Chapter 2. Oscillator Waveform Modification

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The so-called subtractive approach to sound synthesis uses filtering of these basic waveforms to provide control over the harmonic content of a sound. A sinewave has only a single harmonic, and therefore filtering does not alter its harmonic content at all, merely the amplitude of the sound. A triangle wave has more harmonics, but with very low amplitudes, so they too are not too useful as sources for subtractive synthesis. The most useful waveforms for subtractive synthesis are the sawtooth and square (or pulse) waveforms. These waveforms have strong harmonics, suitable for alteration through filtering. A sawtooth waveform contains all harmonics of the fundamental pitch of the waveform, while the square waveform contains just the odd harmonics. A pulse waveform, with a duty cycle different than 50% also has all harmonics, even and odd. Changing the pulse width alters the relative strength of the individual harmonics, with the harmonic levels become more equal as the pulse narrows.

In the subtractive synthesis approach, variations in the spectrum of the sound (or timbre) are usually generated by varying the characteristics, such as cutoff frequency and resonance, of the filters that the oscillator waveforms are passed through. While this is effective for synthesizing many pleasing and interesting sounds, the result is limited somewhat by the rather sparse choices of spectra for the oscillators. Because of this, synthesizer manufacturers and patch programmers have come up with a number of ways in which to expand the spectral possibilities beyond those of the basic oscillator waveforms. In this chapter we will examine some of these approaches and show how they can be implemented in the Nord Modular.

2.1 Oscillator Sync

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I will give here a soft-sync technique that was not covered in the tutorial. This technique was used in the RMS (Rivera Music Systems) synthesizer as well as in the Wiard synthesizers. In this technique, the synced oscillator produces a triangle wave (which could be passed through a wave shaper to produce other waveforms). To this triangle waveform a narrow pulse from the syncing oscillator is added. The sum is compared to a reference level. If the sum exceeds this reference level, the synced oscillator is reset, or the triangle wave reversed. The probability of a syncronization event being created can be seen to be related to the amplitude of the syncing pulse as well as the amplitude of the triangle wave output of the synced oscillator. If the syncing pulse has a low amplitude, then syncing will occur only if the syncing pulse happens very close in time to the peak of the triangle waveform. Otherwise no syncing will occur. At the other extreme, if the syncing pulse has a very large amplitude, then syncing will occur on every pulse, no matter what the phase of the triangle waveform. In this case, the sync action will be that of regular hard-sync. By adjusting the amplitude of the syncing pulse one can vary the sync effect from soft to hard. This technique is illustrated by the following Nord Modular patch. It is not exactly as in the RMS and Wiard synths, since the triangle wave is reset on reception of a sync signal, rather than being reversed. But for those intrepid patchers who want more authenticity, follow the instructions in Rob Hordijk's above referenced softsync tutorial, where he describes how to reverse the slope of a triangle waveform on the Nord Modular.

Figure 2.1. Patch illustrating a softsync technique (J. Clark).

For the technical minded among you, there are some ways to make sync patches sound even better. The NM, as we all know, is a digital system at heart, even though it might sound "analog" from time to time. Because of the finite sampling rate used in the NM, the syncing waveform is actually quantized in time. This means that the synced oscillator is not only synced to the syncing waveform but also to the sampling clock. At low frequencies of the syncing waveform the effect of the sampling clock sync will not be too noticeable, but at high frequencies it becomes quite significant. Rob Hordijk suggests a way to remedy this problem. The jitter caused by the syncing to the sampling clock is most noticeable at the transient that inevitably appears at the point where the syncing signal causes the synced waveform to switch. One can apply a mask, or envelope, over the synced oscillator waveform, to suppress this transient. A simple way to do this is to use a downward sawtooth as the syncing waveform and also use it as an amplitude envelope that modulates the synced waveform. This has the drawback of distorting the synced waveform, of course. An improvement can be had by crossfading two such synced waveforms, each out of phase with respect to each other. This will minimize the distorting effect of the masking. The following patch by Rob Hordijk illustrates the approach.

Figure 2.2. Patch illustrating a masked hardsync technique (R. Hordijk).

2.2 Frequency Modulation

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Figure 2.3. Patch illustrating a simple FM approach to obtaining a brass-like sound (J. Clark).

Instruments with cylindrical bores, such as the clarinet, have mainly odd harmonics. To obtain this type of spectra we set the frequency of the modulator to be twice that of the carrier. Conversely to the case of brass instruments, the timbre of the clarinet starts out bright then becomes duller during the attack phase. So use an inverted envelope generator to adjust the modulation level.

Figure 2.4. Patch illustrating a simple FM approach to obtaining a clarinet-like sound (J. Clark).

To model the sound of the Bassoon and similar instruments using FM, set the frequency of the modulator to a sub-multiple of the carrier frequency. This will weaken the strength of the fundamental relative to the higher harmonics. All harmonics (odd and even) are present in these types of instruments, due to the tapered bore.

Figure 2.5. Patch illustrating a simple FM approach to obtaining an oboe-like sound (J. Clark).

FM synthesis is capable of much more than the simple examples shown here. The output of one carrier oscillator can be used as a modulator waveform as well, and one can create modulation chains, where the spectrum gets more and more complex as one moves down the chain. Modulating the amplitude of the carrier signals in such a chain using envelope generators or LFOs can produce quite expressive and complex time-varying sounds. The following patch shows a three-stage FM chain. The sound is more lively than the simple clarinet patch given above.

Figure 2.6. Patch illustrating the use of multiple modulation stages in an FM patch (J. Clark).

One can use non-sinusoidal waveforms for the carrier and modulators, providing an even greater variety of harmonic spectra. There is no reason one cannot use modulator and carrier frequencies that are non-harmonic. Using these will result in waveforms that have non-harmonic spectra, good for metallic sounds such as bells and gongs. The following two patches illustrate the use of non-sinusoidal modulator and carrier waveforms. The harmonic structure is more complex than in the simple case of sinusoidal waveforms.

Figure 2.7. Patch illustrating non-sinusoidal modulation waveforms in an FM patch (J. Clark).

Figure 2.8. Patch illustrating non-sinusoidal carrier waveforms in an FM patch (J. Clark).

Feedback is a technique that always produces interesting, if not musical, results, and this is true for FM synthesis as well. One can modulate the first oscillator in a modulation chain with the output of one of the other oscillators (or with its own output, for that matter). The results of applying feedback can be very difficult to analyze mathematically, even in the case where all oscillator frequencies are nominally harmonic. For this reason it is advisable to experiment, and see what comes about. Some general rules can be found, however. First is that a little bit of feedback goes a long way - too much feedback causes instability and even chaotic behaviour. Of course, this might be just what you are looking for! Likewise, feedback from an oscillator output closer to the oscillator being modulated gives a more gentle effect than feedback from an output further down the chain. Inserting time delays into the feedback patch can create very unpredictable shifts in the timbre.

Figure 2.9. Patch illustrating the effects of feedback back to the first oscillator in an FM modulation chain (J. Clark).

In the above patch Knob 2 selects the source of the feedback signal, so that you can compare the effects of each. Selecting position 1 gives no feedback, position 2 gives feedback from the first oscillator (i.e. self-modulation of this oscillator), position 3 gives feedback from the second oscillator in the chain, while selecting position 4 gives time delayed feedback from the second oscillator. Play around with the amount of feedback by adjusting the OscSlave1 FM Amount control (Knob 1). If it is set too high a chaotic noisy waveform will be the result.

The oscillator modules on the Nord Modular are specifically designed to permit useful Frequency Modulation synthesis. There are four types of frequency modulation inputs on these modules - "Mst", "Pitch", "A" and "B". The "Mst" and "Pitch" frequency modulation inputs are logarithmic, while the "A" and "B" inputs are linear. The Pitch input is typically used to set the basic pitch of the oscillator, for example, from a keyboard note value. The "Mst" input is usually connected to the "Slv" output of a master oscillator.

To understand the difference between the different input types, consider a typical approach to FM synthesis, where we have two sources determining the frequency of an oscillator. The oscillator adds these two sources together to determine the actual frequency of the oscillator. Typically, one source value is a `stable' one that controls the basic pitch of the waveform (e.g. the "note" being played). The other source is some waveform that varies at an audio rate. The `stable' input should be logarithmic (=non-linear) to accomodate a keyboardvalue on the NM. If the other modulation input (the one connected to the audio rate signal) is also logarithmic the waveform shape, and therefore the timbre, changes if the pitch is increased. A low pitch will be relatively more modulated (have more harmonics) than a high pitch. Sometimes this is desirable, for example when making a fat bass sound, but it makes it not very useful for synthesizing a `physical instrument'-like sound. When modulating with a linear modulating input the harmonic content will remain stable over the complete pitch range.

But what if we selfmodulate a sine-oscillator with the FMA-input? Opening the FMA-knob will not only change the timbre, but also the pitch of the sound, since the average value of the sine output is non-zero. This means that the pitch of the sound will no longer track the note value being fed into the logarithmic pitch input. Not a good thing for emulating standard instruments! Of course, you might be looking for just that off-key type of sound... The FMB input is designed to prevent such a shift in pitch when a waveform with a non-zero average value is used as a frequency modulation source. This input is designed to mimic the form of frequency modulation used on the Yamaha FM-synthesizers. On these synthesizers, such as the DX-7, the pitch doesn't change when selfmodulating. This is because Yamaha used `phase modulation' rather than frequency modulation. In this technique, it is not the frequency of the oscillator that is modulated but it rather the phase of the wave. If the modulating waveform has a non-zero average value, there will only be an (inaudible) shift in phase, and not a shift in pitch. On the Nord Modular we can obtain Yamaha-like FM patches by means of the FMB input. The FMB input does not directly modulate the phase of the oscillator. Instead, the FMB input is passed through a highpass filter (inside the module) and the filtered result is used to frequency modulate the oscillator. The highpass filter removes the DC component of the modulating signal, thereby ensuring that the average value is zero. Hence no pitch shift will occur. Also, consider that a highpass filter acts like a signal differentiator for signals whose frequencies are on the low frequency cutoff slope region of the filter. Frequency modulation with the derivative of a signal is similar to phase modulation with the signal itself (since frequency is the time derivative of phase). Thus, signals applied to the FMB input result in something similar to phase modulation, as in the Yamaha systems. For modulating signals with frequencies above the cutoff frequency of the highpass filter, the modulation will be of frequency and not of phase, so the character of the sound created by the modulation with high frequencies will be different than that for modulation with low frequencies. One might want to lowpass filter the modulating waveform to eliminate the high frequency components that will adversely affect the sound.

Due to the great success and popularity of the Yamaha DX-7 FM synthesizer, there has been much written on the programming of FM synths. There is also a large number of patches available for study. Wout Blommers has created a vast collection of Nord Modular patches based on Yamaha DX-7 patches. These can be found, along with additional information about FM synthesis in the section on Emulating Classic Synths. Rob Hordijk has also written a nice tutorial on FM synthesis which can be accessed at http://www.clavia.se/nordmodular/Modularzone/FMsynthesis.html.

2.3 Wave Shaping

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Figure 2.10. Patch illustrating waveshaping with nonlinear elements (J. Clark).

Note the use of a compressor after the nonlinear elements. The reason for this is to even out the changes in overall amplitude that can result from the application of the nonlinearities. Remember, the purpose of the nonlinear elements here is to modify the shape of the waveform, but not its amplitude. You want to use a VCA/Envelope Generator after the compressor to provide the dynamics for the patch.

The waveforms that you use as input to the nonlinearities are not too crucial but they do make some difference. In particular, you should avoid using pulse or square waves as input, as their shape is not affected by nonlinear elements, only their amplitude. In this regard, they are much like sine waves with respect to linear filtering operations - the shape of a sine wave is not affected by linear filtering, only its amplitude. In mathematical jargon, sine waves are eigenfunctions of linear time invariant systems (which is a fancy germanic way of saying that the shape of sine waves are not altered by linear filters) and pulse (two-level) waves are eigenfunctions of (memory-less) nonlinear mappings. Typically triangle waves or sawtooth waves are preferable, as their linear slopes makes predicting the effect of nonlinear operations easier than for other waveforms. In some synthesizers, many waveforms are derived from easy to produce triangle waves. For example, one can use a nonlinear element that has a gradual decrease in gain with amplitude to round off the corners of the triangle waves, producing an approximation to a sine wave. This is also an example of a case where application of a nonlinear process results in fewer harmonics than exist in the input!

2.4 Vector Synthesis

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Figure 2.11. Vector synthesis patch (J. Clark).

The crossfading between the four oscillators is done via the mathematical relation:

One can iterate the vector synthesis approach, so that the basic waveforms used in these approaches are themselves vector synthesized waveforms. If you have enough computational resources, you can carry this iteration deeper and deeper, leading to fractal waveforms in the limit. The patch shown in the following figure implements a 2-level vector synthesis algorithm, using 16 oscillators (4 groups of 4 oscillators) with five 2-D waveform crossfade controls (4 to do the second level waveforms, and one to do the top level waveform). LFOs and envelopes are used to generate the crossfade controls, but, of course, any source of control signal could be used.

2.5 Wave Sequencing

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Figure 2.12. Wave Sequencing patch (J. Clark).

Note the use of oscillator sync in the above patch. This is to model the fact that in wavetable instruments such as the Prophet VS and Korg Wavestation, the generation of the waveforms is implemented by scanning of waves in the wavetable, and crossfading between waveforms is acomplished by interpolation between different wavetable entries. Thus each waveform is effectively synchronized, and start at the same point.

In the patch above a 3-bit binary counter is constructed from clock divider modules. This counter controls the crossfading of the waveforms in a binary tree. Unfortunately, this approach requires that the crossfades be done quickly, so there is no possibility for smoothing of the crossfades. The next patch, designed by Rob Hordijk, uses a different scheme for crossfading, which permits smoothing of the crossfades. The amount, or smoothness, of the crossfading from one step to the next can be set by the programmer. An abrupt crossfade will give more high frequency components than a smooth crossfade.

Figure 2.13. Wave Sequencing patch that allows smoothing of crossfades from one wave to the next (R. Hordijk).

The crossfader module can be used to interpolate between a current and a previous output of a control sequencer to create a smooth wavetable oscillator. It proved that using the first quadrant of a sine^2 function to interpolate the transitions sounds the most smooth. Nice, as now only three DSP-cheap sinewave slaveoscs, a gain controller, two S&H's, an inverter, the crossfade mixer and a control sequencer to hold the table are needed to do the trick.

Using two (or more) control sequencers allows for fades between waveforms. Its quite a cheesy sound, so to make the sound a bit grittier a fuzz is added.

The control sequencers give up around E5 or so, from there up the patch starts to freak out. Which can be quite nice when e.g. controlled from an external sequencer.

2.6 Audio-Rate Crossfading

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Figure 2.14. Two examples of waveform modification using audio-rate crossfading.

A patch that implements this simple audio-rate crossfading idea is given below. The two input oscillators are synced, so you can alter the harmonic structure by adjusting the frequency of the synced oscillator. Try changing the oscillator waveforms to hear the sound produced in the various cases.

Figure 2.15. Audio-rate crossfading patch (J. Clark).

2.7 Wave Terrain Synthesis

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Figure 2.16. A simple Wave Terrain patch (C. Singer).

In this patch the two 1-D waves which define the orbit on the terrain function are enveloped sine waves. The resulting orbit traces out two circular paths connected by a spiral on the terrain surface. At the beginning of the envelope, the path follows the outer circle, then quickly moves to the inner circle during the sustain phase. Upon the release of envelope the orbit moves to the center of the circular paths and stays there. The circles are offset somewhat from the center of the terrain (see the "Offset" control modules in the patch). This adds some assymmetry to the sound and makes it more interesting.

Figure 2.17. The Terrain function used in the above patch, with a typical circular orbit.

2.8 VOSIM

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The following patch, designed by Pere Villez, implements a basic FOF synthesis system. A slightly more complex patch, FoFGen01Cello02.pch by the same designer, shows the ease with which FOF synthesis can model string instruments.

Figure 2.18. A FOF patch (P. Villez).

2.10 Granular Synthesis

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Figure 2.19. A Granular synthesis patch (P. Dahlstedt).