Thanks for the great infor/refs.
Things sure get confusing with PM/PD/FM! The most basic concepts seem simple enough, but my mind really struggles to grasp all the nuances.
I guess there is a whole lot of trigonometry involved with understanding all this, and maybe that is why I struggle. I feel I have internalized algebra and calculus, but for some reason my mind rebels at all the transformations possible with trig.
Blasphemy.
For some reason, when talking about FM/PM/PD, the “math” pops up.
When talking about the more common subtractive synthesis, the discussion seldomly steers towards math (e.g. the math behind filters affecting frequencies, amplitudes, phases and the general spectrum).
Maybe because we humans just have more trouble grasping the non-linear/un-natural/non-intuitive nature of various “wave shaping” techniques.
Luckily we do not really have to understand how stuff actually works in order to be able to use it. We just need to know how and when to use it. Compare it to a car or a cell phone or even basic tool like a hammer or screwdriver. Just tools for some job. Be it for intended use or (more interesting) exploiting their full potential with unintended use.
Basically
In this case we’re not working the spectrum (e.g. subtractive or additive synthesis) but we’re changing the waveshape directly.
Generally, more modulation (change) leads to more “effect”. And yes, each form of modulation has its own “effect”. This goes for oscillator sync, FM, PM, PWM, WS or any other form of direct manipulation or the wave shape. Just try and see what happens. No need to know “why” it happens (unless you want to really exploit it).
All we need to know is what “tool” to pick for what “job”. And which “knobs” to turn to get it to do what it does.
Choosing and using the right tool for the job is challenging enough, though. May take a lifetime or more to get to grips with how stuff actually works “under the hood”.
This makes an absolute monster of a keyboard synth with a MIDI > CV V/Oct input and the gate going through an ASDR. I used Sonos Harmony to broaden the sound.
I’m having endless fun changing the octave on the Squinky VCOs and the levels in the VCAs.
Glad to inspire some fun.
As said, this was just a basic proof of concept. A basic 3 operator synth with a 3OP stack for an “algorithm.” (3 -->2–>1). But introduce a mixer before OP1 and you can connect both OP3 and OP2 directly to OP1.
A for the octaves. the represent ratio’s. You can have integer ratios (e.g 1 : 2 : 1 or 11 : 1 : 2. But also fractions. Remember that it’s the ratios that count. E.g 3 : 1 : 2 = 1.5 : 0.5 : 1 = 4.5 : 1.5 : 3 (just in different octaves). Since its about sums and difference, 1:1 will give you all frequencies. And 2 : 1 ro 4: 1 will give you odds (trianglish/squarish)
As for the envelopes. For the VCO('s) that actually output audio, the VCA just controls “volume”. For any osc/signal modulating another one (the 2 on the left) the VCA level controls the amount of phase modulation (the spectrum). I was just lazy and used an saw LFO to trigger the sound and emulating an AD envelope. Basically gate/envelope you can replae with any other gates/envelopes/modulators.
BTW
You can patch any oscillator to be a “modulator” (control signal at audiorate) or a “carrier” (audio signal to mixer/amp), or both (at the same time). Including feeding it back into the chain from/to any point. Or itself. Watch out, feedback can soon get “out of control”. Way to much modulation = noise.
Im modular, you are also not limited to any fixed amount of operators (generally in fixed gear 2, 4, 6 or 8) or fixed “algorithms” (various pre-wired configurations).
Also, you can have any module introduced anywhere in the signal and/or control patch. Including filters, folders, ringmod, distortion or even effects like reverb, chorus, phaser etc.
All this started from a “simple” question on how to fully invert the phase for any oscillator. Ending up with controlling the amount of phase modulation to get any angle, even at audiorates.
you are describing my automatic chaining TZOP and TZEN 
Actually, currently, I can’t get my head around the “automatic” bits of your much appreciated FM (TZ) modules. But I have to admit I never checked any manuals.
Maybe I’m just suffering from a severe case of RTFM.
Somehow FM/PM got a bad rep for being “complicated”. Which it really isn’t. No more complicated then chaining other audiorate modulations.
But, due to the (many) generated frequencies (sidebands), it just needs a bit of extra care and attention to prevent it all ending up in dissonance and/or noise.
Especially when your stepping away from using sines (just 1 frequency in the spectrum). Any additional frequency in the spectrum of carrier AND modulator will also generate its own sum and difference sidebands. Using rich spectra (e.g. SAW/SQR) will soon explode into way to many sideband frequencies to control.
But, ignoring FM/PM alltogether means missing out on a world of unique sounds.
simple carrier + modulator
simple CARRIER <-MODULATOR ← MODULATOR
complex CARRIE ← MOD <-MOD with Envelope for Modulation, and envelope for the output
(yellows are virtual connection in virtual INNER sockets)
I’ve forgot
the gate (trig) can be attached only to the TZEN on the left (it’s optional)
Triggering navigate left to right
Thanks for clarifying.
On TZOP Just wondering: Are the LINEAR FM INPUTS actually doing Phase Modulation (instead of FM)? Since stuff still work when (manually) changing the frequency to 0 Hz (which is impossible in FM). There aren’t many actual (prefab) FM solutions due to the inherent complications/limitations like DC offset and no 0 Hz.
BTW, it seems 0 Hz in TZOP is not saved with the patch.
Generally Controlling the “shape” and “speed” of a “Phasor” reading a “Wavetable” seems a very flexible FM/PM (or more general oscillator) solution. Since you can then have “unlimited” and “morphing” waveshapes. Maybe have “the same” wavetable/waveshaping options for both “Phasor” and/or “Wave” or even the position in the Wavetable. Currently Valley Terrorform already offers much of these concepts…
Maybe veering way to far off course here…
(it was a joke a very convoluted one in italian… regarding a loved post proto punk band “CCCP” )
regarding the waveforms:
if the tx81z waveforms were used I had fight a lost battle with antialiasing ! 
yes PM
you should use the version 2 0 15 (the 2 0 10 from library is buggy)
Yeah…FM/PM and aliasing. It has been mentioned before. I actually stick to sines most of the time. So you can actually hear stuff morphing in the spectrum. Not obfuscated by clouds of distracting morphing amplitudes of too many harmonics/frequencies.
Even more on Phase (Shifting):
Relative Phase (Shifting).
You can do controlled static Phase shifting of a (rich) spectrum against another (e.g. 2 Saws or Saw vs Square). This will effectively produce a Comb Filter effect. This way you can fine tune a static spectrum. And/or to modulate the relative offset a little (random/LFO) to create some liveliness, without the full cycling effect of detuning.
Compare the effect to picking one specific spectrum out of the whole cycle of spectra generated by detuning 2 “rich” oscillators.
Detuning will result in cycling the relative phase offset at the rate of the difference in frequency. Phasing frequencies in and out, wherever they are present in both spectra. At the rate of each specific frequency within the spectrum. Where higher frequencies/harmonics will phase in and out faster then lower frequencies/harmonics, thus chancing the spectrum allover but at different rates.
Interlude…
Preventing this Phasing/Combfilter Effect is why oscillator sync was “invented”. Two or more free running oscillators at unstable frequencies would never sound the same when mixed because they would be at different relative phases in their respective cycles all the time.
Syncing oscillators (a “primary/master” forcing a “restart” of the “secondary/slave” every cycle) would make their relative phase and therefore their mix predictable and static. Leaving undiscussed the “abuse” of this feature by changing the relative frequencies (screeching oscillator sync, another story).
Ironically we now crave for these analog imperfections in the perfect world of digital…
To continue…
Frequency and Phase are related. Detuning will just cause one cycle to get ever further behind another. So, effectively an ever increasing phase shift (until they eventually reach “sync” and the whole cycle starts all over).
All this assumes that both spectra have overlap in their spectrum (common harmonics/frequencies). No common frequencies within the spectrum means no frequencies will phase cancel each other where phase canceling happens when the phase shifting is introduced.
E.g. phase shifting a pure Saw (all ODD+EVEN harmonics at 1/rank amplitude) against a “pure” square (only the ODD harmonics) will leave the EVEN harmonic untouched and will give you control over the amplitude of ODD harmonics
Since “mixing” to oscillators is effectively “summing” not only the relative phase is relevant, but also the relative amplitude. Amplitude 50% can only phase out a 100% amplitude for 50%. Giving you even more control/options.
Another way to control where the “phasing” will occur is to bluntly introduce a fixed tuning offset. That will determine where in the spectrum both spectra might overlap (and frequencies in the first might phase cancel frequencies in the second when shifting). E.g. tuning #2 an octave up above #1, will leave the root of #1 untouched.
I give this specific example because normally the dominant harmonic in the spectrum determines the “note”. And generally this is the first harmonic in the spectrum. Not always…e.g. not for filtered noise (not just 1 harmonic) or some resonator instruments (not the first).
So…by choosing / controlling both spectra carefully, you can control where and when and how much the phase canceling occurs.
Hope these (way too many) words about Phase (Shifting) will help and inspire those who are interested in the many marvels of audio and synthesis.
My Kitchen Sink is a PM VCO with other waveforms. It uses internal oversampling to make the aliasing imperceptible. Sounds quite good, some people say.
@Squinky I like and use Kitchen Sink (and other Squinky Labs modules). I assume the LFM input is actually (the more common) PM (intended for audiorate phase shifting)?
Yes. To quote the manual:
"A pedantic note about FM vs. PM
Most modules and synthesizers that claim to do “FM” really use phase modulation. Even the DX-7 used PM. The advantages of phase modulation of frequency modulation are:
So we use phase modulation, as do most “FM” VCOs"
I go into that in huge detail in this article (which has lots of examples, and amazingly not using any of my modules): Demo/docs/fm.md at main · squinkylabs/Demo · GitHub
And again, I’m getting caught out…I did not RTFM…
Thanks. I’m definitely gonna check that out.
