By James Clark

Additive synthesis is a technique which builds sounds from the bottom up, by incrementally adding simple waveforms together to achieve the desired results. Contrast this with the more usual subtractive synthesis approach, where a complex waveform is incrementally whittled away to produce the wanted sound. To use an artistic analogy, subtractive synthesis is like sculpting marble - the marble is chipped away to reveal the statue beneath, whereas additive synthesis is like sculpting with clay, where the clay is molded and clumped together to create the desired shape.

Additive synthesis can be used to very accurately model almost any musical instrument, given enough computational resources. Computational resources are limited on the Nord Modular, however, implying that perfect emulations of instruments will not be achievable. But, very good results can be obtained in some cases!

Besides the high demand on computational resources, additive synthesis has some implementational quirks which can affect the quality of the synthesized sound. One of these arises from the fact that a large number of sound sources are being added together. Since each of these sound sources has some noise, adding them together will neccessarily increase the noise level. Compare this to the situation in subtractive synthesis where the output noise can be less than the noise level of the input waveform, due to the filtering. Thus, instruments created using additive synthesis can be noisy. Another factor to consider is the relative fragility of the harmonic structure of the additive sound. If the user has some control over some aspect of the additive process, say in the amplitude or frequency of some overtone, the complex harmonic structure that provides a certain type of sound can be destroyed, leading to a thin sound. In fact, one of the main criticisms of additive synthesizers is that they produce rather thin sounds. This is not an problem with additive synthesis per se - additive synthesis does have the capability to make rich powerful sounds. The problem is that the characteristics of the overtones must be carefully and precisely set in order to achieve such sounds. It may be difficult to do this just by manually adjusting the sound parameters. Resynthesis based on mathematical analysis of the target sound is usually required to obtain good results. Another problem with additive synthesis is that transient sounds are very difficult to synthesize. This is because transients require a large number of rapidly varying overtones to obtain accurate reconstructions. The phase relationships between the various oscillators must also be carefully controlled to get a sharp attack. Even though the harmonic structure of a sound does not depend on the relative phases of the individual oscillators (except in rare cases of complete cancellation), temporal events such as rapid attacks or decays are very much dependent on the oscillators phases. Noisy instruments, such as drums, flutes, and cymbals, are also hard to synthesize, again due to the large number of overtones required. One can overcome this difficulty somewhat by modulating the overtone frequencies and amplitudes with noise signals. This has the effect of spreading the sinewave frequency, which is itself just an impulse in the frequency domain, to a broader range of frequencies. In general, however, additive synthesis is at its best when synthesizing sounds that are quasi-periodic.

The basic additive synthesis patch is shown below.

Figure 1. A basic patch that can be used as a template for additive designs (J. Clark)

Having a separate frequency and amplitude envelope for every sinusoid in the set of overtones is very resource intensive. This is a serious problem in systems like the Nord Modular in which these resources are limited. One way in which to increase efficiency is to group sets of overtones and associate just a single amplitude/frequency envelope pair to each group. This grouping makes the design process more difficult, as the designer must now identify those overtones which have similar frequency and amplitude dynamics. One commonly used grouping technique is to identify groups consisting of overtones that are harmonically related. The advantage of such groups is that they can be generated using single, filtered, non-sinusoidal, waveforms. This approach can be thought of as a hybrid of additive synthesis and subtractive synthesis, where the subtractive process controls the spectral properties of the individual groups, and the additive process combines different groups to obtain the overall desired result.

Synthesizing Transients and Random Variation

One can modulate the frequency or the phase of the partials with random signals.

Additive synthesis may be used to great effect to synthesize a single type of sound created by an instrument, but has difficulty in synthesizing changing sounds. Most musical instruments are expressive, which means that their sounds can be altered by the performer. Many synthesis techniques, such as FM synthesis, can be controlled with a few parameters. These parameters affect the sound in a global fashion. In additive synthesis, the parameters determining the sound are the partial amplitudes, phases, and frequencies. Changing a single one of these will have little effect on the sound, unlike the changing of a single parameter in an FM synthesised instrument. In order to get an expressive change in the sound of an additive synthesized instrument, a large number of parameters need to be changed at once - a difficult task for the performer!

A solution to the problem of adding expressiveness to additive synthesis is morphing, where the performer controls a morph, or smooth transition, between two different sets of additive synthesis parameters. One could also use an approach similar to wave-terrain synthesis, in which the performer specifies a one-dimensional trajectory or curve in the high-dimensional space of additive synthesis parameters. Different points in this space correspond to different sounds, and in this way the performer can move from one sound to the next.

A patch which implements such a morphing approach is shown below. It uses key velocity and aftertouch as independent morphing signals.

A subharmonic is a sound component with a frequency that is an integer fraction (rather than an integer multiple) of the fundamental. The subharmonic series is the inverse of the harmonic series, that is, it is comprised of frequencies of F0, F0/2, F0/3, F0/4, F0/5... where F0 is the frequency of the fundamental.

Very "old school" effect that was used to fatten one VCO synths. The usually way it was implemented was to simply square the main waveform (actual just use the squarewave out) and drive a digital counter which would divide by 2, 4, 8, and 16. Most of the time the divide by 2 was just used. The resulting "sub" wave(s) where then mixed in to taste. On the NM this is trivial to implement using SLAVE OSC as they have built in frequency ratioing that allows sub-harmonic ratios, both integer and non integer which can also0 produce some interesting effects. The most commonly used subwaveforms though have only odd harmonics. Sines are good to add for a deep bass bottom, but squarewaves are also particularly effective and as pointed out above are the more "authentic" effect. Now, this subharmonic stuff may have its origin in an unproved psychacoustic effect related to the concept of beats (not rhythm, but rather the physical phenomenon of beating waves), which shows that for any two waves playing simultaneously there are extra harmonics perceived although analysis with an oscilloscope shows that they're not actually there. It's an aural illusion. These extra "harmonics" are summation and difference tones and they're easily calculated. Obviously, subharmonics fit precisely under the difference tone category. Supposedly, when playing two sines a major third apart, a third sine is perceived two octaves under the lowest of the two -- because the difference in frequency of two tones a major third apart, in just temperament, is 1/4 of the frequency of the lowest tone. If you play with more complex waves, there's a whole array of this type of phenomena occurring, since all partials of one wave are interacting with all partials of the other, besides interacting among themselves.

Your discussion on the "phantom" harmonic is generally true but I think the phenomena has more to do with the ear expecting a harmonic series then upon "beat frequencies" When you play the the major third you are actually generating the upper harmonics of an implied harmonic series (1, 2, 3...). The ear "expects" this to mimic things from nature which tend to not have missing fundamentals so it fills in the blanks with the phantom 2 octaves below. At least this is my understanding of the phenomena. Regardless of the reason we both agree that the ear likes to invent subharmonics.

Indeed, we totally agree. But the difference tone supposedly heard has the same frequency as the amplitude frequency of the beat--the beat tone itself being the average of the tones involved. The first subharmonic (f/2) can be generated by some loudspeaker cones.

Which Oscillator to Use?

The slave sinewave oscillator is good for additive synthesis for a number of reasons:

- It produces a single sinusoidal harmonic.

- It is the single oscillator that uses the least DSP cycles (3%), and uses just marginally more DSP that 1/6 of the sine-bank.

- It has an FM input, which can be used to provide dynamic variation of the pitch of the partial. Use this carefully if you are also using oscillator sync.

- It has an AM input, which can be used to control the amplitude of the partial.

The slave FM sinewave oscillator is also very good for additive synthesis. It is slightly more expensive in terms of DSP usage than the non-FM slave sinewave oscillator (3.2% vs. 3.0%) but has one important advantage, that is, it has oscillator sync. This can be used to ensure that all harmonic partials will have a fixed relative phase. Don't use oscillator sync on non-harmonic partials, as this will introduce extra harmonics which will be hard to control (of course, these extra harmonics might just give you that sound you are looking for!). The FM sine slave oscillator also has an FM input, which can be used to provide dynamic variation of the pitch of the partial. Use this carefully if you are also using oscillator sync, as the FM then will not cause a change in pitch, but will create extra harmonics.

The number of oscillators that we can use in a Nord Modular additive synthesis patch is limited by the available DSP resources. But what if your grand scheme for the greatest patch ever requires more oscillators than this? Well, one way to get a larger number of oscillators is to go cheap and dirty and use LFO modules as the partial generators. An example is shown in the patch below, which uses 64 slave LFOs. You could probably squeeze in more oscillators, but I think you get the point. Triangle oscillators are used as they use the least DSP cycles of any of the LFO slave oscillators. Triangle waves have some low-amplitude higher harmonics, which limits the types of sounds that can be acehived. This slight drawback is insignificant in face of the other drawbacks of using the LFO slave oscillators - the aliasing that is present at higher frequencies, and the limited control one has over the frequency of the harmonics. This patch seems best for generating noisy clangourous sounds! One might question the use of audio mixers in this patch. Wouldn't it be cheaper to use control mixers? Yes, it would, but it would not make any difference to the overall number of oscillators or polyphony that we can achieve, since this is set by the zero-page usage rather than by DSP cycles. And using 8-input audio mixers is much more convenient and simpler to wire up than the 2-input control mixers that are available. You need 7 of these control mixers to construct an 8-input mixer, and the amount of wiring is considerable. Note that group synthesis is used in this patch - the oscillators are partitioned into groups of eight, and each group is amplitude modulated by a single envelope generator.

Figure 1. Additive synthesis patch using LFOs as partial sources (J. Clark).

On AH people were asking about the maximum number of oscillators you can get on the Nord Modular, so I propose a challenge:

- what is the largest number of oscillators in a single patch?

- what is the largest number of oscillators in an unexpanded NM To get things rolling?

One uses lfo's and gets 35 oscillators but only uses 25% dsp so you can get 16x35=560 oscillators total (although I only get 8 voices for some reason) You might think lfos are cheating, so the second patch uses the sinebank oscillatorsand gets 33 oscillators but uses up 98% dsp. So one only gets 4 voices and 132 oscillators total. Can anyone beat this? (I am also including a patch which uses logic elements to make oscillators just to remind you to think sideways once in a while).

Well JJ My modular comes in a few days...so I will give it a shot... BUT...just for the record... I AM NOT buying this for polyphony... I would much rather have 100 lfos and 1 audio oscillator...than the other way around I look at this like a trombone, my first instrument, where you play each single note with MAXIMUM expression... I'll take that anyday over 128 stale samples, simultaneously pissing their way thru acrappy orchestration.

This might be a stupid question, since you have a nord and (until tomorrow, according to reuse it) I don't.. but isn't this a moot point? I seem to remember this conversation onAH long before the nord came out.. something about, after 4-6 oscillators you didn't really hear much of a difference... something about phase cancellations (although, I don't see why the phase alignments would have to be off.. maybe this was a particularly analogue thing).. ohh well.. I agree with mr marko.. I'd rather have 1 osc and 100 LFOs than vica versa, but I'd also like to see more patch points as well J ..

Actually, my nord is staying on the bookshelf until I hand in my thesis. J I suppose it is partly a question of what model you have in mind, oscillator-wise. If I am doing straightup additive synthesis, I'd like nothing better than around ten kazillion oscillators. Errr ... all sinusoidal, and possibly fixed frequency, though not necessarily. Lots of early computer music systems used the numberof oscillators available for additive synthesis as a benchmark of their powerfulness. Some still do, I guess.

This thread has prompted me to wade deep into the jungles of the AH archive (on marks site) to locate this thread I can barely remember (mainly to re educate myself on multiple osc usage.. it never was much of a priority in the past).. Here are two bits that pertain to what we are talking about... for all I know, this information could be wrong, but I trust the sources..

More can be better, if you're going after sonic complexity. To reiterate a post by Marc Paping, when detuning more than one osc, have the detuned osc be quieter than the original so the fundamental doesn't cancel entirely. The strength of multiple oscs is to either create interval stacks or to create a single sound with multiple harmonic aspects, such as having two or more sync'd oscs set to different harmonics, etc. Poly instruments with a unison mode are a great example of what you'd expect from multiple oscs. You get a huge, rich sound.That's what more means!" - Robbie Sinclair, Jim Henson's "Dinosaurs"On the other hand, multiple oscs, when detuned, which is probable when using analog technology, will generate beat frequencies that cancel and reinforce the waveform. Using a single oscillator on bass sounds is a good thing, as it allows total stability in the sound's volume. You can create a more complex sound by mixing all the waveforms together from that single oscillator, since they are all in phase. Oscillators with phase locking (Moog 921b, etc.) allow multiple oscs to lock together in perfect phase, thus creating a rich sound without beat cancellation. Or, sync'ing a second osc, tuning it a bit higher in pitch, and not sweeping it can add size without cancellation.

> We all know (hopefully) how much bigger the sound of a machine with more than 1 oscillator (like a minimoog) sounds so much more powerful than say a 101 or a 303 (well, i can feel the flames coming my way already). People on this list with modular machines, do you often use more than say 4 or 5 oscillators?

There is a point of diminishing returns with more oscillators. Each one you add makes a smaller difference than the last. I usually only 2-3 is usually good, and less often 4. One reason to use 3-4 is for fatter sounds in multiple octaves (maybe 1 at 32", 2 at 16', 1 at 8'). Another is to have different treatments on each: Example: (this doesn't sound anything like a TB-303) One at "center" pitch Second slightly detuned. Third modulated by EG to slide up into unison Fourth with slight pitch modulation from LFO or S/H.Another reason to have more is for use as modulation sources. VCO to VCO FM, VCO to VCF Cutoff FM, as LFO, etc. I sometimes use many oscillators (10-20) in unison for fun but the sound is not necessarily any fatter than 2-3, and often it is not as fat as 2-3. With more than 3 oscillators there is more potential for phase cancellation. So having many oscillators often gives an effect similar to a flanger or phaser, although it is different, maybe more ubtle.There are so many possibilities that others might find other ways to use many scillators. It never hurts to have the option. But 4 is enough for most applications and or many great sounds you only need 1, 2, or 3.

> [.v/jek.] wrote: This might be a stupid question, since you have a nord and (until tomorrow, according to reuse it) I don't.. but isn't this a moot point? I seem to remember this conversation on AH long before the nord came out.. something about, after 4-6 oscillators you didn't really hear much of a difference... something about phase cancellations (although, I don't see why the phase alignments would have to be off.. maybe this was a particularly analogue thing).. ohh well...

Seems in a *real* analogue, each oscillator would have it's own characteristics, like

1. kbd tracking variations and non-linearities

2. stability variations over time

3. waveform asymmetry variations

4. q variations in amplitude that would make the oscillators sound "phat".

When the 44 module limit is lifted, seems you could "build in" these variations through different techniques to get that same "analogue sound"? 1, 2, and 4 seem like they could be implemented. 3 seems like it couldn't, because the waveforms ar computed Also, it seems like it would be nice if you could TELL the NM where you want your poles and zeros on the filters...this would enable huge variation in filter types for thos engineering types. (A macro type filter that would let people "download" the coefficients?) (Since I've never done DSP filters, am I out to lunch here?)