Since FM-X is meant to be "compatible" with the DX7 - at least from the standpoint of a simple sine wave modulator/carrier 1:1 pair - I would suggest exploring the source code to Dexed to see how that software relates to the question. That software has an engine that is said to most closely emulate the DX7.

Yamaha's FM implementation is actually phase modulation (makes operator feedback stable), not frequency modulation. Some quick searching suggests the maximum modulation index on the DX7 is 13 radians, so you might want to start there.

Yamaha's FM implementation is actually phase modulation (makes operator feedback stable), not frequency modulation. Some quick searching suggests the maximum modulation index on the DX7 is 13 radians, so you might want to start there.

Brain Brian!!!

THANK YOU!!!

This mentally resolves it for me. And makes perfect sense.

Part of my curiosity had always wondered "Where's the phase shift?" when looking at the options for FM-X, and I'd never come up with any reasonable explanation as to why it wasn't there.

@Jason, cheers! Will look into it. Wasn't aware of this historical view accuracy. etc.

Brian, you're a GEM!

Just did some experiments in my code... yep, I can replicate the sound/timbre of the MODX with phase modulation!

And I see why they did it, it's at least an order of magnitude less computationally expensive. Maybe 2.

Brian, if you're still out there... I spoke too soon.

One thing I did created one timbre the same.

Everything else is just annoying, and not the same.

And my searching only leads to incredibly hard to fathom, often seemingly more incorrect, assumptions about how the DX FM is working/doing its thing.

What is being phase shifted? By what?

Probably the same stuff you've already researched ...

https://gearspace.com/board/electronic-music-instruments-and-electronic-music-production/902271-fm-patents-am-synthesis.html
https://moinsound.wordpress.com/2011/03/04/frequency-modulation-or-phase-modulation-synthesizer-technologies
https://www.reddit.com/r/synthesizers/comments/45cdm8/im_trying_to_get_my_head_around_the_difference/
Visualization of a math/audio framework (may help)
https://www.youtube.com/watch?v=zWdjeS9pTrA
https://scsynth.org/t/coding-fm-synthesis-algorithms/1381
https://www.native-instruments.com/forum/threads/what-is-the-formula-for-fm-synthesis-with-2-sine-waves.191734/

@Jason

Can you, after all these videos and readings, answer the questions: What is being phase shifted, and by what?

I, like you, have gone down this rabbit hole.

And am sillier than when I started.

Not sure why this of such interest, when, as far as we "musicians" are concerned all that matters is the output "Audio" result (and its harmonic content in relation to its similarity to sounds we might want to approximate i.e. synthesise).

I am no expert on FM-X. However, AM, FM and PM (Phase Modulation) were covered in my Engineering Degree in relation to Telecommunications. In this case, the "Carrier" was the transmission medium. Always at extremely high frequency (example Radio), because "Audio Frequencies" do not propogate through air very well, and they don't propogate through copper (electrical) or Fibre (Optical) at all.

The Carrier is called the Carrier because it Carries the information you want to transmit (think of a cross country Postal Train carrying thousands of letters and parcels). The information it carries is actually the Modulating Signal or Signals. For efficiency, FM/PM transmission systems were designed to Carry hundreds/thousands of Modulators (Channels) worth of information.

The net result (modulation) on the carrier between FM and PM was the same, except the implementation was slightly different. Each of PM and FM had it's own set of pros and cons, allowing more options depending on communication requirement. PM was/is also known as Indirect Frequency Modulation, and by example was better suited for high bandwidth data (Digital).

Anyway, the terminology "Frequency Modulating" can be confusing.

The Frequency of the Carrier is NOT being Modulated by the Frequency of the Modulator. The Frequency shift in the Carrier is Proportional instead to the AMPLITUDE of the Modulator. As the Amplitude of the Modulator changes over time, so does the Frequency change in the Carrier. This is known a The Index.

The Ratio dictates how frequently the Frequency of the Carrier is being modulated, per second.

You would assume that the Fundamental Frequency of the Carrier (Pitch) would be changing over time, but in fact it isn't (lots of maths). When the Ratio is 1 or greater, the fundamental stays constant and instead, the number, the energy content and dispersion of frequency "side bands" changes, as the Ratio increases. We know sidebands as Harmonics. These sidebands are where the information is stored for transmission, like saddle bags on a Pony.

This makes sense. FM would be of no use as a Communication Transmission system if the Carrier "Centre" frequency changed. Otherwise the far end receiver would not know "where" to look for it, or pick it up. Instead, for example, the Receiver knows the Carrier can be found on 2.83GHz radio frequency.

So when you play a C#, the listener (receiver) will hear a C#. Whether that C# is musically pleasant is a different matter.

Don't know if any of that helps, or is even relevant to this discussion. May be of interest to others though.

If you ultimately wanted to create code as an exercise in understanding the math behind FM-X (a marketing term, not a scientific description) then I go back to pointing you towards open source projects that others have already decided sufficiently "nail" the DX7 for a broad range of Sysex "patches". At the very least a two operator sine wave stack which would be the starting point for any such investigation.

I haven't written an algorithm that simulates DX7-style "FM" so I don't have a silver-platter answer for you.

How the envelopes (first order, to me is AEG) work is also a big contributor. The open-source software also emulates this.

Its Christmas, I'm on Holiday and have time to kill.

Although I have no interest in open source and DIY design, may be I can assist helping to visualise the concepts.

The way I got through my Engineering life was to try, at first, to understand the "mechanics" (visualise) of what was going on, before I could understand (or accept) all the maths.

For now, just accept that Phase Modulating a Carrier Wave and Frequency Modulating a Carrier Wave have the same Net affect on the Wave.

They are two different "means" to the same end result.

In Frequency modulation, you are specifically "instructing" the Carrier Wave to attain a specific frequency at a specific point in time.

In Phase Modulation you are instructing the Carrier Wave to attain a Specific Phase "Offset" at a specific point in time.

The Phase Offset is relative to the Start Phase (Zero Time) or some other reference (for example a 2nd, identical, unmodulated wave... or a "clock").

I have seen Phase modulation diagrams where the picture of the waveform "cuts" and then suddenly reverses direction.

This is not what is going on. The Phase requirement, per consecutive instant of time is constantly changing smoothly (Analog). In order to constantly meet these phase requirements, the Carrier must alternately "retard" and "hasten".

That is, the Carrier must constantly and alternately slow down and speed up. In effect, it must constantly vary its Frequency. Which is exactly what is happening to a Frequency Modulated wave.

This is why Phase Modulation is also known as Indirect Frequency Modulation.

To visualise a "freeze frame" of a Carrier Wave undergoing FM or PM modulation, picture a "Slinky Spring" stretched, but at rest on some rough "high friction" surface, say a carpet.

If you look at the Spring, you will see the coils alternately closer together (Squashed, Higher Frequency) and further apart (Separated, Lower Frequency). But there is an "average" frequency (Wavelength or distance between coils) for the whole Spring which is its "Fundamental" frequency.

The process of alternating "Retardation" and "Quickening" is a slightly different process than direct control of the Speed. Imagine a Car. Retardation is Brakes On, Quickening is Brakes Off. The Retardation (Frictional) force is being controlled rather than the "Direct" Speed of the Engine. FM is control of the Accelerator... Go Faster, Go Slower.

The net result in each case is the car going alternately slower and faster. Its the same.

Briefly Phase Modulation is used for greater accuracy (we are talking telecommunications now). Better accuracy means more bandwidth (a greater volume of information can be transmitted through identical media).

In PM, the information being carried (in The Carrier) is stored in the Phase relative to the Reference. If both the Carrier and Reference are transmitted through the *same* media, the frequencies of each may be distorted (identically) by the media but their relative Phase remains intact (not distorted).

In FM, the explicit frequency of the Carrier at a moment in time contains the information. But the exact Frequency may or may not have been distorted (changed, drifted) during transmission. But regardless, there are cases where FM is still the preferred option.

So maybe, for whatever reasons, Yamaha have decided to implement Indirect Frequency Modulation (PM) over pure FM. Who knows.

What matters is, we don't "hear" PM any differently than FM. The end result is still the same and all of Dr Chowning's discoveries remain the same.

For reference, I'm not trying to understand FM-X for coding's own purposes. Instead, to understand it from/for increasingly complex sound design ideas, along with motion of some of the parameters, wherein I've reached the limit of what I can do through trial-and-error, and now need to be sure of what's actually going on, so as to know what to move, where, how, and why, to get at various whats.

Exploratory works will continue, but it's time to figure out what's actually going on.

You can hear some of the extremities of this in the file I recently uploaded in another conversation, this one... attached here, below, is a launch pad, for me. It's got no motion, yet. This is a single FM-X Part, playing a simple arpeggio, and I've been heading out further from here... but have started to realise that FM-X is not actually FM, nor is it quite pure Phase Modulation, and there's nowhere I can see to discern what's going on in terms of ranges of relationships.

So a good starting point, I've decided is a 1:1 of mod and carrier, at 99 amp each. Is it possible to figure out exactly what's going on at this point, and use this as a start point for dissecting what happens at divergences from it, and get accurately, predictively, responsively creative?

Or are we stuck with FM-101 discussions of theory and biography impertinent to the actual base mechanics of FM-X?

What you need is an FM Recipe Book.

There are a lot of basic "known" standards, but very little of that seems to written down and indexed.

I've asked for exactly what I need:

insight into exactly what's happening within a 1:1 ratio (both) carrier and mod, at full amplitude of both.

It'd then be nice to know what happens at ratios of 2 for carrier and 1 for mod, and the inverse, again, both at full amplitude.

Casio called their method Phase Distortion, and I think that's the first time it became apparent to the world that Yamaha's FM isn't pure frequency modulation, but instead is some form of modulation of the phase of modulators, and that modulators don't actually modulate the carrier. Instead it feels as though there's summing of the "modulators" and "carriers" occurring.

But this can't be right.

Have you seen this? (See URL).

FM Ratio Overview

You asked about the effect of Ratios (including 1:1). There is no simple explanation, but at least the article condenses it.

If you need to get your head around Side Bands, I recommend playing your test sounds into a plugin Frequency Spectrum Analyser and watch whats happening with different Ratios.

Maybe start with a 2 Sine-Op Stack (1:1) with Mod Level set to get a Saw, and feedback on Carrier to get a "Sharper" Saw.

YouTube demo here:-

https://youtu.be/WwVQXItUi1A

Starting with a Saw at least gives you a familiar starting point picture of "all the harmonics" in a neat descending line.

As soon as you start altering the Ratio, that line will stop "being neat". But the harmonic spikes (sidebands)will stay roughly in the same places (with integer Ratios), they'll just change in height.