This
article is hosted by the Yamaha DX1 worldwide information center

because of it´s quality explaining the complexity of FM Synthesis if
you are

willing to get deeper inside programming a Yamaha DX - Synthesizer.

**C****opyright
for article, design and presentation belongs to
T. Yahaya Abdullah**

If
you want to contact the author, his personal webpage with

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Synthesizers,
Music & Broadcasting

©
T. Yahaya Abdullah

Contents

- Definitions
- What is FM?
- What is DX-FM?
- What are Operators and Algorithms?
- Critical relationship between "M" and "C"
- Non-coincident and Coincident Series
- Series generated by M:C
- Modulation Amount
- Two Modulators
- Tips for using F.M.
- FM Synthesizers

- 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

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".

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".

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.

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:-

- There will always be a harmonic at "C", the Carrier frequency.
- 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.
- 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.

| | | | | | | | | | | | | | | | | |

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:-

- 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.
- 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.
- 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.

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.

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

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
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).

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.

- When there is no modulation, only the Carrier frequency exists (no sidebands).
- 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"). - From here, small increases in modulation make the bell-shape wider (ie more sidebands).
- 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 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].

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.{--------- 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

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-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-->--CThis 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.

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.

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 |

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
paramenter like 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.

**Copyright
by T. Yahaya Abdullah**