The Envelope Generator is used to describe change over time. The Amplitude Envelope Generator describes how loudness changes over time. You also have Envelope Generators for the Oscillator (Pitch EG) and the Filter (Filter EG)

It is important to have a clear understanding of how the EG parameters are used to describe/emulate the behavior of musical instrument tones. There are two main categories of musical amplitude shapes.
1) Percussive - sound dies out
2) Self-oscillating - sound can continue as long as pressure is applied

Percussive envelopes are generated by instruments that hammered, struck, or plucked.
Self-oscillating envelopes are generated by instruments where pressure is applied by bowing or blowing

And while these are not the only descriptions of sound, they do describe the majority of musical applications.
‘Hold’ forever, is a fictional situation but one that must be considered as well.

Amplitude Envelope Generators
Let's take a look at the AEG (Amplitude Envelope Generator) - which determines how the sound is shaped in terms of LOUDNESS. How it starts (attack), continues (sustain) and disappears (release). Basically, the envelope describes what happens from KEY-ON and beyond KEY-OFF until silence. It will 'shape' the sound as it exists. Part of what your ear/brains uses to recognize and identify sounds is the loudness envelope.

The AEG (Amplitude Envelope Generator) consists of the following components:
Initial Level
Attack Time
Attack Level
Decay 1 Time
Decay 1 Level
Decay 2 Time
Decay 2 Level
Release Time
These are the parameters that the synth uses to describe what in old analog synthesizers was referred to as the ADSR (Attack-Decay-Sustain-Release). It uses a series of TIME and LEVEL settings to describe the loudness shape of the sound.

Drum and Percussion sounds are trigger events and significantly cannot be held. A Whole Note to a percussionist is more an instruction to not play during that time, than it is an instruction to "hold" anything.

A typical musical (Non-Drum/Percussion) instrument described (think a piano note, hammered string). Here a Whole Note means to hold the sound for that length of time. You can hold a musical note, while a traditional drum/percussion instrument cannot hold a note. Notice the finger showing "Releasing the key (Key off)". Normal musical instrument require Key Off, traditional drums and percussion mostly ignore Key Off. True, the piano is, technically speaking, a percussion instrument, but it's behavior as to loudness contour and its sustain function make it unique among this family of instruments.

The ATTACK is easy to understand as the start of the sound. ATTACK TIME is how quickly the loudness travels to ATTACK LEVEL — in percussive instruments this usually is the biggest burst of acoustic energy, and causes what is referred to a ‘transient peak’. Transient because it is a momentary burst of energy that quickly dissipates. This immediate drop off is described by the “Decay 1 Level”. If you strike a piano note, the Attack Time and Attack Level are registered, after the transient loudness peak, the sound dies down a bit to this secondary Level (Decay 1 Level) where it endures for the musical timing value of the note...

If DECAY 2 LEVEL is 0, you have a percussive sound. And as you will discover, the piano always has this behavior - while a sustain slows down silence, the sound continues to decay towards silence. If DECAY 2 LEVEL is not 0, you have a sound that will be dependent on KEY-OFF to begin the RELEASE portion... to return to a level of 0. We will emphasize this point Because it is a key to understanding musical behavior. TIME can be understood as how long it takes to move from LEVEL to LEVEL. If DECAY 2 LEVEL is anything but 0, the sound will never go away while a key is held... Eventually as long as a key is held the level will reach the value as set by Decay2Level. If it is 0, then the sound will die out, eventually.

By the way: RELEASE LEVEL is always going to be 0 (that is why the parameter is not included).

In a Piano (and all percussion family instruments) the DECAY 2 LEVEL is 0, you can delay the eventual return to level = 0 with the sustain pedal or by holding the key... but it will stop vibrating eventually. (The Sustain pedal does not hold the sound of the piano FOREVER _ as piano players we know that because we grew up with the sustain pedal's natural behavior on the instrument). Of course, if you do release the KEY at any time in the envelope, the sound proceeds directly to the RELEASE TIME.

RELEASE TIME on a piano is not immediate - you don't often appreciate it until it is wrong. But there is a definite slope to how the sound disappears - if you set the RELEASE TIME so that is too fast, it is just not comfortable. If you set it so it is too slow, it is not right either. Experiment with a PIANO Voices Release time and you'll hear immediately what I mean.

The following article goes in depth into the Yamaha Envelope Generators — parameter of Time and Level are used to describe how loudness changes, it will also apply to the MONTAGE/MODX AMW2 engine as well.
In the FM-X engine an AEG is apart of each Operator, where the movement is described in Rates and Levels... again how quickly the envelope moves from one Level point to the next. In FM-X “Rate” is a function of Time but is a value, 0-99, where 0 is slow movement and 99 is fast. Contrast this with AWM2 where “Time” 0 is immediate, and 127 is very slow (long time).

Link — Learning About Envelopes

Perhaps I'm not making myself clear.

I know exactly what these envelopes do, in all synths... normally.

I'm struggling with the Yamaha variation on these envelopes.

They don't behave in a way that makes sense (to me), and I can't find how to benefit from them.

They're not gelling.

Everything I try to with them is a battle, and only through luck do I get results that I'd ordinarily easily dial in on any other synth.

There's something odd about the way they behave.

I get the instrument thing etc.

But back to the questions, which look vague, but are not. Consider the above, and think about these as serious questions, in consideration of the manner in which Yamaha is doing this is different to all others (but ignore that, if you fixate on that, you're thinking about these questions wrong...)

These stand alone, but within the context of the fact that Yamaha's way isn't... common:

What problem are they trying to solve by this method of shaping sound?

Why this approach chosen?

This is my interpretation, not Yamahas. I am a guitarist, all too familiar with the exponential (non-linear) attack/decay curves of guitars, which cannot be well represented by 4 stage (ADSR) linear "triangles". As guitarists, we typically put a lot of effort into re-shaping our Amplitude Envelopes with compressors, drive, delay and reverb pedals (also check out the eBow, Fernandes Sustainer). We don't have envelope generators. But if such a thing did exist, I suspect most guitarists would be immediately unhappy with a 4 stage ADSR arrangement... it wouldn't sound right.

I suspect non-linearity is probably true of most instruments.

Briefly, using guitars as an example again, Pitch and Timbre also naturally change over time. We often find ourselves trying to tame (change) an electric guitars "ice-pick" attack - loud, very high pitched, unpleasantly so.

What problem are they trying to solve by this method of shaping sound?

1) AWM2 + FMX Carriers - More stages allows for better approximation of complex, non-linear behaviour. In the case of emulation, this would allow more natural sounding results. In the case of original/unique sound design, allows for more "character" or unusual behaviour to programmed into the end result.
2) FM-X Modulators - allows for more natural "timbre" changes over time to be programmed. A good example is Woodwind instruments, where the initial "Breath" (white noise) can be programmed on Attack/Decay1, and the resultant tone can be programmed on Decay2 and Release. This applies to both Amp and Pitch. It also allows for other cool modulation when sound designing (Timbre change over time).

How is this of benefit to sound design?
See previous.

What other benefits are there to this approach?
Check out Manny's FM-X Tutorials where both Modulator and Carrier (akin to AWM-2) envelope generators and their applications are studied in detail. There is another series by Manny in which he builds an Acoustic Grand Piano from FM-X. This is where the benefits come into their own for advanced sound design.

How is it not beneficial (other than the obvious troubles I'm having with them)?
For "day-to-day" users, who have become accustomed to traditional ADSR parameters, the need to program an additional 2 parameters (Decay 2 Time, Level) is an extra burden. However, in use, they can largely be ignored if desired, setting Decay2 Time to Zero, and Treating Decay2 Level as your Sustain will get you back to familiar territory.
Something I picked up from Manny.... if you will need to set multiple similar shaped envelopes, it is best to set for 1 element/operator, copy that element/operator multiple times to the other elem/op slots you will be using. And only after doing this, start editing (fine tuning) the other element/operator parameters.... level, pan, fine tune, etc.

Why this approach chosen
My 2c... old style ADSR was a technological limit which has now been overcome. Yamaha have evolved the technology and applications.

Antony, all of what you say would be true if the manner in which time tracks through the Yamaha envelopes wasn't so utterly destroyed by early release of key presses, and exponential curves weren't available in the Motion Sequencer.

Since the Yamaha envelopes get completely and utterly 'lost' if the user releases the key before the completion of the Attack cycle, the two decays get even more 'lost' if this occurs.

Since Yamaha knows how to make an interface for, and sampling of, exponential curves, they can make an ADSR with exponential easings, and get a better representation and reality approximation through a 'traditional' ADSR that users exponential curvings.

Without these things, simply adding another decay has made things odder, to me, not better.

So while you're theoretically correct, I've not yet seen a single example, nor found a predictable manner, of using the second decay that makes me go "argh, that's why!"

Have you?

These are the types of curves that I'm doing with the Motion Sequencer, which has the added benefit of ignoring the duration of the keyed note input.

Sometimes this is desirable, sometimes not.

It would be nice, if possible, to have the Motion Sequencer abbreviate a step, if desired, so if the player inputs and releases quicker than the Attack time, something like this occurs:

But there are also many other options that'd be nice, with regards handling shorter inputs and after touch that'd be most welcome, too!

Ignore this post in this thread if you are easily distracted, or will tend to fixate on these examples. Consider them nothing more than a demonstration that triangle/linear based ADSR's are a limitation of the past, and that the limitation of straight lines in the envelopes of the Montage/MODX aren't solved by "adding" a "second" decay.

There are cases where the second "Decay" can/is actually used as a second Attack. The Decay 2 level is higher than the Decay 1 level, and has a positive slope (like Attack).

I will refer you to Manny's articles and FM-101 articles for reference and examples. Like you, I had wondered about the relevance of the second decay, and it was all explained and demonstrated in these articles.

Using examples from my familiar Guitar world, we are aware of an instance of a "second attack portion" which we know as "Bloom".

This features quite a lot in hard driven Tube Amps. Its known as "Amp Sag and Bloom" and is a peculiar type of compression caused by plate voltage starvation. The hard struck guitar results in a softer and lower volume attack than one might expect (sag) and as the amp "recovers" the sound gets louder, and is very rich in harmonics (bloom). It happens very quickly, but it is never-the-less a double attack if looked at from an envelope perspective.

So there's at least one definitive application. I'm sure there will be more.

Incidentally, you can negate Decay2 by setting D2 Time to 0, and Decay 2 Level = Decay 1 level. I'm sure you already know this.

Regarding short key presses...
I'm not sure of the resolution in time (milliseconds) of Attack and Decay times, but I know small values are very rapid, and do make a difference. Examples can be found in Manny's recreation of an Acoustic Piano. Extremely short Attack and Decay times are used to create a different "Attack Timbre". The values seem so small as to be rendered negligible and one might think the time taken to program this triviality would be unnecessary. But the end results are quite striking.

Manny's stuff is worth going through if you find the time. He doesn't just cover the FM science, but also the "analog" science too... as in what's happening at the audio output (your classic ADSR AEG, PEG, FEG and LFOs). He uses LFOs, Control Assigns and Motion Sequences as additional envelope generators... very clever mind blowing stuff. From memory he uses this to recreate pseudo Pulse Width Modulation, Hard Sync etc... not strictly available in FM... but he manages to pull it off.

All this is way beyond my current level of needs, but I like to know what's available and why, just in case I might need it in the future.

Since Yamaha knows how to make an interface for, and sampling of, exponential curves, they can make an ADSR with exponential easings

Sampling of exponential curves? A sampled envelope would have no bearing on what ADSR processing were available.

@anotherScott

Replace "sample" with "evaluate". They're interchangeable, in this case, and unrelated to what you know as audio sampling.

Think of it as "looking up" where a curve would be in time.

eg time = 0.15 seconds since key struck... where are we to sample/evaluate/lookup the value on the ADSR curve at this time?

That kind of "sample" is what's meant here.

Antony, I've already gone through Manny's stuff, and learnt a lot.

But it didn't resolve, for me, what the other benefits of the double decay mechanism are, nor make them comprehensible and relatable for other explorations, outside of the way he uses them.

I agree, those subtle and not so subtle timbre changes (which are much more exaggerated when applied to FM's relative pureness) of very fast changes to the attack are interesting. And offer some interesting results, too!

But put that aside, and think about attack times that are slightly longer, and have much more dramatic effects, but are also just sufficiently long enough that a fast player might not be holding the key long enough to fully ascend the entirety of the attack, some of time. In this situation (and this is ONLY ONE example of many) there are issues with the way the Yamaha envelopes operate that can't be, and aren't creatively fun to deal with...

But don't fixate on that one issue. It's just one example.

I'm trying to get a holistic understanding of WHY the Yamaha envelopes are the way they are, and a very honest understanding of their limitations and benefits, so that I begin to use them in a more natural/comfortable and appropriate way, rather than constantly fighting in trial-and-error, where most things aren't working right or don't work in ways that make much sense...

Which is why I eventually turned to the Motion Sequencer to act as a better, curvable (including very strong exponential curves) and highly customisable ADSR, for all sorts of things.

I can innately and intuitively get the curves I want out of the Motion Sequencer. Getting the subsequent right ratios upon ratios... that's a whole other design problem in the Yamaha world... which Jason has elucidated very well, so I won't bore you with the myriad of issues down that path.

I use amplitude AEGs on my Yamaha gear a lot and haven't had much problems with the basic machinery. I can't necessarily help with the struggle since I can't readily identify how Yamaha's approach is much different than the typical ADSR.

If using AEGs on a sample that doesn't loop and decays quickly - that'd make it difficult to get much out of the AEG since the sample itself has an "aggressive" AEG built-in. Picking an organ sound or something you know loops forever helps this. That's all I can think of as a wrinkle.

FM-X and AWM2 use slightly different AEGs. To avoid fumbling around (on my side) it may help to nail down the engine you're having issue with the envelopes. Also, although I can assume amplitude envelopes (AEG) - the question was only stated more generally as "envelopes". So nailing down amplitude vs. filter vs. pitch would be good to do as well.

I'm not sure if you've reviewed the pictures - but this doc:

https://usa.yamaha.com/files/download/other_assets/1/812531/synthesizer_en_pm_c0.pdf

Page 33 shows pictures of the AEG. Difference between AWM2 and FM-X can be seen.

As far as why decay1 vs decay2 - I've used these two parameters to try to approximate curves rather than straight slopes. Something where decay1 decays quickly 1/2 way down then decays more gradually with decay2. I just have straight lines to paint with - so the 2 decays allow for some shaping. Or sometimes I want to do something different on the attack so attack will not be the maximum amplitude - but decay 1 will be the peak and considered (by me) part of the attack. This allows some shaping of the attack vs. a straight line.

Jason, narrow it down to the aspects of these envelopes that you find are NOT beneficial.

Attempting to deal with an isolated example is the exact opposite of what is being attempted here.

So think holistically.

Describe what is wrong with these envelopes.

That will likely reveal much.

I'm out.