Additive synthesis may be the key to those tones your sampler and analogue synths won't give you - it may even help straighten some of your samples.

WHETHER YOU REALISE it or riot, we are now firmly in the digital age. Nowhere else is this more apparent than in the music world, with sounds and songs being stored on floppy disks rather than on patch charts and manuscript paper. With samplers at affordable prices, the AD/DA conversion software and hardware is omnipresent to process and create sounds entirely in the digital domain. It's reached a point where the major limitation of our sonic freedom is pretty much down to the software available.

Additive synthesis (the creation of complex sound waves from a combination of simpler waves) has been with us in one form or another for a while (on machines like the Fairlight, PPG Waveterm, Synclavier and so on) but only recently has it become available to the masses. It's getting easier to find software that enables you to create waveforms or strings of waveforms on a computer and transfer them to a sampler for playback. From there the sound material is treated as a sample with the accompanying characteristic of its getting shorter the higher it's taken from its "root" pitch and longer the further it's taken lower. This "munchkinisation" is a drawback, but if you've grown used to mapping multisamples for realism you're already living with it comfortably enough. Truly additive synthesisers keep the timing of a waveform progression (wavetable) constant no matter what pitch is played, but require a separate hardware or software oscillator for each harmonic.

For the past several years I've been creating additive waveforms and wavetables on my PPG Waveterm (with its 32 harmonics) using them as powerful digital sounds in their own right, as well as mixing them with sample material, and have started experimenting with additive synthesis software for the Akai S900. Take it from me, designing waveforms can be both rewarding and frustrating. Setting out to create a wave that sounds anything like a "real" instrument may prove disappointing for all but the most simple acoustic events. It's much faster and easier to sample the real thing. Besides, the character of many instruments changes drastically even over short periods of time. This means a lot of work with little guarantee of producing anything useful. (And there's multisampling to consider if you don't own a proper additive synthesiser.)

Designing sounds in the digital domain can be more fun than playing Pacman. Remember to save as you experiment, holding onto tones that you like and maybe even those that you don't like so much. Single waveforms take up minuscule amounts of memory space and interesting multisample and wavetable transition material may often be found in waveforms you came up with on your way to discovering "the biggie". I've used additive synthesis to create interesting variations on existing instrument family "themes", and intermediary "bridging" tone colours impossible with subtractive or FM synthesis alone.

Ignoring the naturally-occurring decrease in amplitude in higher harmonics makes building super-bright waveforms possible. This is unique to working in the digital domain, and you won't find sounds quite like these anywhere else. Loop your single waveforms, make use of the Digitally Controlled Filter and Amplifier on your sampler and you have one monster of a synthesiser on your hands. Remember, single waveform loops take up almost no storage space. Hundreds of custom waveforms can be stored on floppy disk. Even complex waveform multisamples, with slightly different waveforms under each key, will probably take up less space than an average sample.

Taking waveform building one step further; stringing individual waveforms together into "Wavetables" enables you to create acoustic events, rather than static tone colours. Outrageous harmonic and filter-sweep events are surprisingly simple to put together from separate waveforms. Correcting or adding new tone colours to existing samples can be accomplished with digital sample mixing. Specific portions of a sample can be enhanced or reinforced by keeping track of key timbre and amplitude transition points and building a wave progression to match. Adding new and distinctive attack transients to percussive sounds doesn't consume very much time at all; all you need is an interesting "blip", "plick", "flick" or "swoof' for the beginning of a sound. Perhaps most importantly, seamless loops can be constructed: wavetable loops can be designed to smooth out existing sample loops.

If you're into simulating acoustic instrument tones, harmonic analysis of instruments can be found in many musical acoustics books (be wary of the accuracy of early works) and especially back issues of the Journal of Audio Engineering and Computer Music Journal. You'll find the harmonic spectra for an oboe and clarinet sounding the same pitch analysed at one second into the respective instrument's notes included in Table 4. Although much more realistic than subtractive synthesis imitations, you'll find, as with samples, if taken too far from their root pitch, things start sounding strange. The bottom line is that there isn't a "universal" waveform that will depict an acoustic instrument throughout its entire playing range.

Since additive synthesis is based on dealing with harmonics and the harmonic series, before we go any further we need to look at what these are.

The Harmonic Series

HERMANN HELMHOLTZ AND Jean Baptiste Fourier have a lot to answer for. Between them, they deduced that complex periodic sound waves could be broken down into combinations of much simpler sound waves, called harmonics, at any given time during a sound and that each harmonic, a pure sine wave, existed as a whole number multiple of a given sound wave's fundamental frequency.

Some pretty weird intervals, in relation to the fundamental, turn up above the 20th harmonic - see harmonics 23, 27, 29 and 31. With modern technology producing computers and samplers capable of calculating harmonics well up into the hundreds in the twinkling of an electric eye, these two scientists' research into the nature of sounds has been proven beyond all doubt. Without their efforts, sampling might not be with us as we know it today.

Harmonics and Partials

THE DIFFERENCE BETWEEN harmonies and partials confused me for quite a while. I couldn't understand why overtones were referred to as partials in one musical acoustics book and harmonics in another. They are different ways of looking at the simple overtones found in more complex sounds.

Harmonics: The fundamental is the first harmonic of the harmonic series and all subsequent harmonics fit into neat little slots that are whole number multiples of that fundamental. A fundamental of 100Hz would have a second harmonic of 200Hz, third harmonic of 300Hz and so on.

Partials: While a harmonic series may or may not be present, partials take non-harmonic overtones into account. This is the only way of analysing complex sounds like car crashes, those produced by instruments with strong resonances like kettle drums, bells and cymbals, and performance artifacts.

Non-harmonic partials seem to be the culprits holding up foolproof sample resynthesis software and full flexibility over additive synthesis. Computers like dealing with nice round numbers and can cope quite easily with any harmonic recipe or progression you might choose to ask of them, but throw them a few strong non-harmonic partials and the software required gets prohibitively complex. It looks like we're going to be working with harmonic additive synthesis based around numbers that computers digest easily for a few years to come (which may be no bad thing as we've barely touched the tip of the iceberg with what can be done with "simple" harmonic additive synthesis).

I know it's easier to use a synthesiser for standard oscillator waveforms, but it really is a good exercise to experiment with programming these into your additive software to get a feel for waveform building. Besides, with comprehensive DCF and DCA envelopes on your sampler, you may never need to touch an analogue synthesiser again.

Although much more symmetrical than waveforms sampled from life, oscillator-type waveforms demonstrate the underlying principle that a harmonic's amplitude decreases as its distance from the fundamental increases - something that you'll find pervading all digitally-created sounds.

Note that the sawtooth, square (50% pulse) and 25% pulse waves are all based around the same mathematical progression; the fundamental amplitude is divided by a given harmonic number. The triangle wave is similar to the square wave in that only odd numbered harmonics are present, but radically different, as the fundamental's amplitude is divided by the square root of a harmonic's number.

Here are a few tables to get you going.

- Synth Bell is very simple and one of my favourites. Its harmonic amplitudes follow the formula for the sawtooth wave but with progressively more space between harmonics as they move upwards.

- The oboe and clarinet waveforms were analysed from samples with the amplitude of the loudest harmonic adjusted to 100%. Both were sampled at D# 311.126Hz (the D# above middle C on the piano). To my ears they sound most realistic when played within a fifth of the root. The oboe is a good example of the fundamental not always being the loudest harmonic. Be adventurous, try bridging the gap between instruments by averaging their harmonic values for an "Obonette" tone colour.

- Monster sawtooth demonstrates how three octaves of sawtooth wave can be layered upon themselves in the same waveform, loudest harmonic again adjusted to 100%.

Although the fundamental is the lowest tone in the harmonic series and usually the pitch we hear, it isn't always present in appreciable amounts. Sounds from life rarely have fundamentals as prominent as synthesiser tones do, especially brass, string and percussion waveforms.

Triangular Problems

I'VE FOUND IT difficult to produce a true triangle wave with only 32 harmonics. Unless I'm doing something terribly wrong, there don't seem to be enough upper harmonics to make the waveform resemble a sampled analogue triangle wave with its perfect angularity. Not that triangle waves produced with less than a hundred or so harmonics aren't musically useful. On the contrary, they're great for beefing up, or using as building blocks in synthesising a whole spectrum of sounds like flutes, pan-pipes, woody and tuned percussion, ethereal vocals and bass.

Waveform Clipping

YOU MAY FIND when the total combined harmonic amplitudes add up to more than your software has "headroom" for, that the resultant waveform distorts. This is not necessarily a bad thing. The effect is very much like the clipping used in fuzz and overdrive effects pedals and adds upper harmonics. Depending upon the severity of the "overdrive" and how your software deals with waveform clipping, the result may turn out sounding like anything from a very bright version of your waveform to digital noise. Controlled clipping of waveforms (and samples) can be a very useful tool. For software with a limited number of harmonics it can be used as a crude method of generating upper harmonics. With the proper amplitude adjustment, clipped waveforms can be used to roughen up the bow scrape portion of a violin wavetable or imitate the chiff of flute and panpipe notes.

Sampled Sine Waves

EVEN IF YOU don't have software for additive or wavetable synthesis you still have plenty of scope to experiment in this area using sampled sine wave progressions. FM synthesisers have either four or six sine waves available at the same time that can have their own frequencies and, admittedly limited, amplitude envelopes. Eight-voice, four-oscillator multi-timbral FM synthesisers such as Yamaha's TX81Z can stack up to 32 harmonics in one pass. The same applies to almost any multi-timbral synthesiser able to play its voices in a stacked or "mono" mode.

Work out your progression for the first few harmonics, sample these, then repeat the process for the next batch of harmonics. Once you've sampled these separate "clusters" of harmonics they can be mixed, as samples, via software. If you don't have mixing software but have the creative urge, you can record and mix harmonic clusters using multitrack tape and sample the composite wave.

This sampled harmonic method makes experimenting with non-harmonic partials and the pitching of harmonics a breeze, especially with FM synths where oscillators can be detuned, given a pitch envelope, or even modulated. Rapidly modulating the pitch of selected harmonics or harmonic/partial clusters gives a fuzzy feeling to sounds and can be of help in simulating vocal or instrument breath, bow scrapes, even coloured noise. When you find a composite progression that you really like, remember to backtrack and record a multi-sample of it to keep the time/pitch ratio more constant.

Don't underestimate the power of this method of sound synthesis. Although it is laborious, I don't think the computing power and programs needed to execute wavetables are quite with us yet. Someone, somewhere must have, or be working on, a piece of software that gives more freedom with operator envelopes, maybe even software allowing something like a TX81Z to be used as a 32-harmonic additive synthesiser. It'd be nice.

Free Fundamentals

WHICH LEADS US to Free Fundamental Additive Synthesis. Until now, we've been dealing with simple sine waves as harmonics which contain no overtones whatsoever. Wolfgang Palm with his PPG Waveterm may have been the first to introduce the concept of building complex waveforms using any waveform for its harmonics. The alternative waveforms available in the Yamaha TX81Z should give you room to experiment in this area.

Formants and Resonance

ONCE WAVEFORMS ARE in your sampler, they can be treated in the same way as samples: with looping, DCF, DCA, vibrato, automatic and manual pitch-bend, detuning between layered sounds and so on. All the while taking up insignificant amounts of memory space. If you've found a tone colour you really like, you might want to take the time to construct a multi-sample of it, spreading copies of that waveform across the keyboard, each with its harmonic amplitude adjusted to simulate formant or resonance bands... much like a built-in graphic equaliser. Multi-sampled waveforms have considerably more warmth and character than single waveforms covering the entire keyboard range. For the example below start with a waveform reminiscent of a vocal sound to demonstrate formant multi-sampling.

Remember that "sampled" waveforms can be combined with sample-mixing software to make composite tone colours. Different versions of the same waveform can be designed to take advantage of the "loud and soft" sample switching facility on many samplers. Rapidly sweeping past high concentrations of upper harmonics with a lowpass filter will give a cutting edge to percussive waveforms and is useful in creating synthesised mallet percussion and plucked/struck string sounds.

While this article hasn't touched upon every aspect of additive synthesis and the various permutations of it available in software systems, it may have whetted your appetite enough to experiment. Yeah, it's a time consuming process but at the end of the day, you'll have custom-built waveforms unavailable anywhere else.