Yes, we can. And this is a great method to learn FM synthesis, by learning to build the basic wave shape families. Any Algorithm where one Operator is stacked above another. Which on the original DX7 was any Algorithm except number 32

An audible sine wave can be created by any Operator that is a "Carrier". Carriers are audible, they output sound... And are found at the bottom of an Operator stack. "Modulators" are Operators that are not audible directly, you only hear their influence/affect on the Operator that they influence that is a Carrier.

An Operator is like an analog oscillator, in that it outputs a signal, but unlike an analog oscillator each Operator also includes an Amplitude Envelope Generator to define its role in the resulting output. Rather than sharing an AEG with other signal sources, in FM each signal source has its own attack-decay-sustain-release engine.

Each Operator can be tuned to a frequency within the range of potentially audible frequencies. FREQUENCY determines the pitch. It is often represented as a RATIO. This is where a lot of musicians unnecessarily check out, but shouldn't. It's all basic math that represents musical relationships. RATIO or FIXED are your choices. FIXED is easy, if you fix the frequency of an Operator to 446.7Hz, every note across the keyboard will produce the same pitch... Precisely 446.7 cycles per second. RATIO is easy too because it's MUSICAL. You don't have to figure it out, you play with it!!!! It's when musician try to figure it out separate from what they already "hear" that FM seems difficult. It's really not. Ratio simply explains how the pitches get mapped to Keys!

The FREQUENCY of the "A" above middle "C" is 440 Hz. When the RATIO = 1.00, pressing that Key will reproduce 440 times 1.00 or 440.
If the FREQUENCY RATIO is set to 2.00, pressing that same Key will reproduce 880 or (440 x 2.00) and so on. When set to Ratio 1.00 if you play music on the keyboard it will sound equal temperament tuned as you expect. You know where to find what you need (even if you don't know the math)!

The Ratio number is the "harmonic" when it is a whole integer (1.00, 2.00, 3.00, 4.00, 5.00) these numbers are the harmonic series: whole number multiples of the Fundamental, 1.00)

Okay, that said.
Sawtooth waves are produced by a minimum 2 Operator stack (Modulator:Carrier) when M:C ratio is 1 to 1 (1:1) that means when the Carrier (the Operator on the bottom) is 1.00 and the Modulator above it is 1.00. As you increase the Output Level of the Modulator you will increase its influence on the audio timbre of the Carrier. As you reach about 3/4 of the output level of the Carrier (if Carrier output is 100, set Modulator output at 75) you will start to recognize that the tone starts to change - you will hear an increase in harmonic content. Each whole integer multiple of the Carrier's fundamental will begin to be produced (side bands) - giving a very "sawtooth" sounding wave. All harmonics are included in the sawtooth family.

Square Wave is achieved by increasing the RATIO between our two Operators. By increasing the RATIO to 2:1... Modulator = 2.00; Carrier = 1.00 again at about 3/4 of the output level you begin to hear every other harmonic (just the odd numbered harmonics 1, 3, 5, 7 etc., etc.) this is your square wave - every other harmonic (only odd harmonics).

As you continue to increase the RATIO between the Modulator's pitch and the Carrier's pitch, the narrower the resulting pulse wave. All square waves are pulse waves, a square wave simply is the 'perfect' case where the ON portion equals the OFF portion. If the ON portion is one third of the OFF portion, every third harmonic will be missing, and the pulse wave is considered "narrower". If ON is one fourth of the OFF, every fourth harmonic will be missing, and the wave will be narrower still. And so on.

Sine waves are pure tones like a penny whistle or flute
Square waves are hollow, woody ...like a clarinet, marimba
Pulse waves are nasal, and pinched ...like an oboe, clavinet

Creating any complex tone on a DX7 FM required a short stack (two Operators) in a top:bottom arrangement. This stack indicates the modulator is influencing the carrier. The carrier is the bowed string... You hear it's pitch audibly as it disturbs the air. The modulator is the violinist's left hand which is adding "vibrato" (modulation) to the same string. The left hand is modulating the frequency of bowed string, the Modulator has a frequency (a rate of speed) and so does the Carrier. You do not hear the Modulator directly, you hear it's influence on the Carrier.

When the Modulation speed (frequency) equals the speed of the Carrier you get a Sawtooth wave (each harmonic is generated)
When the Modulation speed is twice that of the Carrier, you get a Square wave (only odd harmonics are generated)
When you continue to increase the speed of the Modulation, the more harmonics go missing, and the narrower the pulse wave that is generated.

Now if you modulate a Modulator with a third Operator stacked above the first two, you can begin to see how Pulse Width Modulation can be generated. You are using yet another Operator to modify the first Modulator's influence on the Carrier.

You can use any Algorithm (a fancy word for the arrangement of the Modulators and Carriers. Any Algorithm with a two Operator stack will allow you to build the basic wave shapes. An Algorithm where all Operators are Carriers (usually listed as the highest number Algorithm) you have a situation called "additive" - each source is an audible sine wave generator, exactly like the drawbars of a tone wheel organ.

16' is the sub harmonic of the fundamental; Ratio = 0.50
5 1/3 is the sub harmonic of the 3rd harmonic; Ratio = 1.50
8' is the Principal (Fundamental); Ratio = 1.00
4' is Ratio = 2.00
2 2/3' is Ratio = 3.00
2' is Ratio = 4.00

A six drawbar organ can easily be built with the DX7's six Carrier Operators in Algorithm 32

You can identify the harmonic produced by an Operator by its number. 1.00 is the fundamental, 2.00 is the second harmonic, 3.00 is the third harmonic, etc. It is the relative volumes of the audible harmonics that make an identifiable sound.

What about all those numbers in between?
The fractions (all those values between harmonics, 1.00 and 2.00, for example) are what represent sounds that function as non-musical. If you consider music "organization" a good argument could be made its because things compliment and harmonize, if "noise" is considered the antithesis of "music", and is disorganization, the region between harmonics is used to describe all sounds non-musical, chaotic, random, detuned... From slightly out of organization to the extreme of random chaos. Everything from so-called "bell tones" to random noise can be generated by combining non-whole integer frequencies... While some bells are clearly musical, others are freighteningly disturbing, as are gongs, and other sounds described by these fractions include explosions and sounds rated completely non-musical.

Hope that helps. Looking at FM from the standpoint of the math of music is only one method to create sound. An approach where you build sounds based on musical relations that are supported by the math is not so intimidating. If you can build the basic wave shapes. Sawtooth and Pulse you have the same source waves as you do on any analog synth. This is a great starting point. Now you just have to suffer with all the "modular" control - each oscillator has its own ADSR, and in the original DX-style FM, you had no filter, so timbre change was done by artful sculpting of the Amplitude Envelopes of the Modulators. If you wanted a sound to get brighter, you use a slowly rising envelope on a Modulator so it's influence increases over time. This is where additional Operators can be extremely useful... Adding movement and depth within a sound over time.

Sorry for repeating some of what you probably already know, but with the reface DX, FM is again something to talk about.

Thanks for this small refresher course on FM synthesis, Bad Mister!
Who would know that this knowledge would once again come in handy?
I'm really hoping for an advanced and user-friendly computer editor for FM synthesis for the Montage.
There's so much that can be done there to finally tame for what has been such a beastly method of sound programming.
An excellent editor can be combined with both spectrum and wave displays, and much more.
It can even contain and be a teaching/training/educational tool.
This is very interesting stuff.

Bad Mister wrote:

Yes, we can. And this is a great method to learn FM synthesis, .........
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. FM is again something to talk about.

Awesome tutorial, Phil!