.lastname@example.org> no brackets obviously, and I'll give you better contact info through there - it's a high profile website, don't want to be posting my real email haha. It'll be open for 24 hrs. – im so confused Oct 24 '12 at 14:28
As previously discussed in Section 1. sine waves can be considered the building blocks of sound. In fact, it was shown in the 19th Century by the mathematician Joseph Fourier that any periodic function can be expressed as a series of sinusoids of varying frequencies and amplitudes. This concept of constructing a complex sound out of sinusoidal terms is the basis for additive synthesis, sometimes called Fourier synthesis for the aforementioned reason. In addition to this, the concepts of additive synthesis have also existed since the introduction of the organ, where different pipes of varying pitch are combined to create a sound or timbre.
Figure 6.1. Additive synthesis block diagram.
A simple block diagram of the additive form may appear like in Fig. 6.1. which has a simplified mathematical form based on the Fourier series:
Constructing common harmonic waveforms in additive synthesis Edit
Where a 0 <\displaystyle a_<0>> is an offset value for the whole function (typically 0), a n <\displaystyle a_
Figure 6.2. The first four terms of a square wave constructed from sinusoidal components (partials).
If we know the amplitude weightings and frequency components of the first x <\displaystyle x> sinusoidal components or partials of a complex waveform, we can reconstruct that waveform using an additive system with x <\displaystyle x> oscillators. The popular waveforms square, sawtooth and triangle are harmonic waveforms because the constituent sinusoidal components all have frequencies that are integer multiples of the fundamental. The property that distinguishes them in this form is that they all have unique amplitude weightings for each sinusoid. Fig. 6.2 demonstrates the appearance of the time-domain waveform as a set of sines at unique amplitude weightings are added together; in this case the form begins to approximate a square wave, with the accuracy increasing with each added partial. Note that to construct a square wave we only include odd numbered harmonics- the amplitude weightings for a 2 <\displaystyle a_<2>> . a 4 <\displaystyle a_<4>> . a 6 <\displaystyle a_<6>> etc. are 0. Below is a table that demonstrates the partial amplitude weightings of the common waveshapes:
A conclusion you may draw from Fig. 6.2 and the table is that it requires a large amount of frequency partials to create a waveform that closely approximates the idealised mathematical forms of the waveforms introduced in Section 5. For this reason, it should be apparent that additive synthesis techniques are perhaps not the best method for producing these forms. The strengths of additive synthesis lie in the fact that we can exert control over every partial component of our sound, which can produce some very intricate and wonderful results. With the constant modification of the frequency and amplitude values of each oscillator, the possibilities are endless. Some examples of ways to control the weightings and frequencies of each component oscillator are illustrated:
There is, however, the major consideration of computational power, however - complex sounds may require many oscillators all operating at once which will put major demand on the system in question.Additive resynthesis Edit
In Section 1 it was mentioned that just as it is possible to construct waveforms using additive techniques, we can analyse and deconstruct waveforms as well. It is possible to analyse the frequency partials of a recorded sound and then resynthesize a representation of the sound using a series of sinusoidal partials. By calculating the frequency and amplitude weighting of partials in the frequency domain (typically using a Fast Fourier transform), an additive resynthesis system can construct an equally weighted sinusoid at the same frequency for each partial. Older techniques rely on banks of filters to separate each sinusoid; their varying amplitudes are used as control functions for a new set of oscillators under the user's control. Because the sound is represented by a bank of oscillators inside the system, a user can make adjustments to the frequency and amplitude of any set of partials. The sound can be 'reshaped' - by alterations made to timbre or the overall amplitude envelope, for example. A harmonic sound could be restructured to sound inharmonic, and vice versa.
Additive synthesis refers to a number of related synthesis techniques, all based on the idea that complex tones can be created by the summation, or addition, of simpler tones. As we saw in Chapter 3, it is theoretically possible to break up any complex sound into a number of simpler ones, usually in the form of sine waves. In additive synthesis, we use this theory in reverse.
Figure 4.1 Two waves joined by a plus sign.
Figure 4.2 This organ has a great many pipes, and together they function exactly like an additive synthesis algorithm.
Each pipe essentially produces a sine wave (or something like it), and by selecting different combinations of harmonically related pipes (as partials), we can create different combinations of sounds, called (on the organ) stops. This is how organs get all those different sounds: organists are experts on Fourier series and additive synthesis (though they may not know that!).
The technique of mixing simple sounds together to get more complex sounds dates back a very long time. In the Middle Ages, huge pipe organs had a great many stops that could be "pulled out" to combine and recombine the sounds from several pipes. In this way, different "patches" could be created for the organ. More recently, the telharmonium, a giant electrical synthesizer from the early 1900s, added together the sounds from dozens of electro-mechanical tone generators to form complex tones. This wasnt very practical, but it has an important place in the history of electronic and computer music.
This applet demonstrates how sounds are mixed together.The Computer and Additive Synthesis
While instruments like the pipe organ were quite effective for some sounds, they were limited by the need for a separate pipe or oscillator for each tone that is being added. Since complex sounds can require anywhere from a couple dozen to several thousand component tones, each needing its own pipe or oscillator, the physical size and complexity of a device capable of producing these sounds would quickly become prohibitive. Enter the computer!
Excerpt from Kenneth Gaburos composition "Lemon Drops"
A short excerpt from Kenneth Gaburos composition "Lemon Drops," a classic of electronic music made in the early 1960s.
This piece and another extraordinary Gaburo work, "For Harry," were made at the University of Illinois at Urbana-Champaign on an early electronic music instrument called the harmonic tone generator, which allowed the composer to set the frequencies and amplitudes of a number of sine wave oscillators to make their own timbres. It was extremely cumbersome to use, but it was essentially a giant Fourier synthesizer, and, theoretically, any periodic waveform was possible on it!
Its a tribute to Gaburos genius and that of other early electronic music pioneers that they were able to produce such interesting music on such primitive instruments. Kind of makes it seem like were almost cheating, with all our fancy software!
A Simple Additive Synthesis Sound
If there is one thing computers are good at, its adding things together. By using digital oscillators instead of actual physical devices, a computer can add up any number of simple sounds to create extremely complex waveforms. Only the speed and power of the computer limit the number and complexity of the waveforms. Modern systems can easily generate and mix thousands of sine waves in real time. This makes additive synthesis a powerful and versatile performance and synthesis tool. Additive synthesis is not used so much anymore (there are a great many other, more efficient techniques for getting complex sounds), but its definitely a good thing to know about.
These soundfiles are examples of sentences reconstructed with sine waves. Soundfile 4.2 is the sine wave version of the sentence spoken in Soundfile 4.3, and Soundfile 4.4 is the sine wave version of the sentence spoken in Soundfile 4.5.
Sine wave speech is an experimental technique that tries to simulate speech with just a few sine waves, in a kind of primitive additive synthesis. The idea is to pick the sine waves (frequencies and amplitudes) carefully. Its an interesting notion, because sine waves are pretty easy to generate, so if we can get close to "natural" speech with just a few of them, it follows that we dont require that much information when we listen to speech.
Sine wave speech has long been a popular idea for experimentation by psychologists and researchers. It teaches us a lot about speechwhats important in it, both perceptually and acoustically.
These files are used with the permission of Philip Rubin, Robert Remez,
and Haskins Laboratories.
As weve said, additive synthesis is an important tool, and we can do a lot with it. It does, however, have its drawbacks. One serious problem is that while its good for periodic sounds, it doesnt do as well with noisy or chaotic ones.
For instance, creating the steady-state part (the sustain) of a flute note is simple with additive synthesis (just a couple of sine waves), but creating the attack portion of the note, where there is a lot of breath noise, is nearly impossible. For that, we have to synthesize a lot of different kinds of information: noise, attack transients, and so on.
And theres a worse problem that wed love to sweep under the old psychoacoustical rug, too, but we cant: its great that we know so much about steady-state, periodic, Fourier-analyzable sounds, but from a cognitive and perceptual point of view, we really couldnt care less about them! The ear and brain are much more interested in things like attacks, decays, and changes over time in a sound (modulation ). Thats bad news for all that additive synthesis software, which doesnt handle such things very well.
Thats not to say that if we play a triangle wave and a sawtooth wave, we couldnt tell them apart; we certainly could. But that really doesnt do us much good in most circumstances. If angry lions roared in square waves, and cute cuddly puppy dogs barked in triangle waves, maybe this would be useful, but we have evolvedor learned to hear attacks, decays, and other transients as being more crucial. What we need to be able to synthesize are transients, spectral evolutions, and modulations. Additive synthesis is not really the best technique for those.
Another problem is that additive synthesis is very computationally expensive. Its a lot of work to add all those sine waves together for each output sample of sound! Compared to some other synthesis methods, such as frequency modulation (FM) synthesis, additive synthesis needs lots of computing power to generate relatively simple sounds.
But despite its drawbacks, additive synthesis is conceptually simple, and it corresponds very closely to what we know about how sounds are constructed mathematically. For this reason its been historically important in computer sound synthesis.
Figure 4.5 A typical ADSR (attack, decay, sustain, release) steady-state modulation. This is a standard amplitude envelope shape used in sound synthesis.
The ability to change a sounds amplitude envelope over time plays an important part in the perceived "naturalness" of the sound.Shepard Tones
One cool use of additive synthesis is in the generation of a very interesting phenomenon called Shepard tones. Sometimes called "endless glissandi," Shepard tones are created by specially configured sets of oscillators that add their tones together to create what we might call a constantly rising tone. Certainly the Shepard tone phenomenon is one of the more interesting topics in additive synthesis.
In the 1960s, experimental psychologist Roger Shepard, along with composers James Tenney and Jean-Claude Risset, began working with a phenomenon that scientifically demonstrates an independent dimension in pitch perception called chroma, confirming the circularity of relative pitch judgments.
What circularity means is that pitch is perceived in kind of a circular way: it keeps going up until it hits an octave, and then it sort of starts over again. You might say pitch wraps around (think of a piano, where the C notes are evenly spaced all the way up and down). By chroma, we mean an aspect of pitch perception in which we group together the same pitches that are related as frequencies by multiples of 2. These are an octave apart. In other words, 55 Hz is the same chroma as 110 Hz as 220 Hz as 440 Hz as 880 Hz. Its not exactly clear whether this is "hard-wired" or learned, or ultimately how important it is, but its an extraordinary idea and an interesting aural illusion.
We can construct such a circular series of pitches in a laboratory setting using synthesized Shepard tones. These complex tones are comprised of partials separated by octaves. They are complex tones where all the non-power-of-two numbered partials are omitted.
These tones slide gradually from the bottom of the frequency range to the top. The amplitudes of the component frequencies follow a bell-shaped spectral envelope (see Figure 4.6) with a maximum near the middle of the standard musical range. In other words, they fade in and out as they get into the most common frequency range. This creates an interesting illusion: a circular Shepard tone scale can be created that varies only in tone chroma and collapses the second dimension of tone height by combining all octaves. In other words, what you hear is a continuous pitch change through one octave, but not bigger than one octave (that’s a result of the special spectra and the amplitude curve). It’s kind of like a barber pole: the pitches sound as if they just go around for a while, and then they’re back to where they started (even though, actually, they’re continuing to rise!).
Figure 4.10 Bell-shaped spectral envelope for making Shepard tones.
Shepard wrote a famous paper in 1964 in which he explains, to some extent, our notion of octave equivalence using this auditory illusion: a sequence of these Shepard tones that shifts only in chroma as it is played. The apparent fundamental frequency increases step by step, through repeated cycles. Listeners hear the pitch steps as climbing continuously upward, even though the pitches are actually moving only around the chroma circle. Absolute pitch height (that is, how "high" or "low" it sounds) is removed from our perception of the sequence.