The answers to the modulation shape questions are scattered throughout the forum, and somewhat incomplete.  I’ll do my best to compile much of what I know about them, but I don’t have any insight into the official Eventide implementations.  (Please feel free to correct any errors in logic).

It sounds like you already have a handle on the basic waveforms.  Tempo Off will cycle at the current time in milliseconds or Speed / Rate scale.  Tempo On will cycle at a note value based on the current BPM.  I’ll include the image below for completeness:

Note that the Sawtooth waveform above is a “Ramp-Up” (found in the HarModulator algorithm, for example).  Most of the algorithms use “Ramp-Down” as a modulation source, for more of a percussive attack feel.

I’ve read nickrose [Eventide] here describe Peak as a “differentiated square wave”.

Again, it’s a cyclical waveform shape, but with a ‘lopsided’ characteristic.  I use it when I need a repeating LFO waveform that has a little glide & sloppiness to it.  Sort of a cross between Square & Ramp shapes.  It gets even better with the Mod Delay double waveform implementation.  Think ‘Uni-Vibe’ here, or vintage analog circuits with slew.  Here’s an exaggerated example using the Vibrato algorithm that makes it easy to hear the effect of Peak modulation:

 

Sample and Hold is most reminiscient of synthesizer-type effects.  You’ve definitely heard it applied to pitch, and often to filter effects.  There’s a common misconception with Random and/or Sample & Hold modulation.  Random modulation is truly ‘random’: gliding among modulation values without respect to strict tempo or rate.  Sample & Hold will store a level (sample), then hold it for a specific length of time.  Sample frequency and Hold time will be the current SPEED / RATE value.

A Random waveform is (many times) used as an input, and a stepped sequence of modulation levels is produced at the output.  This is how it works in Eventide pedals: 

Where would you use it?  I put together an entire thread on using Random or Sample & Hold modulation in the H9 / ‘Factor pedals:

http://www.eventideaudio.com/community/forum/stompboxes/some-random-h9-presets

But to get started, here’s a demonstrative preset to show the differences.  You might want to dial down the SPEED control to 1/4.  Then switch between the default SAMPHOLD under SHAPE, and the RANDOM modulation.  Under the right circumstances, it becomes more obvious on when & where you would pick each type of modulation.

 

Up until now, none of the modulation shapes above relate directly to your input signal.  Envelope and ADSR are both affected by input, but in different ways.  Sometimes it’s difficult to hear the difference, because certain settings in either modulation type will produce a similar result.  But here goes:

Envelope is a peak (envelope) follower.  It takes your input signal, and smoothes it out before becoming a source of modulation.

ADSR is another borrowed synthesizer tool that creates an ‘envelope’ over an oscillator where there was none before.

Yes, I know.  The terms get confusing.

In the H9/ ‘Factor algorithms, the most important setting is the SENSITIVITY control.  Too little, and your Envelope / ADSR will never get triggered.  Too much, and they’ll always be maxed out.  Envelope truly ‘follows’ your input signal: overall level; pick attack; string decay; even the volume knob on a guitar.  [Test that out!  Modulation from your pinky finger!]

ADSR is a fixed set of attack, decay, and release times (and I don’t know what those fixed time lengths are).  It’s a triggered, yet exactly repeatable mod shape.  ‘Event Onset’ in the image above is when your input note crosses the SENSITIVITY threshold.  ‘Gate Duration’ is how long your input stays above that threshold (with sustaining notes).  When your input drops below the SENSITIVITY threshold level, the Release segment takes over.

Again, here is one more preset to show the contrast.  Switch from the default ENVELOPE setting under SHAPE, to the ADSR setting.  You may have to adjust SENSITIVITY, but 50 is a good starting point that generally works on most any input.

When to use Envelope & ADSR?  All the time.  You can create any kind of triggered wah, phaser, flanger, pitch bender, … Pick Envelope when you want nuanced modulation control from your fingertips.  Select ADSR when you need a similar effect that doesn’t depend so heavily on your playing technique.  Now, did I miss anything?

EXP PED (expression pedal) should be self-explanatory. Heel = low limit modulation.  Toe = high limit modulation.  And I need to add:  You can mix & match two modulations [SHAPE and MOD SOURCE] together to produce – among many other things – envelope-controlled speed ramping, super-deep changes, and modulation rates-within-rates.