Copyright by T. Yahaya Abdullah
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.
Copyright
for article, design and presentation belongs to
T. Yahaya Abdullah
If
you want to contact the author, his personal webpage with
additional interesting material can be reached following one of these links:
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http://tyala.freeyellow.com/2fmsynth.htm
Synthesizers,
Music & Broadcasting
©
T. Yahaya Abdullah
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.
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 | 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 |
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:-
| 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.
Certain series have a "x2" or "x3" on them. It is the same series except that
it is transposed upward by that amount.
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.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.
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).
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 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.
As the modulation is increased to moderate levels, the distribution becomes
more bell-shaped (centred around "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].
{--------- 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-->--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.
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 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.
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.
Note - Elka EK-44 and EM-44 fall under the DX-21 category.
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
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
