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F.M. Synthesis


Contents
  1. Definitions
  2. What is FM?
  3. What is DX-FM?
  4. What are Operators and Algorithms?
  5. Critical relationship between "M" and "C"
  6. Non-coincident and Coincident Series
  7. Series generated by M:C
  8. Modulation Amount
  9. Two Modulators
  10. Tips for using F.M.
  11. FM Synthesizers
FM Synthesis - Printer Version
FM DX Supplement - A deeper look at FM and DX synths
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Definitions

Oscillator
A device for generating waveforms
FM
Frequency (or Pitch) Modulation - Where the pitch or frequency of an oscillator [the Carrier] is modulated by another oscillator [the Modulator]
DX-FM
DX Synthesizer FM - Where both oscillators use Sine-waves and are "musically-tuned" frequencies generated from a keyboard
"C"
Carrier Frequency- The frequency of the oscillator which is being modulated
"M"
Modulator Frequency - The frequency of the oscillator which modulates the Carrier

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:-

  1. There will always be a harmonic at "C", the Carrier frequency.
  2. To the right of "C" (harmonics greater than "C"), there will be harmonics following the series C+M, C+2M, C+3M, C+4M etc.
  3. To the left of "C" (harmonics less than "C"), there will be harmonics following the series C-M, C-2M, C-3M, C-4M etc.
So if "C"=9 and "M"=2 (i) there will be one harmonic at 9, (ii) on the right, there will be the harmonic series 11, 13, 15, 17 etc, and (iii) on the left, there will be a similar series 7, 5, 3, 1 etc.

Harmonic Series - "C"=9 and "M"=2
| | | | | | | | |
1 3 5 7 9 11 13 15 17
C-4M C-3M C-2M C-M Carrier C+M C+2M C+3M C+4M

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).

Now, it is important to note the following:-

  1. If "C" was detuned down to 8.5, then the whole harmonic spectrum would be shifted down by 0.5! So detuning "C" shifts the entire spectrum.
  2. If "M" was detuned down to 1.5, then the Sidebands would move in closer and be separated by 1.5! So detuning "M" compresses or expands the Sideband separation.
  3. IMPORTANT - Of the left-hand-side Sidebands, there comes a point where the Sidebands go beyond zero. Sidebands with negative values are "reflected Sidebands" (reflection point = zero, silence). Don't worry about this... ignore the minus-sign and treat it as another Sideband.

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)

In M:C = 2:3, the reflected Sidebands are coincident with the non-reflected Bands. In this case, there are two components in each Sideband.
In M:C = 3:5, the reflected Sidebands do not coincide with those of the non-reflected. In this case, each Sideband stands alone.
In M:C = 1:1, the Sidebands are coincident except that there is a Band at the Zero frequency. Obviously, you cannot hear this particular frequency as it is silent.

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.

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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.

SERIES generated by Modulator-to-Carrier combinations ("M"=columns; "C"=rows)
C\M 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1:1 2:1 3:1 4:1 5:1 6:1 7:1 8:1 9:1 10:1 11:1 12:1 13:1 14:1 15:1 16:1
2 1:1 1:1 x2 3:1 2:1 x2 5:2 3:1 x2 7:2 4:1 x2 9:2 5:1 x2 11:2 6:1 x2 13:2 7:1 x2 15:2 8:1 x2
3 1:1 2:1 1:1 x3 4:1 5:2 2:1 x3 7:3 8:3 3:1 x3 10:3 11:3 4:1 x3 13:3 14:3 5:1 x3 16:3
4 1:1 1:1 x2 3:1 1:1 x4 5:1 3:1 x2 7:3 2:1 x4 9:4 5:2 x2 11:4 3:1 x4 13:4 7:2 x2 15:4 4:1 x4
5 1:1 2:1 3:1 4:1 1:1 x5 6:1 7:2 8:3 9:4 2:1 x5 11:5 12:5 13:5 14:5 3:1 x5 16:5
6 1:1 1:1 x2 1:1 x3 2:1 x2 5:1 1:1 x6 7:1 4:1 x2 3:1 x3 5:2 x2 11:5 2:1 x6 13:6 7:3 x2 5:2 x3 8:3 x2
7 1:1 2:1 3:1 4:1 5:2 6:1 1:1 x7 8:1 9:2 10:3 11:4 12:5 13:6 2:1 x7 15:7 16:7
8 1:1 1:1 x2 3:1 1:1 x4 5:2 3:1 x2 7:1 1:1 x8 9:1 5:1 x2 11:3 3:1 x4 13:5 7:3 x2 15:7 2:1 x8
9 1:1 2:1 1:1 x3 4:1 5:1 2:1 x3 7:2 8:1 1:1 x9 10:1 11:2 4:1 x3 13:4 14:5 5:2 x3 16:7
10 1:1 1:1 x2 3:1 2:1 x2 1:1 x5 3:1 x2 7:3 4:1 x2 9:1 1:1 x10 11:1 6:1 x2 13:3 7:2 x2 3:1 x5 8:3 x2
11 1:1 2:1 3:1 4:1 5:1 6:1 7:3 8:3 9:2 10:1 1:1 x11 12:1 13:2 14:3 15:4 16:5
12 1:1 1:1 x2 1:1 x3 1:1 x4 5:2 1:1 x6 7:2 2:1 x4 3:1 x3 5:1 x2 11:1 1:1 x12 13:1 7:1 x2 5:1 x3 4:1 x4
13 1:1 2:1 3:1 4:1 5:2 6:1 7:1 8:3 9:4 10:3 11:2 12:1 1:1 x13 14:1 15:2 16:3
14 1:1 1:1 x2 3:1 2:1 x2 5:1 3:1 x2 1:1 x7 4:1 x2 9:4 5:2 x2 11:3 6:1 x2 13:1 1:1 x14 15:1 8:1 x2
15 1:1 2:1 1:1 x3 4:1 1:1 x5 2:1 x3 7:1 8:1 3:1 x3 2:1 x5 11:4 4:1 x3 13:2 14:1 1:1 x15 16:1
16 1:1 1:1 x2 3:1 1:1 x4 5:1 3:1 x2 7:2 1:1 x8 9:2 5:2 x2 11:5 3:1 x4 13:3 7:1 x2 15:1 1:1 x16

] and [ denote non-coincidental reflected bands.
If "C" appears on any "]", then "[" will be a reflected Band (and vice-versa).
][ denotes coincidental reflected Bands.

Series of Harmonics generated
Series 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
1:1 ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][
2:1 ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][
3:1 ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [
4:1 ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [
5:1 ] [ ] [ ] [ ] [ ] [ ] [ ]
6:1 ] [ ] [ ] [ ] [ ] [ ]
7:1 ] [ ] [ ] [ ] [ ]
8:1 ] [ ] [ ] [ ] [
9:1 ] [ ] [ ] [ ]
10:1 ] [ ] [ ] [ ]
11:1 ] [ ] [ ] [
12:1 ] [ ] [ ]
13:1 ] [ ] [ ]
14:1 ] [ ] [ ]
15:1 ] [ ] [ ]
16:1 ] [ ] [
Series 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
5:2 ] [ ] [ ] [ ] [ ] [ ] [ ]
7:2 ] [ ] [ ] [ ] [ ]
9:2 ] [ ] [ ] [ ]
11:2 ] [ ] [ ] [
13:2 ] [ ] [ ]
15:2 ] [ ] [ ]
7:3 ] [ ] [ ] [ ] [ ] [
8:3 ] [ ] [ ] [ ] [
10:3 ] [ ] [ ] [
11:3 ] [ ] [ ] [
13:3 ] [ ] [ ]
14:3 ] [ ] [ ]
16:3 ] [ ] [
Series 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
9:4 ] [ ] [ ] [ ] [
11:4 ] [ ] [ ] [
13:4 ] [ ] [ ]
15:4 ] [ ] [
11:5 ] [ ] [ ] [
12:5 ] [ ] [ ] [
13:5 ] [ ] [ ]
14:5 ] [ ] [
16:5 ] [ ] [
13:6 ] [ ] [ ]
15:7 ] [ ] [
16:7 ] [ ] [

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.

  1. When there is no modulation, only the Carrier frequency exists (no sidebands).
  2. At low levels of modulation, the amplitude distribution (the heights is the harmonic spectrum) is sort of tent-like with the apex being "C".
    As the modulation is increased to moderate levels, the distribution becomes more bell-shaped (centred around "C").
  3. From here, small increases in modulation make the bell-shape wider (ie more sidebands).
  4. As the modulation is increased to much higher levels, the distribution changes into a pair of bell-shapes at the middle orders with a "spike" at "C".
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.

The examples given are (i) M:C = 1:7 [with no reflected sidebands], (ii) M:C = 3:4 [with reflected sidebands which are non-coincident], and (iii) M:C = 1:1 [with reflected sidebands which are coincident].


{--------- M:C = 1:7 ---------} - {------------M:C = 3:4 -------------} - {--- M:C = 1:1 ---}
                                -                                       - |           
               |                -          |                            - |  |        
               |                -          |                            - |  |        
               |                -          |                            - |  |          
            |  |  |             - |        |        |                   - |  |  |       
            |  |  |             - |        |        |                   - |  |  |       
         |  |  |  |  |          - |  |     |        |        |          - |  |  |  |         
      |  |  |  |  |  |  |       - |  |     |  |     |        |        | - |  |  |  |  |      
2  3  4  5  6  7  8  9 10 11 12 - 1  2  3  4  5  6  7  8  9 10 11 12 13 - 1  2  3  4  5  6  7 
                                -                                       -    |         
               |                -          |                            - |  |         
               |                -          |                            - |  |  |      
               |                -          |                            - |  |  |        
            |  |  |             - |        |        |                   - |  |  |  |     
         |  |  |  |  |          - |  |     |        |       |           - |  |  |  |     
      |  |  |  |  |  |  |       - |  |     |  |     |       |      |    - |  |  |  |  |      
   |  |  |  |  |  |  |  |  |    - |  |     |  |     |  |    |      |    - |  |  |  |  |  |   
2  3  4  5  6  7  8  9 10 11 12 - 1  2  3  4  5  6  7  8  9 10 11 12 13 - 1  2  3  4  5  6  7 
                                -                                       -       |      
                                -                                       -       |      
               |                -          |                            - |  |  |  |   
               |                -          |                            - |  |  |  |     
         |     |     |          -    |     |                 |          - |  |  |  |  |  
      |  |  |  |  |  |  |       - |  |     |  |     |        |        | - |  |  |  |  |  
   |  |  |  |  |  |  |  |  |    - |  |     |  |     |  |     |        | - |  |  |  |  |  |   
|  |  |  |  |  |  |  |  |  |  | - |  |     |  |     |  |     |  |     | - |  |  |  |  |  |  | 
2  3  4  5  6  7  8  9 10 11 12 - 1  2  3  4  5  6  7  8  9 10 11 12 13 - 1  2  3  4  5  6  7 
                                -                                       -          |     
                                -                                       -          |     
                                -                                       -       |  |  |  
               |                -          |                            - |     |  |  |     
      |        |        |       -    |     |                 |        | - |     |  |  |     
   |  |  |     |     |  |  |    -    |     |  |        |     |        | - |     |  |  |  |  
|  |  |  |     |     |  |  |  | -    |     |  |        |     |  |     | - |  |  |  |  |  |  | 
|  |  |  |  |  |  |  |  |  |  | - |  |     |  |     |  |     |  |     | - |  |  |  |  |  |  | 
2  3  4  5  6  7  8  9 10 11 12 - 1  2  3  4  5  6  7  8  9 10 11 12 13 - 1  2  3  4  5  6  7 
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.

For more details on the calculating the amplitudes, see FM DX Supplement. To look at FM amplitudes graphically, see FM Spectrum Graphs (contains animated GIFs).

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).

~ Two-into-One ( M1 + M2 : C )

     M1-->-+->--C
     M2-->-+
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 :

For M2:C = 2:5, you will get -     5   ,  7   ,  9  , 11  , 13  ...
                                          3   ,  1  , (1) , (3) ...

For M1:C = 3:5, you will get -     5   ,  8   , 11  , 14  , 17  ...
                                          2   , (1) , (4) , (7) ...

The end result will be both these added together. 
Where M1 + M2 : C = 3 + 2 : 5 -    5   ,  7   ,  8  ,  9  , 11  ...
                                          3   ,  2  ,  1  , (4) ...

~ Two Modulators In-Series ( M2 : M1 : C )

     M2-->--M1-->--C
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 :

For M2:C = 2:5, you will get -     5   ,  7   ,  9  , 11  , 13  ...
                                          3   ,  1  , (1) , (3) ...
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:-

Synth DX-7, 5, 1 DX-9 DX-21, 27, 100 CX-5, 7, 11
- TX-7, 816, 802 - TX-81Z FB-01
Mod.Output Orig (0~99) X (0~99) CX (0~127)
Parameters Rate/Level ADSDR
Algorithms 6-op 4-op CX 4-op
Note - Elka EK-44 and EM-44 fall under the DX-21 category.

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 classic 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.


For a deeper look at DX synthesizers and calculating Harmonic Spectrum Amplitudes, see FM DX Supplement.

For a familiarisation of a selection of synthesizers, see Synthesizer Layouts.

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