What is FM?

Frequency Modulation (FM) is where the output of one oscillator is used to modulate the pitch of another, the oscillators being called Modulator and Carrier respectively. "Modulate the Pitch"... that's the key phrase! The pitch of the Carrier is being changed (modulated) in tandem (in sync/ going up and down at the same time) by the Modulator.

Think of it as one person singing and another person grabbing the throat of the first and shaking him in a rhythmic manner; the singer being the Carrier and the throttler being the Modulator.

In analogue synthesizers, you can use an LFO (Low Frequency Oscillator) to modulate a VCO (Voltage Controlled Oscillator). Let's take a slow LFO and modulate the VCO... what happens is that the slowly rising and falling LFO makes the pitch of the VCO rise and fall also, giving you a sort of wobbly sound (referred to as VIBRATO). Increase the modulating LFO Amount and there's more wobbling. Increase the modulating LFO Speed and the wobbling gets faster. This is also commonly called "Pitch Modulation".

What is DX-FM?

On DX synthesizers (DX-FM), the only real difference is that the Modulator is a "musically-tuned frequency" (whose frequency is determined by the notes actually played on the keyboard). The other difference is that DX-FM oscillators are all Sine-waves.

Imagine an old analog synth with 2 VCOs... When you play the keyboard, both the VCOs will emit their respective waveforms, taking its pitch by reference of the notes played on the keyboard. Now imagine rerouting VCO1 into the modulation input for VCO2... Play the keyboard and both VCOs will play their respective notes but now the pitch of VCO2 is changing exactly in time with the frequency of VCO1. And there we have it ... one FM synth (VCO1=Modulator; VCO2=Carrier). Some synths already have this facility except it's commonly called "Cross-Modulation".

What are Operators and Algorithms?

Operators are just Oscillators. Your FM synth will have either 4 or 6 Operators. Why so many Operators? Because the sounds from one Modulator & one Carrier aren't exactly that overwhelming.

Algorithms are the preset combinations of routing available to you. Note that the Carriers are always the last Operators in any Algorithm chain and all other Operators are Modulators.

Critical relationship between "M" & "C"

Let's look at the one Modulator & one Carrier set-up.

[MODULATOR] ------> [CARRIER] -------> [sound output]

The carrier frequency "C" and the modulator frequency "M" will together determine which harmonics will exist (or have the possibility to exist) in the harmonic spectrum. The harmonic spectrum is a graphic representation of frequencies where "1" is the fundamental frequency and the other harmonics are just multiples of the fundamental.

The rules determining which harmonics can exist are as follows:-

What is happening is that the energy of the modulation is transformed into "Sidebands" (the series of harmonics on both sides of the Carrier).

The appearance of Sidebands is always in pairs on each side of "C". These Sideband pairs are ranked by their "order" of separation from "C" (eg 1st pair is "M" distance apart from "C", 2nd pair is 2x"M" distance apart from "C"... etc).

Non-Coincident and Coincident Series

Let's look at a few examples. Reflected Sidebands are denoted by brackets. These examples only give frequencies up to the 6th Sideband but, of course, the number of Sidebands is, in theory, infinite. In general, the intensity of the higher Sidebands will decrease in intensity to a point where they become inaudible. At this point, don't worry about the heights (amplitude)of the harmonics because they haven't been determined yet.

Examples

M : C Sidebands

2 : 3 5 7 9 11 13 15

: 1 (1) (3) (5) (7) (9)

3 : 5 8 11 14 17 20 23

: 2 (1) (4) (7) (10) (13)

1 : 1 2 3 4 5 6 7

: 0 (1) (2) (3) (4) (5)

When the sidebands are coincident, you'll notice that the separation between them is regular. With non-coincidental sidebands, you'll have an alternating separation (eg 1,2, ,4,5, ,7,8... etc). This sort of harmonic arrangement cannot be obtained using normal subtractive synthesis.

IMPORTANT NOTE - if you replace the Carrier value with that of any Sideband (reflected or not), you get the same Series. Try it!

Also note that detuning the Carrier Frequency (C) produces quite a remarkable change in the series. In M:C = 1:1 (with coincident sidebands), if we detune the Carrier to C=1.01, the unreflected bands will be at 2.01, 3.01, 4.01, 5.01 etc and the reflected bands will be at 0.99, 1.99, 2.99, 3.99, etc, so they no longer coincide.

Series generated by M:C

Below are 2 tables. In the first table, use "M" and "C" values to find out what Series is being generated. Then go to the second table to see the harmonic spectrum of that Series.

Certain series have a "x2" or "x3" on them. It is the same series except that it is transposed upward by that amount.

Also note the series from "5:2" onward share one strange property in that they do not have a harmonic at the fundamental frequency (ie 1).

Modulation Amount

The Modulation Amount determines the loudness/ amplitude of the Carrier and each "order" of Sidebands. "Order" refers to the sideband's ranking from the Carrier (ie 1st, 2nd, 3rd sideband) and they are always in pairs. The resultant FM output will be symmetrical for each "order" of sideband pairs.

The exact amplitudes are very difficult to calculate and, quite frankly, you only need to know how the bands are affected (rather than go through the messy calculations).

Very basically, as you increase the Modulation Amount, more and more sidebands will appear. The way in which the sidebands appear is what gives DX-FM its characteristic sound.

The table below shows the distribution changes as the Modulation Amount is increased (Top graph is least modulation and bottom graph is most modulation). The graphs serve only as guides and are not accurate.

FM Spectrum

In DX-FM synthesizers, the modulation amount is controlled by envelope generators so quite dramatic timbral changes can be achieved. Having a visual picture of how the modulation amount changes the amplitude distribution helps us understand what is going on.

The actual amplitudes are a calculated using a Bessel Function and based on the strength of the modulator output measured as a Modulation index. For more details on the calculating the amplitudes, see FM DX Supplement.

If we disregard amplitudes, there are basically 2 types of Series; (a) with coincidental reflected sidebands, and (b) with non-coincidental reflected sidebands (reflected sidebands are phase inverted). If we examine the amplitudes, the effect of the reflected sidebands depends on the modulation amount (ie how much the sidebands spread out) as well as the Carrier frequency "C" (as "C" is the centre point of the modulation). The modulation frequency "M" determines the separation of the sidebands.

• M:C = 1:16 ~ This is a good example of the FM spectrum without the interference of reflected sidebands. It shows the basic spread of sidebands as modulation is increased. C=16 is high relative to M=1. In this case, LS#16 (the 16th lower sideband) is at F1, LS#17 is at F0 (ie silent) and LS#18 is inverted at F1 (LS#18 is the first refected sideband). The sidebands spread out on both sides of "C" symmetrically. The effect of reflected sidebands (ie LS#18 onwards) only occurs at high modulation amounts.

• M:C = 1:8 ~ This example also shows the symmetrical spread but has some interference of coincident reflected sidebands. C=8 is medium relative to M=1. In this case, LS#8 is at F1, LS#9 is at F0 (silent) and LS#10 is inverted at F1 (LS#10 is the first reflected sideband). The sidebands spread out symmetrically up to a point at moderate modulation amounts where the effect of reflected sidebands (ie LS#10 onwards) occurs.

• MC = 1:1 ~ This example shows the dramatic effect of the interference of coincident reflected sidebands. C=1 is low relative to M=1 (a very popular ratio for string sounds). In this case, LS#1 is at F0 (silent) and LS#2 is inverted at F1 (LS#2 is the first reflected sideband). The effect of reflected sidebands (ie LS#2 onwards) occurs at low modulation amounts.

• M:C = 2:1 ~ Another example of the effect of the interference of coincident reflected sidebands. C=1 is low relative to M=2 (a ratio for wind instruments perhaps). In this case, LS#1 is inverted at F1 (the first lower sideband is already reflected). The effect of reflected sidebands (ie LS#1 onwards) occurs at low modulation amounts.

• M:C = 3:1 ~ This is a good example of the effect of non-coincident reflected sidebands. C=1 is low relative to M=3 (perhaps for wood percussion sounds). In this case, LS#1 is inverted at F2 (the first lower sideband is already reflected) but is not coincident with another sideband. The effect of reflected sidebands (ie LS#1 onwards) occurs at low modulation amounts.

For M:C = 4:1 or 5:1 or 6:1 or 7:1 etc, the amplitudes would be the same as M:C=3:1 but the separation of the sidebands would be different. In general, series with non-coincident reflected sidebands are easier to predict.

Two Modulators

So far, we have only dealt with M:C ; single sine-modulator to single sine-carrier. When there are two Modulators, they can either be "Two-Into-One" (M1 + M2 : C) or "In-Series" (M2 : M1: C).

This is where there are 2 separate Modulators, "M1" and "M2", both modulating the the only Carrier "C".

Since the Modulators are separate, you will basically end up with "M1:C" and "M2:C" added together.

Let's look at an example where M1=2, M2=3 and C=5 :

This is where one Modulator "M2" is modulating "M1" which is, in turn, modulating the Carrier "C". This is a lot more complicated because "M2:M1" will produce one complex waveform. From that complex waveform, each and every sine-frequency (in the harmonic spectrum) will act as a sine-modulator into "C".

Let's look at an example where M2=2, M1=5 and C=1 :

Now, imagine every single one of those frequencies as modulating the Carrier. As you can appreciate, the "In-Series" modulators calculation can become very complicated and perhaps confusing too.

Tips for using F.M.

Programming FM synths, can be daunting indeed. As such I have come up with a few tips which you may find useful.

Tip#1 - If you're using an identical pair of M:C (ie 3:1 and 3:1) with the Carriers slightly detuned to fatten up the sound... you can usually short-cut this into a "one-into-two" (ie 3:1+1 with detuned "C"s). It may not sound exactly the same as the original.

Tip#2 - If you're using a pair of M:C where C is the same (ie 7:1 and 9:1), you can usually short-cut this into a "two-into-one" (ie 7+9:1)... especially useful if you're running out of operators. It may not sound exactly the same though.

Tip#3 - Fixed frequencies can be useful as an LFO. For "chorused" sounds, you can make one Modulator as a fixed low-frequency and it'll sound like an LFO at work. This is commonly used with "in series" combinations (eg Fix:M:C), although "two-into-one" combinations will also work (eg Fix+M:C).

Personal Sidenote - Personally, I find the timbre of "in-series" modulators to be less exciting than the "two-into-one" (or many-into-one) combinations. I normally only use the "in-series" like 1:1:1 for producing string-type timbres. I find the "many-into-one" produces more impressive timbres.

Actual DX algorithms can be found in article Synthesizer Layouts.

FM Synthesizers

FM synthesizers (mainly by Yamaha) underwent 3 stages of evolution.

It started with the classic DX-7 and DX-9. These were intricate synths and were designed for performance. The parameters available were very flexible allowing subtle nuances to be controlled. However, they were very complex to programme.

Next came the affordable DX-21 and DX-100. They were designed to have a wider variety of sounds and simplified parameters. Programming was easier but the finer detail was lost.

Finally came the CX-5 and FB-01. They were FM for computers and a few minor design changes only. These designs were later used for computer sound-cards.

We can analyse the design differences into basically 4 types of FM synthesizers, as follows:-

MOD. OUTPUT - This is the output level of the Modulator into the Carrier. Basically, there are 3 types (I've made up the names). The Orig (0~99) could output a Modulation Index from 0~13.1 (Mod.Index is the scientific measurement of the Modulator output value). The X (0~99) could output a higher range 0~25.1 Modulation Index. The CX (0~127) was similar to the Orig with a range 0~12.6 Modulation Index but the bias was different.

PARAMETERS - The classic FM synths used Rates and Levels for most of their parameters. The subsequent generations were simplified to the more "normal" synthesizer parameter-set using ADSDR for envelopes.

ALGORITHMS - Algorithms are the combinations of Modulation and Carrier Operators available on the synth. The classic FM synths used 6-operators and had 32 algorithms. The exception was the DX-9 with 4-operators and 8 algorithms. This 4-op design was carried forward onto the subsequent synths. The CX/FB computer range also used the same 4-op design except that the operators were numbered in reverse order.