Personally I would track down the Principles of FM synthesis first.

If you know how you can make Vibrato with help of an Oscillator and an LFO, you are off to a good start.

Just have the LFO Modulate the Pitch (Frequency) of the Oscillator. Then Adjust the speed and depth (level) to your liking.
The difference between this and FM synthesis is that you replace that LFO with an Oscillator. An oscillator, unlike LFO, track to the keyboard and operates at audio frequencies. Then you adjust the level for brightness, and adjust it's linear frequency controls such as Coarse and Fine to change the timbre. The Coarse changes the harmonic frequencies, Fine changes the inharmonic frequencies.
The clue is that most instruments have both harmonic frequencies and inharmonic frequencies, but for the sound to reproduce a pure tone,
the dominating factor should be be the Harmonic Frequencies. If you change the Coarse to 2 on the modulator, you get a square wave (similar). And you can make a flute easilly

An operator however have three components basically, An Oscillator, AMP and an Envelope Generator to control the AMP.
When an Operator is a Carrier (the one that you hear), the Envelope Generator and level, will change the Volume over time. When the operator is a modulator, (modulates the carrier, as an LFO would). The Envelope Generator and level would affect the brightness of the sound. While the frequency settings such as Coarse (basically a multiplier of the base frequency) will affect the timbre.

Best way to start of, is by using two Operators, a Carrier and a Modulator. Find an algorithm where you have two Operators above eachother, the bottom one is the carrier. Adjust the levels of those two operators, and you will get the hang of it.

The is no correct way to approach learning about FM digital synthesis. The fact that you're curious is enough to begin learning. Start anywhere, just start!

To quickly answer your question, on an FM-X Part you can view the Algorithm
Press [EDIT]
Press [PART SELECT] of the FM -X Part
Touch "Common" in the lower left portion of the screen (or the lower [COMMON] button)
Touch "Part Settings" > "Algorithm"

An algorithm is a recipe. A recipe for constructing different sounds. The different arrangement of the Operators can very much influence the sound you get as a result. Operators can be Carriers (sound makers) or Modulators (timbre influence). The number of Carriers can greatly influence the result. If you have one Carrier all the timbre influence goes to make a single complex tone. If you have multiple Carriers the sound can take on multiple identities. For example, a synth lead, a bass sound might use one Carrier, while a two Carrier Algorithm would allow for a complex attack, for example, one Carrier and its Modulators create the percussive attack, while the second Carrier and its Modulators create the body of the sound. The more Carriers the more separate components you can construct.

Carriers are always shown on the bottom of the recipe (algorithm) Modulators are always stacked above another Operator.

You select Algorithms based on what you need a sound to do. As you begin exploring FM-X, the relationship between Modulator and Carrier is the core of deconstructing a sound. If the Carrier is sounding a Sine Wave, increasing the output level of a Modulator directly above it in the recipe, cause more and more complex harmonics to occur. The output level increase is the "Modulation Index".

When you select an Algorithm that is all Carriers, you have basically a drawbar organ with 8 draw bars. Only the pitches are totally tunable. Organs combine Sine waves in this very fashion to build complex harmonic tones. The drawbars on an organ are just Volume Sliders for each harmonic.

Once you find the Algorithm that makes the sound you are exploring. Isolate just the Carriers. Use the fourth row of buttons to Mute and unMute the Operators. Then turn on a Modulator (stacked above the Carrier) adjust the output level of the Modular to hear how the amount of Modulator changes the timbre (harmonics) in the resulting output from its Carrier.

Hello Robin - thank you for the pointers, and yes, I have had a go at just the thing you describe - but the yawning chasm between 2 operators, and 8, is a bit daunting - will I live long enough? So I figured to short-circuit the learning curve, and start with the basic shapes of various instruments. I have a basic organ sound (8 carriers) to play with (the 'drawbar' settings were explained in the article concerned), and have a few bell-like sounds too, and can play with those to see what effect moving sliders and knobs have ... but what about violin, trumpet, piano, clarinet etc:? I thought if I had a basic shape, I could then play with those to much greater effect than thousands of bell-like tones (which is what I get most of the time!). I will persevere along your lines anyway, out of curiosity ...

Hello BM, and your procedure matches mine exactly - only 'Algorithm' isn't there - it's a box named 'Pitch' instead. I notice though that you say to use the Part number of the FM Part (Part Select) - I've been using single Part FM sounds as being the simplest and even I can't go wrong - but have I? If I used a multi-Part sound with (at least) one of them being FM, and selected that Part instead of Part 1, would I then get the 'Algorithm' box back? I'm away from Montage right now, so can't just try it - I will later! But I wonder if that will give me what I'm looking for, which is the basic algorithm shape for a trumpet (say). I can see that different algorithm shapes may produce a trumpet-like sound using different slider and knob settings, but I imagine that experience has produced the 'best' basic shape?

Rod wrote:

Hello Robin - thank you for the pointers, and yes, I have had a go at just the thing you describe - but the yawning chasm between 2 operators, and 8, is a bit daunting - will I live long enough? So I figured to short-circuit the learning curve, and start with the basic shapes of various instruments. I have a basic organ sound (8 carriers) to play with (the 'drawbar' settings were explained in the article concerned), and have a few bell-like sounds too, and can play with those to see what effect moving sliders and knobs have ... but what about violin, trumpet, piano, clarinet etc:? I thought if I had a basic shape, I could then play with those to much greater effect than thousands of bell-like tones (which is what I get most of the time!). I will persevere along your lines anyway, out of curiosity ...

Which algrorithm to choose is kind of hard to imagine unless you have more experience. But the good thing is, that the algorithm arrangements are made in a way, that you can switch to another one that is similar, but might give you what you need.

Don't think about how many Operators there are, just focus on the ones you need. First you need think of what you want to make,
then go one operator at a time. Sometimes there are two carriers, each with a stack of modulators, with this, you can make a sound of two layers. Strings and Piano etc. it's simply 4 operators x2. With this simple guide, I will help you understand why you stack modulators on top of eachother, why two modulators can modulate a single carrier etc.

Spend some time with 3-operator setups, with different arrangements.

1.
Modulator>Modulator>Carrier

2.
Modulator>Carrier
Modulator>

3.
Modulator>>Modulator>Carrier
Modulator>

When programing, ask yourself. "Do I need the sound to be more complex?" Do I need the sound to have more flavors of Harmonics? Do I need articulation? How does the timbre of the instrument I'm making, change over time?

Take a look at nr.1:
It's similar to what you've done till now. But you have yet another modulator, that modulates the modulator, that modulates the carrier.
This setup allows you to get a more complex sound. For each operator added in height, the complexity encreases, and mostly you don't need more then 4 on top of eachother, but that is up to you. Try fiddling with the level knobs, and also try with envelopes. Think of a sound but more complex then a flute, maybe a trumpet? Think of how a Trumpet sounds. It's louder at the beginning (Adjust the carrier envelope), it's also brighter (encrease levels on the modulators with envelopes at the beginning). Near no release, it stops when you stop blowing, use reverb effects instead. 🙂
You can also use the pitch envelope to articulate the sound.The beginning of the trumpet sound is very essential for it's character.

Take a look at nr.2:
Two modulators, modulates a Carrier. This one is less complex then nr.1, but you have more low complexity options.
On one modulator, use high coarse and short attack and decay settings to make a "Click" sound.
On the other, use a more slow decay. Now you can get something like a Toy'ish Electric piano. Or maybe a Rhodes.

Nr.3:
This is a more complex version of Nr.2. Try to make an EP. But more like Clavinet.
Clavinet sounds dirty'er and has more going on in the sound. So going for more complexity might be the right thing to do.

For me I usually imagine the sound I want to create first, and then try to make it.
With an analogue synth, misstakes happen and it might be a good one. With FM, it's the otherway around.
FM is in many ways opposite of Subtractive synthesis. You add Harmonics instead of removing them.
But FM gives you alot more control over the harmonics.

Remember:
Higher stacks of Operators encreases complexity.
Level encreases brightness. (also controlled by the Envelope generator).
Wider stacks of Operators adds more of the same complexity.
Coarse & Fine changes the Timbre.

Not very complex:
Woodwinds
Rhodes
Whistle

More Complex:
Brass
Clavinet
Guitar

Very complex:
Noise
Explosions
Wind
Breathe
Clicks
Percussion

Hope this helps.

Hello BM, and your procedure matches mine exactly - only 'Algorithm' isn't there - it's a box named 'Pitch' instead. I notice though that you say to use the Part number of the FM Part (Part Select) - I've been using single Part FM sounds as being the simplest and even I can't go wrong - but have I?

See page 132 of your Reference Manual

Hello Robin - and a big Thank you for the trouble you have taken to condense a complex subject so concisely! I'm sure you'll have a lot of fans following your guidelines - not just me! I will play with things in the progressive manner you suggest, and see how far I get - I agree it would be better if I arrived at a conclusion myself, rather than picking one off the shelf, and you may have just made that possible!

Hello BM too, my mistake (Good Lord!). I imagined that I could choose two Attributes, so went 'FM-X' and 'Single Part' ... when it dawned on me that I can't do that I went 'FM-X' only and picked out the green 'uns from the resulting screen. That works! 'Algorithm' is back where it should be. I'm looking at Algorithms 15, 16, and 17 (which are all exactly the same) and their associated settings. I'm surprised at how very sensitive they can be, both sliders and knobs. I shall continue twiddling, but in a somewhat more organised manner ...

Hello Dmitry as well - that looks a very useful treatise! Thank you! I have downloaded it, and hope to match up my experiments with the relevant parts as I progress, hopefully filling in the theory gaps with the practical results. One can dream ...

I'm looking at Algorithms 15, 16, and 17 (which are all exactly the same) and their associated settings. I'm surprised at how very sensitive they can be, both sliders and knobs. I shall continue twiddling, but in a somewhat more organised manner ...

Please look closer. Why on earth would there be three exactly the same? - please look closer, and then let your curiosity be your guide. This will introduce you to that other word you see on this Algorithm screen, that you don't understand yet: "Feedback".

Your on your way... 🙂

Rod wrote:

Hello Robin - and a big Thank you for the trouble you have taken to condense a complex subject so concisely! I'm sure you'll have a lot of fans following your guidelines - not just me! I will play with things in the progressive manner you suggest, and see how far I get - I agree it would be better if I arrived at a conclusion myself, rather than picking one off the shelf, and you may have just made that possible!

I'm very happy for your appreciation. I hope people find the infomation usefull. I love to teach, but sorry for my, not so good english.

I will give you a bonus experiment.

Select an Algorithm, where every Operator is a Carrier. And no, we are not making an organ.
I will introduce you to additive synthesis, well Organs are additive synthesizers, so there you go.
We are making a Sawtooth wave.

Remember, every sound can be made with the help of sine waves.

Algorithm 1:

Parameter setting for a Sawtooth wave using additive synthesis method:

Operator 1: Coarse 1, Level 128.
Operator 2: Coarse 2, Level 100.
Operator 3: Coarse 3, Level 85,
Operator 4: Coarse 4, Level 60
Operator 5: Coarse 5, Level 50
Operator 6: Coarse 6, Level 45
Operator 7: Coarse 7, Level 35
Operator 8: Coarse 8, Level 30

Adjust the levels or change the Coarse settings to create the saw wave of your liking.
Think of what you can do with this. With envelopes changing the levels, or any other modulation or even the motion sequencer.
You can simply make an automated morphing wave in a totally different way then FM. You can control the end result in interesting ways.

Think about what can be possible with Algorithm 68 and 84.
With 68, you can modulate an Operator by using an additive source.
With 84, you can modulate your saw wave.
And don't forget the automations you can do with LFO's, motion sequencer, envelopes etc.
You can use the Velocity sensitivity to affect the timbre as well, to all your experiments.

It should sound close to a saw wave, to get a perfect one, you might need infinate amounts of sine waves. But, you may wonder how something simple as sine waves, together, can create such harmonicaly rich sound.

I came up with even more things for you to try out.

Velocity:
In a basic modulator>Carrier arrangement. Adjust the Velocity sensitivity, for the modulator. When you strike a key on a piano at different velocities you not only affect the Loudness, but also the brightness and timbre. Use Velocity sensitivity to control the level of the modulator for brightness, and the sound becomes much more interesting to play. Use Velocity sensitivity on the carrier to control the loudness.

Try this on the nr 2. arrangement mentioned earlier. You can make a clav or many variations of EP's and make them much more fun to play by using the Velocity sensitivity to make it feel more like a piano. Add some Distortion and phaser effects to really rock the floors. 🙂

Keyboard level scaling:
You can affect the level of an operator not only by help of the Velocity, Envelope generator, level parameter, LFO etc. You can also use the keyboard. Sometimes the sound doesn't always sound the way you want across the keyboard, so you can change the level according to the notes you play. Sometimes the higher notes may sound awfull, but it sounds great at the lower notes. You can fix this by adjusting the level for the higher notes to your liking.
Like always, Doing this on a carrier adjusts the volume, or adjust the brightness by doing it on a modulator.

Fixed Frequency:
You can make an operator operate at a fixed frequency. With this, you can play any note on the keyboard, and the operator will output the same frequency no matter what. You just dial in the frequency and it stays there. This is usefull, when making percussion sounds. Or it can be usefull when you are making clicks, if you want the click to stay the same across the keyboard. This can also turn the operator into an LFO.

In fact , I can't get into the Algorithm as well , unless I have have to press the Edit button twice to make it flash ... but when the Edit button flash you cannot change the Algorithm ... see my video link below
Thanks

https://youtu.be/tsTR3ypKwz0

The correct way to change the algorithm is to edit a PART, make sure the lower-left "Common" touchscreen box is highlighted in blue or touch the "Common" square to highlight it, then choose the left-side (touchscreen) menu option "Part Settings" -> "Algorithm". Then touch the "Algorithm" box and enter a number, or exit out of that and use the DATA DIAL to spin to a different algorithm, or press the Algorithm Search touchscreen area to select how "wide" you want the algorithm (how many carriers, or mixed outputs) and how "tall" you want your algorithm (how many modulators+carrier you want as the maximum modulator chain "height"). It does require some understanding of what a modulator is, what it does, how that relates to the carrier, how ratios of modulator vs. carrier can create more dissonance or consonance, etc. But really, I'm just trying to address how to select an algorithm without worrying about why you would do this or how anything relates.

When you press "[EDIT]" (the physical button) for the second time, notice what the top of the screen says (upper left) - it says "compare". Feel free to consult the manual on what compare does for more detail. Also look below the [EDIT] button and notice it says "COMPARE" which is the function of the button while flashing. Compare will show you the settings of the performance BEFORE you made any changes to it - so if you edited many things on the screen - but "oops" changed one the settings which was a number from 0-127 - and don't remember what it was - press the [EDIT] button to make it flash and show the settings before you started editing the settings.

Of course while the system is showing you old settings (the STOREd settings) - it doesn't allow you to edit these.

I think the mistake you made in pressing [EDIT] twice is good - because hopefully it lets you learn about that feature which may have otherwise not surfaced without mistaking it as some way to edit Algorithms.

Thanks Jason, but
First: I started a new Init Normal FM and choose Cool Trance as my first sound (part 1) ... then I pressed Edit and the screen showed me no Algorithm but it just a Pitch button show on the screen ... you watch my video again and will see it... it's showed the Algorithm when i pressed second time on edit button ... not like what you stated above.

You are starting to answer your own question - but are not yet armed with the full picture. Hopefully I can help detail this for you.

1) You start with the "Init Normal (FM-X)" performance - which starts with a simple sine wave as PART 1.
2) You begin to modify the performance by changing PART 1 to another preset.
When you do this - ask yourself, am I adding an FM-X preset? Or is this an AWM2 preset? Note that loading "Init Normal (FM-X)" is just a starting point. It starts you with an FM-X PART. But if you replace the FM-X PART with something else, it's not going to prevent you from doing this. I just pose this as an experiment. I'm not at the keyboard to see for myself and am not looking at the docs to try to see.
3) After replacing the PART with a new part - when you press [EDIT] to make it flash -- it will show you the ORIGINAL setting which was the original FM-X part before you replaced it with "Cool Trance". IF Cool Trance is NOT FM-X (is an AWM2 PART) then what you are seeing is the NEW part (Cool Trance) which is AWM2 and does not have that menu switch to the menu when you press [EDIT] which exposes what the part was previously.

So go back and retrace your steps WITHOUT replacing the PART with "Cool Trance". See that you can edit the algorithm when the steps I gave are followed.

Next, find a preset PART you *know* is FM. Maybe select "FM Sweeping Poly" as the PART you replace the Init PART with instead of using "Cool Trance"

You cannot edit an algorithm of an AWM2 PART because AMW2 doesn't have algorithms like FM-X does. AWM2 is sample playback with waveforms, loop points, velocity/range mapping, etc.

Hello Tho - you did what I thought initially! You have to start with an FM sound - no point going to 'Init' and then to another category as you only switch to the other category. Your 'Cool Trance' is composed of several Parts, of which only the Part 1 is FM ... that should be OK though, except I don't see you pressing the 'PART 1' button to get into the algorithm - just pressing the 'PART SELECT - COMMON' button won't do the job. You must then select the Part that is FM - in this case Part 1 - the lower 'COMMON' button also lights up - and you'll find that the 'Algorithm' box has replaced the 'Pitch' box. At least I hope so ... good luck! Frightening, ennit?