Our waveshaper VCO.
In release 16.4 we added a lot of features, breaking patch compatibility, so now there are two versions. The old one is still available, and will load when old patches are loaded. It has been renamed [old] Chebyshev. These instructions are for the new one.
Here is a tutorial video by Omri Cohen: Squinky Labs Chebyshev - VCV Rack Tutorial
Chebyshev polynomials have been used to generate complex tones since the early days of computer music. This special math discovered by Mr. Chebyshev enables digital generation of waveforms with any overtone structure using very little computer power. In addition, it is easy and computationally inexpensive to vary the spectrum over time.
Eventually, this form of synthesis fell out of favor, as FM could provide a wider variety of timbres with acceptable CPU usage. Now, however, the distinctive sound of this form of synthesis provides another unique source of sounds for VCV Rack users.
Lately they ability to control individual harmonic levels under voltage control has come back in the form of synthesizer modules called Harmonic Oscillators.
The magic of the Chebyshev polynomials is that if a sine wave with amplitude one is put into a Chebyshev polynomial, the output will still be a sine wave, but multiplied in pitch by an integer.
In our implementation we include the first ten Chebyshev polynomials to generate the first ten harmonics of the harmonic series. These are then mixed together based on knob settings and control voltages to give an output tone with complete control over ten harmonics.
In addition, FM synthesis may be added to the mix via the LFM input.
Many of the controls in this module allow different ways of mixing together these ten harmonics. In addition external signals may be shaped by the waveshaper, with folding or clipping applied.
Like all our modules, Chebyshev's CPU usage is quite low.
While there are many, many different ways to use Chebyshev, its use as a harmonic VCO will be familiar to some. It is very easy to use Chebyshev as a harmonic VCO, but unless set up deliberately, you may find that Chebyshev is functioning as some other kind of VCO.
It can be a great help learning this module if you patch the output to a Scope module and a frequency analyzer. As you adjust the controls it will be clearer what is going on.
Also, note that we will repeat that when set up normally, each Chebyshev waveshaper is producing a different harmonic of the VCO output. Because of this, we will refer to these as "harmonic levels" and "waveshaper levels" interchangeably.
And - each waveshaper is a perfect harmonic only when driven by a pure sine at exactly 1Vp-p. This is very important, and we will mention this again later.
First there is a sine wave VCO. It has the controls you would expect, as well as a through-zero linear FM input, which allows a minimal DX7-style FM.
The VCO output then goes to a wave folder/clipper with gain controls. This allows for some distortion effects, and keeps the signal in a range that will make the next stage happy. An LED will go red when either of these kick it.
The output of the folder/clipper then goes to ten parallel Chebyshev waveshapers. The outputs of these are then mixed together through a specialized mixer.
When everything is set in a typical manner, each of the Chebyshev waveshapers will be outputting a pure sine wave at an integer multiple of the fundamental frequency.Thus, each one will be a discrete harmonic.
The controls in the upper right are all for the sine wave VCO.
- Oct raises and lowers the pitch by an octave. It has a ten octave range. The octave is indicted with a label above this knob.
- Semi raises and lowers the pitch in semitone intervals, with a range of 11 semitones up or down. The semitone is indicted with a label above this knob showing the pitch "name".
- Tune raises or lowers the pitch by up to a semitone.
- Mod controls the modulation (exponential FM) depth of the signal patched to the Mod jack.
- LFM controls the linear FM depth of the signal patched to the LFM jack.
- V/Oct input is where the main control voltage is patched.
Mod and LFM perform different functions. Mod, like the CV input, is an exponential control. If an LFO is patched into the Mod input and the Mod depth is adjusted for a vibrato of one semitone, that vibrato will be one semitone regardless of the base pitch. But if an audio rate signal is patched into the Mod input you will tend to get “clangorous” sounds with inharmonic overtones.
LFM, on the other hand, allows through-zero linear FM. While this is not very good for vibrato it does create complex timbres where the harmonics are in tune, and that consonance will remain as the mod depth is changed. Exponential FM at audio rates can also be tuned, but the tuning will disappear as the mod depth changes, making it impossible to to generate dynamic harmonic sounds.
Chebyshev polynomials are poorly behaved if they see more than one (volt) at their input. So we use a folder/clipper to make sure this doesn’t happen.
The controls and CV of the Folder/Clipper:
- Fold/Clip switch. In clip mode, it is a simple hard clipper. In fold mode it’s a waveform folder.
- Clip LED. The LED will be green when there is signal, and red when the folder/clipper engages.
- Gain. Controls how hard the folder/clipper is driven. Gain knob and CV are combined.
- Gain trim. The small knob below the gain knob is an attenuverter for the gain CV.
- EG. Also combines with the gains, allowing an external ADSR to control the harmonic balance.
- Ext In. When a signal is patched here it replaces the internal VCO, allowing any signal to be run through the waveshaper.
Note that while you can get some cool effects with clipping and folding, they will tend to cause audible aliasing at higher frequencies. Use with care.
In classic waveshaping synthesis an ADSR or similar would be connected to the EG input. By dynamically changing the level of the sine wave hitting the waveshapers a dynamic timbre will be generated.
The output of the folder/clipper drives the Chebyshev waveshapers. The last group of controls all work together to determine how the waveshapers are mixed together.
While it may be counter-intuitive, keep in mind that while the EG input does control the gain of a VCA, that VCA is not on the output of Chebyshev; it is after the VCO/external input, and before the clipper, folder, and Chebyshev waveshapers. As such an envelope applied to the EG input will cause dramatic shifts in timbre.
To get a normal volume envelope applied to the output of Chebyshev, you will need to use an external VCA. But to get dynamic waveshape timbres, the EG input if your friend.
Driving the EG input above 10, either with a hot ADSR, or by dialing up the gain knob, will make the red LED come on, indicating clipping or folding is occurring. Which will sound very nice in some situations.
There are a lot of controls that work together to determine how the waveshapers are mixed. When configured normally, that means these controls determine the ratios of all the harmonics of the VCO.
The small knobs running up the left side individually control each waveshaper/harmonic, with fundamental on top, and harmonic 10 on the bottom.
The input jacks next to them allow the levels of each harmonic to be voltage controlled.
The Preset button toggles all ten harmonics between some good starting points, and also resets the Gain to be exactly 1.
The Even control increases/decreases the level of all the even harmonics together.
The Odd control increases/decreases the level of all the odd harmonics together.
The Slope control will apply a gradual roll-off of the upper harmonics. When it is all the way down the roll-off is 18 decibels per octave. When it is all the way up it’s flat.
Note that the level of the fundamental is not affected by either the Even or Odd control.
The Odd, Even, and Slope controls may be thought of as subtractive. When they are all the way up, they have no effect, and you get the mix you would expect from the individual harmonic levels. When you turn these controls down they will reduce the levels of the corresponding harmonics.
The Preset button toggles between two or three settings. It will always have a setting where the fundamental is full and all other harmonics off, and a setting where all harmonics are up full. In addition, if you started with your own setting of the harmonics, the preset button will eventually take you back there, but with the master gain set back to one.
There is a lag generator on the control voltage input for each of the 10 harmonics. They all have the same rise and fall time. They all have exponential slopes.
The Rise control sets the rise time of all the lag units. There is also a Rise CV input.
The Fall control sets the fall time of all the lag units. There is also a Fall CV input.
When set up correctly, Chebyshev will give you independent control of the levels of the first ten harmonics in the harmonic series. Each the the ten controls / CVs on the left will control the level of the corresponding harmonic.
But this is only true if Chebyshev is driven with a pure sine wave at one volt peak to peak. By far the easiest way to accomplish this is:
- Use the internal sine oscillator. If, instead, you patch something into the Ext In jack, it will be difficult to impossible to precisely control the individual harmonics.
- Use the default gain. If you turn the Gain knob it will be almost impossible to get it back to 1. In this case, pressing the the Preset button three times will get you back exactly where you are, but with the gain at exactly one.
If the gain is more than one the red indicator comes on. If the gain is much lower than one the green light will go off. But there is a range of gains where the green light will be one. So, again, resetting the patch, or using the preset button, is the only practical way to set the gain exactly.
But of course there is a world of other sounds in Chebyshev if don't set it this way, and many will be pleasing. They just won't be exactly a harmonic oscillator.
Aka arbitrary waveform generator.
Turn the Odd, Even, and Slope controls all the way up. Then the level of each harmonic is controller its own volume control. Mix the harmonics to get a pleasing sound, then use as a static timbre, or run it into a VCO.
Or start with the individual harmonics all the way up, manipulate Even/Odd/Slope to start sculpting the harmonics.
Don’t forget the Preset button - it’s your friend here. And don't forget that you need to use the internal sine VCO and the default gain to control the harmonics precisely and independently.
The possibilities for timbral variation seem limitless if you take the time to patch controls signals into the harmonic level CV inputs. Use all the usual suspects here - clocks, LFOs, shift registers, sequencers.
If a very sharp edge is patched the harmonic level input, pops will be generated at the harmonics are abruptly turned on and off. If this occurs, just increase the rise and fall times of the lag units until the pops go away.
Setting the rise and fall times to be longer can round a square LFO until it sounds more like a smooth triangle wave.
An easy way to get dramatic timbral changes is to patch independent modulation sources to each harmonic input. While this can be done with any collection of modules, a module that provides a lot of independent outputs makes this easy. Some particularly good ones are our own Gray Code generator, and Bogaudio's 8FO.
If you try modulating the volumes with clocks, make sure to adjust the Rise and Fall times to keep the pops down, and to provide dramatic variations in the sound.
Use the built-in VCO. Adjust the harmonic mix to something nice and bright. Then connect an ADSR to the EG input. Like the harmonic VCO patch, the dynamic waveshaper is most predictable when driven by a one volt p-p sine, and the gain set to default. And the ADSR should go from zero to ten.
The ADSR will more or less control the brightness. Make sure the the clip LED is just on the edge of clipping when the EG input is at max (which will be the 10 volts mentioned above).
At very low levels the output will be primarily fundamental. At max level it will be determined by the waveshaper mix controls. The timbre will go from dull to bright as the EG input increases, but the evolution of the timbre will not be completely smooth or predictable, and definitely will be different than what you would get modulating a VCF with an ADSR.
The evolution of timbres often sounds "brassy," like a brass instrument. Brass synthesis was indeed a common use of waveshaping synthesis before the era of affordable sampling and physical modelling.
Most conventional VCFs allow a filter of a specific shape to be modulated up and down in frequency. A variable slope VCF lets the shape of the filter be modulated. Modulating the filter slope can be more "natural" sounding.
In the case of Chebyshev we don’t have a filter, but by controlling the harmonic levels directly we can mimic one. Set up a nice bright sound, patch an ADSR into the Slope input, and try out a simulated variable slope filter.
This is certainly no substitute for a Serge variable slope VCO, but can produce subtle timbres unlike any other.
The inclusion of the LFM input allows a simple form of FM synthesis - two operator FM. But with Chebyshev the two operator FM is then run through the Chebyshev waveshaper. There is a universe of sound in this intersection of FM and waveshaping.
Use an external sine VCO, and patch it into the LFM input on Chebyshev. Turn up the LFM knob. Use the Preset button to turn up the fundamental and turn off the other waveshaper outputs.
In FM speak, the external VCO is the modulator and the VCO in Chebyshev is the carrier. The result should be consonant if the frequency of the carrier is a small integer multiple of the modulator frequency. For example, set the modulator an octave lower than the carrier.
Once again, as the modulation is increased more harmonics will be present, so use an external ADSR and VCA to modulate the level of the modulator sine before it’s patched into Chebyshev.
Now start adjusting the Chebyshev mixer controls. For something dramatic, press the preset button until all the harmonics are on. Then adjust slope, even, and odd to tame all the harmonics that will come out.
Try tuning the VCOs so they are a fifth or a fourth apart. Go crazy!
Run something other than a sine wave into Ext In, then process your signal with the folder and/or the Chebyshev waveshapers. Again, the output of the waveshapers can be pretty unpredictable as the input signal becomes more complex.
In the standard configuration there will be little, if any, aliasing. Since the highest harmonic is 10X the fundamental, the Chebyshev module can’t even start to alias until the fundamental gets to 2kHz.
That said, there is no anti-aliasing in this module. The wavefolder and clipper will alias. Normally the LFM will not alias very much, but with high modulation depth and high pitches it will alias quite a lot.
We have an informational article that talks more about aliasing here.