Additive Synthesis
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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 ones. 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. 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 ...). There is no permission for this photo.
This organ has a great many pipes, and they're used exactly like an additve synthesis algorithm.
The technique of mixing simple sounds together to get more complex ones 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 wasn't very practical, but it has an important place in the history of electronic and computer music. Installed
This applet demonstrates how sounds are mixed together.
The Computer and Additive Synthesis
A short excerpt from Kenneth Gaburo's 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! (Yeah, right). It's a tribute to Gaburo's 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 we're almost cheating, with all our fancy-schmancy software.
Soundfile .x
While instruments like the pipe organ (and there have been many such experiments in physical additive synthesis over the centuries) 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 them would quickly become prohibitive. Enter the computer!
If there is one thing computers are good at, it’s adding things together. By using digital oscillators instead of actual physical devices, you 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 it's definitely a good thing to know about.
A Simple Additive Synthesis Instrument
Let’s design a simple sound with additive synthesis. A nice (but sort of silly — we’ll get to why in a minute) example is the generation of a square wave.
You can probably imagine what a square wave should look like! We start with just one sine wave, called the fundamental. Then we start adding odd partials to the fundamental, the amplitudes of which are inversely proportional to their partial number. Huh!?! All that means is that the 3rd partial is 1/3 as strong as the 1st, the 5th partial is 1/5 as strong, and so on (remember that the fundamental is the first partial, we could also call it the first harmonic). The picture below shows what we get after adding 7 harmonics. Looks pretty square, doesn’t it!
Now before you get too excited, we should admit that there's an easier way to synthesize squarewaves: just flip from a high sample value to a low sample value every n samples. The lower the value of n, the higher the frequency of the square wave that's being generated. Although this technique is clearer and easier to understand, it has its problems too; directly generating waveforms in this way can cause unwanted frequency aliasing.
Installed This applet lets you add spectral envelopes to a number of partials. This means that you can impose a different amplitude trajectory for each partial, independently making each louder and softer over time. This is really more like the way things work in the real world: partial amplitudes evolve over time, sometimes independently, sometimes in conjunction with other partials (in a phenonenon called common fate). This is called spectral evolution, and it's what makes sounds live.
Figure The Synclavier was an early digital electronic music instrument that used a large oscillator bank for additive synthesis. You can see this on the front panel of this instrment — many of the LEDs indicate specific partials! On the Synclavier (as was the case with a number of other analog and digital instruments) the user can tune the partials, make them louder, even put envelopes on each one.
This applet lets you add sinewaves together at various amplitudes, to see how additive synthesis works.
Applet x
A More Interesting Example
Okay, now how about a more interesting example of additive synthesis? The quality of a synthesized sound can often be improved by varying its parameters (partial frequencies, amplitudes, envelope) over time. In fact, this is essential for any kind of "lifelike" sound, since all naturally occurring sounds do this to some extent.
Sinewave speech is an experimental technique that tries to simulate speech with just a few sinewaves, in a kind of primitive additive synthesis. The idea is to pick the sinewaves (frequencies and amplitudes) carefully. It's an interesting notion, because sinewaves are pretty easy to generate, and if we could get close to "natural" speech with just a few of them... Sinewave speech has long been a popular idea for experimentation by psychologists and researchers. It teaches us a lot about speech, what's important in it, both perceptually and acoustically. These files are used with the permission of Philip Rubin, Robert Remez,
Sinewave Speech
Regular Speech
Sinewave Speech
Regular Speech
Soundfiles These are examples of sentences reconstructed with sinewaves. If you select the left soundwave of each pair you will hear the sinewave speech version of the soundfile on the right.
and Haskins Laboratories.
Attacks, Decays, and time evolution in sounds
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 it’s good for periodic sounds, noisy or chaotic ones are hard to generate.
For instance, creating the steady state part (the sustain) of a flute note is simple with additive synthesis (just a couple of sinewaves), but 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.
There's a worse problem that we'd love to sweep under the ol' psycho-acoustical rug, but we can't. It's great that we know so much about steady-state, periodic, Fourier analyzable sounds. We've really got those nailed! Unfortunately, from a cognitive and perceptual point of view, we really couldn't 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). Bad news for all that additive synthesis software we just started to write!
That's not to say that if we play a triangle wave and a sawtooth wave, we can't tell them apart. We certainly can. But, that really doesn't do us much good in most circumstances. If angry lions roared in square waves, and cute cuddly puppy dogs barked triangle waves, maybe this would be useful, but we have evolved, or learned to hear (don't ask us which, we're not that smart) attacks, decays, and other transients as being more crucial. What we need to be able to synthesize are transients, spectral evolutions, modulations — and lots of 'em.
Another problem is that additive synthesis is very computationally expensive. It’s 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 FM (frequency modulation) synthesis, additive synthesis needs lots of computing power to generate relatively simple sounds. On the other hand, it is conceptually simple, and corresponds very closely to what we know about how sounds are constructed mathematically. For this reason it’s been historically important in computer sound synthesis.
The ability to change a sound's amplitude envelope over time plays an important part in the perceived "naturalness" of the sound.
Figure .x A Typical ADSR (attack, decay, sustain, release, steady-state modulation). This is a standard amplitude envelope shape used in sound synthesis. We saw some of these in earlier sections.
Shepard Tones
One cool use of additive synthesis is in the generation of a very interesting phenomena 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 phenomena is one of the more interesting topics in additive synthesis.
In the 1960s, the experimental psychologist Roger Shepard, along with the 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 its hits an octave, and then it sort of starts over again.You might say pitch wraps around (think of a piano, the C#s 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 which 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 880 Hz as 440 Hz. It's not exactly clear whether this is "hard-wired," learned, or ultimately how important it is, but it's an extraordinary idea and an interesting aural illusion.
We can construct such a circular series of pitches in a laboratory setting, using synthesized tones now commonly referred to as 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 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 is be created that it 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!)
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 which 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") is removed from our perception of the sequence.
Try clicking on the waveform by the keyboard. After you listen to the file once, click again to listen to the frequencies contue on their upward spiral. Used with permission from:
Figure .x
Susan R. Perry, M.A.
Dept. of Psychology
University of Tennessee
The soundfile to the right is an example of the spiraling Shepard Tone Effect.
Figure .x The Shepard tone contains a large amount of octave-related harmonics across the frequency spectrum, all of which rise (or fall) together. The harmonics towards the low and high ends of the spectrum are attenuated gradually while those in the middle have maximum amplification. This creates a spiraling or barber pole effect. (Information from Doepfer Musikelektronik GmbH)
This piece was composed in 1969. This composition is based on a set of continuously rising tones similar to the effect created by Shepard Tones. The compositional process is simple: each glissando, separated by some fixed time interval, fades in from its lowest note, and fades out as it nears the top of its audible range. It is nearly impossible to follow, aurally, the path of any given glissando, so the effect is that the individual tones never reach their highest pitch.
Figure .x For Ann (rising), by James Tenney. Tenney is an important computer music composer and pioneer who worked at Bell Laboratories with Roger Shepard in the early 1960s.