In my recent post *Is there a perfect synthesizer?*, I of course talked about frequency modulation synthesis, and about the fact that what Yamaha implemented e.g. in their DX7 is in fact phase modulation – which is in that context “essentially the same”.

Some people have challenged that statement, or rather demanded a more thorough explanation, which extends beyond that mentioned in the literature section of the aforementioned post. So here it goes: an entire article about phase modulation, frequency modulation and how they are connected.

# Beautiful Math

## Frequency and Phase

Before we can start modulating them, we first need to get an understanding what “frequency” and “phase” is, and how they are related to each other.

First, assume a function [1] of an angle, y(φ). It may also depend on other variables, but we start with considering that angle for the time being. The angle is not measured in degrees, but in so-called radians, with 2π=360°(=”full circle”). Using those radians makes the discussion a little easier. The function y is periodic with 2π, which means that when the function argument reaches 2π (the *period*), the values start to repeat, or y(φ+2kπ)=y(φ), with k any integer. Apart from that, the function can be anything you’d like it to be. The function’s argument, the angle φ, is called the *phase*.

Next, we’ll introduce the *angular frequency* ω=2πf (unit: radian angles per second), and followingly, we’ll use *frequency* as short for “angular frequency”. The *frequency *here defines how an angle (in that case, our phase angle) increases per time amount, much like the speed of something describes how many metres per second (or kilometers per hour) it moves.

So that frequency is somehow related to the phase, but how? The simple case is that the frequency is constant, in that case the phase is **φ(t)=ωt**. In other words, if the angle progresses at the frequency ω, and it progresses for the time t, then after that time, the angle has increased by φ.

Things get trickier if the frequency changes with time, however. Then we can’t just use multiplication, we need to do an integral so the equation changes to

**φ(t)=∫ω(τ)dt** (1).

People versed in math will now start to point out that this equation misses an additive constant. Yes it does, but in the following discussion, **we’ll just ignore additive constants and also constant scaling factors. We just assume we can work with them with a simple parameter in our synthesis engine.** Please remember that last sentence, especially in the section on modulation, before you start to write to me to point out errors.

### A simple analogy: running on a track

Assume you’re running on a track with a constant speed of 5 m/s. Which distance will you have covered after five seconds?

It’s simple: the distance s is your speed times the time you’re running at that speed, so s=vt=5m/s*5s=25m. And of course, if your speed changes during that run, it’s not so simple anymore – you need to consider for which time amount you’ve been running at which speed or how you’ve been accelerating – mathematically speaking: you need to do an integral.

The analogy here is plain: replace the distance s with our phase φ, the speed v with our frequency ω, and you’ll see it’s exactly the same.

### Summary: Frequency and Phase

We have a phase φ and a frequency ω, which are related by the equation

**φ(t)=∫ω(τ)dt** (1),

or, in the special case of a constant frequency, by the simple equation

**φ(t)=ωt** (1a)

The phase is the argument of a function y(φ) which, with the equivalence (1) above, can be rewritten as

**y(φ)=y(∫ω(τ)dt)** (2).

Now we’ll get on topic quickly, because that function y will later become the output of our synthesizer’s oscillator, and φ and ω are the quantities we want to modulate in phase or frequency modulation, respectively.

## …and Modulation

Modulation is a term from communication technology and describes varying a property (or several ones) of a periodic waveform, called the carrier signal, by another waveform, called the modulating signal. Now we’ve already got one waveform from the discussion before, y, and that will be our carrier signal. This carrier signal has properties, namely phase, frequency, and time, which all somehow depend on each other. As we can’t modulate time, we can look at modulating phase or frequency, and this is just what this article is all about [2].

We still leave the properties of our carrier signal undefined, but we’re gonna pick a simple example for our modulating signal: sine or cosine waveforms [3]. In the discussion, we’ll switch freely between sine and cosine, mostly because in the scope of this discussion, they’re very similar – in fact, with cos(x)=sin(x+π/2), we can quickly transform one into the other.

### Modulating Frequency

In the chapter before, we’ve already considered a frequency varying with time. This can be seen as the sum of the constant part ωC (with C for “carrier”) and the variable, modulated part ω(t). This time dependency now surfaces in the form of the frequency being modulated by a cosine wave. So our frequency becomes a cosine function of time with in this case a constant frequency for the modulator ωM=const [4]. Thusly, our signal frequency is

**ω(t)=ωC+cos(ωM*t)** (3).

Putting that into equation (1) and using the integral of a cosine being a sine, we get

(3) -> (1): **φFM(t)=∫ωC+cos(ωM*τ)dt=ωC*t+sin(ωM*t)** (4).

### Modulating Phase

This time, our frequency only has a constant component, ωC. The variable, modulated component is now added to the phase. We use a sine wave for the carrier (which, as we’ve discussed, is about the same as the cosine we did use before), and keep our modulator frequency ωM constant again.

With that, we get for the phase:

**φPM(t)=ωC*t+sin(ωM*t) = φFM(t)** (4).

# It’s all the same! Really?

The proof is above. The equations are the same, so it’s the same. However, I can just wait for you challenging that, mostly with:

- it is only valid for specific output functions y,
- it is only valid for specific (in that case: sine/cosine) functions for modulating,
- it is no longer valid if you modulate the modulator signal, just like you can do in a proper FM/PM synth,
- I got both a FM and PM synth, and they sound completely different!

### ad 1: “it is only valid for specific output functions y”

This is not the case. As you can see, we didn’t even look at the output function y itself, only at its argument.

### ad 2: “it is only valid for specific (in that case: sine/cosine) functions for modulating”

This seems to be true: only some functions (exponential functions, to be precise, to which sine/cosine are related somehow [5]) derive and integrate into itself or nearly itself. So if you would use something that is not a sine or cosine, it will no longer work. Now wait a minute – didn’t Fourier tell us that all signals with specific properties (in short: all the relevant signals in this context) can be written as a sum of sines or cosines, so we could just break down our arbitrary modulating signal into cosines, then integrate and stay the same?

Obviously not. This is the moment where the scaling factors we so hastily decided to ignore above start to come back into discussion. Before, we could just rescale the entire thing and be fine. But now, due to the scaling factor, each summand gets a different scaling factor, so this no longer works.

So, if you want to use an arbitrary signal for modulating, and can’t create its integral (or derivative) – both is easy, you just need a lowpass or highpass, respectively – then there is indeed a big difference between modulating the phase and the frequency.

### ad 3: “it is no longer valid if you modulate the modulator signal, just like you can do in a proper FM/PM synth”

Also, this is true. As a simple example, if you modulate the modulator frequency in phase modulation with another sine, then you’d need a sum of two multiplied sine/cosine functions to get the same result. It still can be done, only it’s more complicated.

### ad 4: “I got both a FM and PM synth, and they sound completely different!”

In addition to what we explained above, here’s yet another reason, taken from my article about the perfect synth: Even if it’s the same in theory, different implementations of the same synthesis architecture will still sound different. And that’s a good thing.

## Summarizing – do we need both?

We’ve seen, in the first chapter, that in theory and for some special cases, frequency modulation and phase modulation are exactly the same. And in general, you can get the same kinds of sounds with both approaches, only one specific sound might be easier to obtain one way or the other.

# In Synthesizers…

## So why do we need it in the first place?

The vast majority of analogue synthesizers of the 70s worked with subtractive synthesis. One or several oscillator signals rich in overtones (saw, triangle, square, pulse) were sent through one or several filters which would then remove the upper or sometimes lower frequency content, or cut out a segment in the middle. The resulting sounds analogue synths are known (and to this day, highly regarded) for are fat leads, warm strings or throbbing bass sounds. And those adjectives say it all. You usually don’t hear someone raving about an analogue synth doing metallic leads, crystal-clear strings or clangy basses. They just can’t do it – at least not with a number of oscillators and filters that makes sense [7].

The reason has to do with the frequency spectrum of different instruments, especially of idiophonic percussion instruments which are not based on a harmonic frequency spectrum. And so, while the frequency spectrum of those subtractive synthesis examples can look convincingly like the sustained part of a string sound, a church bell sound is completely different. And this is where FM – or PM – comes in. It simply allows us to generate sounds that can’t be done with subtractive synthesis.

But there’s more. While with subtractive synthesis, the result was always very stable with regard to changes in the synth’s parameters (i.e. slowly lowering the lowpass cutoff frequency would make the sound equally slowly more dull), modulating a key parameter in FM – like the modulator’s frequency, the resulting sound will change quite drastically, if sometimes not very intuitively. So for some expressive playing of sounds, this can also greatly add to your sonic repertoire.

The latter (and here I’m coming back to my line of argument from the last article) is also the reason why FM synthesis cannot be simply replaced by using sample-based synthesis: even if you can imitate that electric piano even better with samples than with FM synthesis, you can’t morph the sound as crazily as you can e.g. on a Yamaha FS1R.

## Why the confusion with the names?

FM synthesis was first introduced as a mass product by Yamaha in 1983 in their DX7 synthesizer. This is a well-known fact – but is it true?

The synthesis implemented here was licensed by Yamaha from a patent by John Chowning at Stanford University for a “method of synthesizing a musical sound” [8]. While this application talks of frequency modulation, what the author describes (even though he doesn’t exactly go out of his way to keep his physical units in check) clearly indicates PM.

So did Yamaha simply license that faulty wording with the patent? I’d rather guess that they simply stuck with it because “frequency modulation” and FM were already known household language – FM radio stood for “better, clearer, and more modern” in comparison to AM radio, so it made sense to exploit that existing analogy for marketing this new technology.

Why PM was used for the actual implementation instead of FM is really open for some guessing. Some sources [9] claim that for low frequency (as opposed to radio frequency) applications, PM is easier to implement, especially in a digital synthesizer. Hordijk also points out that PM allows for selfmodulation of an oscillator without changing the fundamental frequency – and that is an important property for doing musical implementations of the concept [6].

# Gloss/Literature

- For the math geeks: this needn’t be a function necessarily. However, sticking to a function makes life easier, and also does not limit our choices because all of the signals we’ll be looking at will be functions.
- Yes, in theory it would be possible to modulate time as well, but no one has ever done this for communcation engineering or use in synthesizers – prove me wrong, please!
- Incidentially, Yamaha used sine waves for their FM synthesis products, such as the DX7.
- But yes, this frequency can also vary with time by being modulated by yet another modulator, and this is also done in FM synthesizers.
- The details here: cosine and sine are the real and imaginary part of the result of a complex exponential function with only an imaginary argument (i.e. exp(jx)=jsinx+cosx). With exp(jx)’=jexp(jx), by looking only at the real part, we get cosx’=-sinx, and looking at the imaginary part, sinx’=cosx.
- Rob Hordijk: FM workshop (from: Nord Modular Tips & Tricks).
- If you move away from the “number of oscillators that makes sense” limitation, you can of course create those metallic, crystal-clear or clangy sounds by adding signals from a large number of (sine) oscillators. This is called additive synthesis, as used by synthesizers like the Kurzweil K150 or the Kawai K5000.
- John M. Chowning “Method of synthesizing a musical sound“, U.S. patent 4018121.
- Unfortunately, I don’t remember those sources – anyone?

Hey whatsup:) thanks for giving me the time and space to ask this question I really appreciate it. Even though it’s gonna be a pretty simple question, it’s the kinda question only someone with a deep technical understanding of this topic will be able to give a precise answer on (and your overqualified:), and in my case right now, I do in fact need a precise answer for reasons I won’t get into but okay here goes, and try to work with me here…. lol, because what I need to do is take all this information and simplify it into a visual example that I’ve come up with for people that are 1) hearing about synthesis for the first time and 2) are not mathmaticians. So in my audio/visual analogy, I want to compare and contrast AM vs. FM vs. PM and I’m using the the ABC’s as an analogy of the carrier that is going to be modulated by a single cycle sine wave across A-Z. I’m happy with my AM example, but now I just want to make sure I’m accurately dumbing down the concept of FM vs. PM by how I’m thinking of doing it. I will post the image on your twitter, its actually an animation and along with it I will for example recite the A-Z getting increasing louder from 0, then really low, and back to 0 just as it looks in the graphic. In spirit of where am I going with this simple example, I was going to use the top set of ABC’s in the image as the FM example and recite them from a normal pitch and go up in pitch then down and back to normal (just like the AM example), and for the PM example I was going to recite them normal pitch, but the time between each letter get increasingly faster then slower and then back to the middle. So my question is… is that an accurate oversimplified representation of FM vs. PM? In my ABC example, would FM correspond to just modulating the PITCH of the voice reciting the ABCs, while PM affected the TIME inbetween each letter being recited?

okay thanks for hearing me out. I really hope I didn’t confuse you. I’ve been inside and out reading all I can on the subject, and as you already know, there so much mixed opinions/missinformation out there about it, or at least missleading info and my questions end up coming back full circle after reading different sources. As you proved yourself, even Yamaha themself has contributed to the confusion.

so i dont want to contribute to confusing the matter more and spreading wrong information, If i can think about it in the terms that i just described tho with the ABC’s and if it makes sense to you, then I think I’ve got the handle I want on the matter. knowing that my example makes sense will pretty much fill in all the blanks I have on the more technical aspects, i just need to know that I can simplify it down to this example to know that i really understand.

Ok, first of all, I’m gonna include the image you’ve posted for reference here:

http://pbs.twimg.com/media/CA5VnzjXEAAZYpo.jpg:large

I’m totally with you that for the target audience you describe, the representation of amplitude modulation is good to understand, because it works well (even though not being fully accurate when going into the details of the sidebands, see below) both if you see your A-Z line as a representation of an audio recording of someone speaking those letters, as well as seeing those letters as variables for 26 samples in an audio recording.

However, I feel that those two ways of understanding your representation does not work so well for the FM/PM example, and simply because if you decide to see this as a representation of an audio recording of the letters, it will lead to a misunderstanding the way you describe it (i.e. one changes the pitch, the other the time between the letters).

I would like to explain it first by sticking with the other way of understanding that A-Z series: each one being a sample in an audio clip.

From my post, we have that for a sine modulator, FM and PM are essentially the same.

In this context: if that A-Z is an audio clip, then when we play it back in an unmodulated fashion, then the sampler simply plays back each sample (A, B, C and so on) with equivalent time steps in between each sample. The phase here is represented by a saw wave – i.e. it steadily moves from start to end at constant speed (eq. 1a). If each letter takes one second, then the frequency of that carrier (and also of your identical-length modulator) is 1/26Hz.

Modulating the frequency with a sine now moves the carrier frequency up during the first 13 letters, then down during the second 13 – and this is exactly what your diagram shows.

Now for the PM part (and here it gets tricky), it is not so easy to intuitively show how this works, because the visual representation you chose has that aforementioned saw wave already implicitly in it, namely because of the order of the samples (B comes after A etc.). So the trick we need to do is to think in frequency (which the distance between the letters represents), and then ask ourselves: how does the frequency get changed when we modulate the phase with a sine wave?

Using Eq.4, we know that the frequency then gets modulated by a cosine – which would mean that it’s like the third line, only the letters are close to each other at the beginning, very far apart in the middle, and close to each other at the end.

In essence, your third line represents FM (not PM), and PM would be as described above. And if you move your sine wave one quadrant to the left, then FM=PM.

Two footnotes:

1. The sidebands I mentioned above: amplitude modulation also does something to the frequency. This is an effect that you don’t notice for very low modulator frequencies in comparison to the carrier (which in our example above with a 1/26Hz modulator over a speech recording, is the case). However, if the modulator frequency gets closer to the carrier, this becomes clearly audible. A very-well known example is a ring modulator (note that what a ring modulator does is not “ring modulation”, but “double-sideband suppressed carrier amplitude modulation”).

1a. The same with the relation of the carrier/modulator frequency is also true for FM/PM. While this is always heard as a change in frequency, for the modulator being slow, we don’t perceive that as generating sidebands, but as the frequency wobbling.

2. There is an important difference between modulating frequency and pitch, with pitch being a logarithm of frequency.

Hope that helps…

Rainer

wow thank you! I appreciate you’re time and energy to write all that thats really cool of you.

okay so this is really helpful in many ways but I still need to digest it a bit more just to make sure.

the good news is that I was with you the whole way. you even brought clarity to questions I’ve been asking myself that I didn’t bring up, so that was great. And then I was relieved to even understand some your detailed concerns such as what you brought up about the sidebands and what I wasn’t considering with the AM interpretation… I’m with you on that, I’m having to put certain details like that aside at this stage in the explination in order to get the initial point across…. but don’t worry I do drill in on those things later.

Now as far as the PM vs. FM which was my main concern, I thiiiink I got it. That was a great explination and thank you for totally putting yourself in my shoes about it and helping me paint the picture the way I’m trying to paint it…. you know… without saying no no no its all wrong lol… let me re-read this a few times and I will come back and run it by you just to make sure I absorbed your explination properly, and then I might need to decide if throwing in PM in this analogy is a good idea or not and just keep it to AM and FM. It might confuse the point if I don’t execute it right.

okay i’ll beeee baaack!

Do you know of a software synth that uses true FM rather than PM? I’ve been looking for a long time, but it’s a difficult thing to search for using keywords.

Of course you can use NI Reaktor, and I’m sure there’s some instruments for that in their user forum. Apart from that (and other usual suspects, e.g. PD, MAX/MSP) I don’t know any.

It might be interesting to also mention phase distortion synthesis while it’s related.

Your math is incorrect. The integral of Cos(x) is indeed Sin(x). But the integral of Cos(ax) is Sin(ax)/a

(Because you have to use the chain rule.) This is simple first semester calculus.

I’d like to point you to the first section of this post (right after eq. 1):

“we’ll just ignore additive constants and also constant scaling factors. We just assume we can work with them with a simple parameter in our synthesis engine. Please remember that last sentence, especially in the section on modulation, before you start to write to me to point out errors.”

As far as I understood it, the factor is not a constant but rather frequency dependent. Off course you can also apply this in your synth engine but my point is just if you implement a naive FM you would get a different result depending on the pitch so a high tone has a different timbre compared to a low tone as opposed to a PM Synth where you get the same result independent of the pitch…

https://en.wikipedia.org/wiki/Phase_modulation says;

– “The phase of a carrier signal is modulated to follow the changing signal level (amplitude) of the message signal. The peak amplitude and the frequency of the carrier signal are maintained constant, but as the amplitude of the message signal changes, the phase of the carrier changes correspondingly.”

– “This shows how {\displaystyle m(t)}m(t) modulates the phase – the greater m(t) is at a point in time, the greater the phase shift of the modulated signal at that point. It can also be viewed as a change of the frequency of the carrier signal, and phase modulation can thus be considered a special case of FM in which the carrier frequency modulation is given by the time derivative of the phase modulation.”

– “For small amplitude signals, PM is similar to amplitude modulation (AM) and exhibits its unfortunate doubling of baseband bandwidth and poor efficiency.

– “For a single large sinusoidal signal, PM is similar to FM”