Through Zero Linear Frequency Modulation


A TZFM module popped up on this thread. It is a contentious topic among the modular synth manufacturing community with many approaches.. I don’t understand it and please don’t explain it unless you have read the muffwiggler thread. What I can do is share a patch I made on the plane just now. Happy to be told what I did wrong, or how I failed to give an example that uses a ‘non-trivial’ approach.

TZFM Tests.audulus (431.2 KB)


I hope you’ll forgive me for skipping over that a little bit.

My understanding is mostly grounded in DSP with only passing familiarity with the analog solutions to TZFM. Basically a negative frequency is when you are running around the unit circle in a clockwise direction and as a consequence your waveforms are inverted.

We can think of periodic waves in terms of period, frequency, and angular frequency.

Angular frequency is great for describing discrete systems like computers because it isn’t tied to a particular sample rate. We can take the ideas about positive and negative frequencies and talk about them in radians rather than seconds.

As a side note let’s separate frequency from pitch for the time being. Pitch is logrithmic in terms of Frequency and consequently will never pass through zero. We don’t have a note for silence, we use a different symbol when writing in western musical notation and we treat zero hertz as the space in the piano roll.

Frequencies however can range in the positive and negative domain and when we are talking TZFM, it is almost always going to be linear through zero FM. When the modulation to the frequency input is strong enough a TZFM oscillator won’t stop at zero like traditional FM but continue to modulate into negative frequencies with an inverted waveform.

I think “ndkent” fron the MW thread summed it up best when he said, “Finally my hint is FM is kind of boring and unimpressive - through zero or not - if it just gets dialed in and sits there. What makes it exciting is changing the modulator amplitude over time typically with a VCA. Then what gets my goat is people just patch one VCO to another cranked at full amplitude and say “X” does great FM if they liked the sound which is partly due to the carrier. Or “I didn’t care for the FM” when they simply crank up the knob on play with the carrier frequency”

Now I’m now quite as passionate as all that, but I do know that a lot of the most interesting FM patches I have made have been a result of enveloping the amplitude of the modulator oscillator. So you might give that a try in your patch.


Do you mean modulating the fm (amount) dial? Or, I suppose you might be saying “treat it subtractively”; not the dial but the whole approach – hence the vca?

What a condescending, smarmy video. Sorry I was done after 30 seconds of that. My Dad is an electrical engineer, we build circuits and acoustic instruments. I have programmed very little, he used to work in fortran, basic, C, etc. I really don’t find the math communities very fun at all. It feels like everything is always backwards and needlessly abbreviated.

I appreciate the help though, for what it is worth. Maybe it has something to do with how Hollywood equates genuis with solving algebra formulas at college.

I also tend to feel that these circular models fail to account for time properly. We don’t all share the same mainstream scientific paradigms. I am partial to Bergson’s concept of time, not Einstein’s.

*And everyone goes, “you fool, what do you expect, after all it is the standard everyone is working in.” The trouble is, there is often a great difference between knowing how to implement solutions on the one hand, versus knowing what might be wrong headed or knowing why a field of study is stuck on a problem. I understood the challenge that spawned the various approaches in the wiggler thread.


The approach you used in your patch appears to be correct. My earlier point regarding through zero FM was that the built in oscillator node is not capable of through zero modulation, but the phasor node is. It all really revolves around the concept of a “negative” frequency. In fundamental terms frequency is the ratio between some number of periodic events and the time over which they occur. In order for this ratio to be negative, it is necessary for one of the terms to be negative. Since the number of events is positive, in order to have a negative frequency the time duration term must be negative. This is typically taken to mean that the direction of time has been reversed which leads to a waveform that is reversed in time with respect to its unmodulated shape.


Now, here’s my hunch. You can reverse time in an equation. Event time, so to speak, can only be perceptually reversed. Since sound can only be experienced over time, there must be a mental buffer that we fill and, in some sense, capture time. This captured time – the buffer – can be perceived apart from regular time or universal time.

When I read Deleuze’s book Bergsonism, it seemed to provide a poetic understanding of the concept of time, apart from ratio based/parameter based applications. The idea that ‘we dive into time’ has always stuck with me. The thing about periodic events is they tend to provide the repititions necessary to ‘anticipate’, so that perspective is possible through comparative change.

It is tempting to wash our hands of this kind of talk and just agree upon a quantified standard of negative values. It is equally tempting to think that Heidegger’s book Being and Time has nothing to do with math/programming. One might also think that Wittgenstein’s later work on meaning has nothing to do with DSP. However, we are deep into the world of logic. It is quite difficult to set aside the claim that understanding is more than just knowing how to continue a series, for example.

If anyone is interested in a more modern thought experiment, John Searle’s Chinese Room problem is straightforward.

If we use Descarte’s coordinates to plot time, like a scope, then freeze the image or tear off the page, it seems like time could be a product. But I don’t think any of this gets at inverse frequencies, which I suspect would make more sense to me if it was conceived in terms of electrical applications. In this case wouldn’t the math be a logical statement such as, ‘let x=?’ Further, if we use it as a modulation source, it then ceases to have anything to do with time, other than as a clock sync.

I don’t want to kick up dust to cover up understanding here. I just wanted to hash out a small portion of how I would be having difficulty with the video drops like @robertsyrett posted above. I really do not think it is so simple, just as I do not think that being able to carry a mathimatical series forward is clear evidence of understanding.

You might then ask, if we can get the equations right, what more do we need? I believe it has a lot to do with AI, and I am happy that the current culture can’t quite figure it out yet.

I have a long drive ahead of me so I have to leave it here.


The polarity of frequency does not affect the flow of time but rather the chirality of our perception.

The circle must be used for understanding, not for reasons of Kahndescention, but because circle is the physical manifestation of a cycle. Positive and negative have almost arbitrary meaning, and you could just as well construct a system where the two terms were swapped. But the circle is there regardless of the construct, transcenting formal syntax. The unit circle is the cosmic zoetrope, sometimes the image may appear to run in reverse and we say that it’s a negative frequency, but the map is not the territory.

More examples to develop an intuition of the common notation:

  • Vinyl LPs have a negative angular momentum when running forward and therefore a negative frequency when viewed from above. When we reverse the direction the record spins, we perceive the sounds as playing backwards.

  • The rotation of an electric motor

So through zero modulation in these contexts would be a varispeed control for scrubbing audio or a switch on a fan that makes it suck or blow.

The reason that it’s useful to know the equations is it provided a common way of describing all these events. It may be cryptic at first, but with an intuitive understanding we can reap the benefits of applying it to other areas, like making cool new waveforms for our synthesizers.



:stuck_out_tongue: Cool or exhaust then. I suppose no self-respecting manufacturer would say their fan sucked. :wink:


Time and periodicity are irrevocably linked. The only mechanism we have to objectively measure time is by counting some periodic event. We started by counting the revolutions of the earth, moon and sun and have progressed to counting the vibrations of Cesium atoms but the fundamental principle is the same.The model of the universe described by Newtonian physics considered time as an invariant which proceded uniformly and was the same for all observers in all locations and at any instant. With the introduction of relativistic physics by Einstein, time becomes a dimension equivalent to the three spacial dimensions of Newtonian physics, and the flow of time is now thought to be relative to the observer and not a universal constant. We perceive time as always progressing from past to future at a constant rate, but we know from observation that this is not actually the case. Time flows at different rates for each observer. Even so, there still seems to be a direction to time. Although much of physics is invariant if you reverse the time axis, this does not appear to be the case in reality. We invariably seem to move in one direction through time. From this perspective, a negative frequency is a meaningless concept. Even the idea of a 0 frequency is questionable. Consider the nature of a waveform with 0 frequency. What is its shape? How can you measure its amplitude? In some sense a 0 frequency wave does not exist at all.
Still, we must at some point bow to practicality, and decide what is the best action to take when the modulating signal controlling the frequency of an oscillator falls below zero. We can simply ignore values below zero, which is the approach the oscillator node takes, use the absolute value of the modulator so that a negative value has the same effect as the positive one, or reverse the waveform in time which is the case with the phasor node. None of these choices are inherently better than the others. It is simply a matter of preference for the designer.


The first video is over my head.

I don’t find this intuitively satisfying. The thing is, quite a bit of the math you guys use all the time was developed by “philosophers.” There is actually no such thing as a circle. A circle is a concept.

Positive and negative are not arbitrary for electricians. A circuit is a cycle but it is not a circle.

The second video is important. It captures the reversibility of flow. If we look deeper into the chemistry of the parts involved in motors/generators, there will be characteristics that share modalities. These structures interact within a field, just as equations represent inverse relationships. If you read about transistors you will find the concept of doping quite intriguing. I suspect that without this ‘barrier,’ compressors won’t work. The transistor creates a buffer in the flow which ‘charges’ the event. There is more energy present in that period.

Now, I am not correcting anyone. I am making conjectures so that I can reveal my hand. I want to stop here. Tell me where I am wrong.


But the rose has teeth in the mouth of the beast.


Is the circle in the center of the animation, that pulls the modulation across the vertical line, representative of phase?


My brain just said that while crossing that vertical threshold puts the parameters into “the past,” you could use such a source for modulating reverbs and delays in “real time,” since the tails are synthetic echos. But that all depends on what that centered circle represents.


Diving further in with the same patch…

TZFM Tests 2.audulus (636.5 KB)


Your comments led me to consider approaching frequency modulation from a different perspective. We typically view FM as rapidly changing the frequency of an oscillator of some type, but it would be equally valid to regard FM as moving an observer back and forth relative to a wave of constant frequency. This is typically referred to as the Doppler effect, but is in fact frequency modulation. Imagine an observer sitting on a platform that can be moved back and forth at a high rate. Now imagine that a constant frequency oscillator of some type is turned on so the the observer can detect it. While the observer is stationary, the frequency of the signal is constant. If the platform moves toward the signal source the frequency will rise, and if the platform moves away the frequency will fall. Vibrating the platform will result in the observer perceiving an FM signal rather than the original constant frequency. As the platform moves away from the source at a velocity approaching the speed of sound, the observed frequency will fall to zero. What if the observer is moving away faster than the speed of sound? The observer will then detect the oscillator waveform in reverse. In essence this is the basis for through zero FM. It also offers an interesting possibility for implementation. Typically we perform through zero FM on periodic waveforms. Because we know their shape in advance, we can easily calculate their time-reversed form. In fact sine, square, triangle and other waves that are symmetric along the time axis don’t need to be reversed, since in their case a time reversal is equivalent to a change in phase. A ramp, however is not symmetric along the time axis and can be either rising or falling. More generally audio signals tend to be quasi-periodic and are not predictable and therefore not easily reversed. However with a suitable buffer that can be played forward or backward at varying rates, which is relatively simple in the digital world, one could apply a form of through zero FM to non-periodic signals. During the positive portion of the modulation one would move forward through the buffer, and during the negative portion one would move backwards. Hopefully A4 will be capable of this type of manipulation.


That, and presets. Here’s the exact patch above, but with a slight variation on the ADSR module.
TZFM Tests 3 - Sweet Spot.audulus (636.5 KB)



I fixed up the mess a bit. Sounds more FM. Not sure if I have gone through zero though. If the modulator is bipolar and the carrier responds to negative waveforms, does that mean all is well?

TZFM Tests 4.audulus (697.9 KB)

This short article got my understanding of TZFM a little further along.


A minor adjustment. I took into account @robertsyrett’s point about amplitude modulating the modulator with a VCA. No great changes in sound yet though.
TZFM Tests 5.audulus (698.8 KB)


TZFM Tests 6.audulus (748.2 KB)

I believe in this one. :no_mouth:


I was running a patch through some reverb and delay, then I realized that it was also getting more reverb and delay through some hardware. It sounded good so I tried this double stage before and after a filter strictly in Audulus. Then I used some lfo’s to animated the 4 pole k-pass. It sort of travels around nicely. Headphones recommended.

TZFM Tests 7.audulus (713.6 KB)