- KVRAF
- 11552 posts since 21 Mar, 2008, from Hannover, Germany

Just noticed that Serum in the Warp modes offers doing FM, AM and RM (ringmod) between the two main Oscillators.

As in DUNE 2.5 you could route the oscillators to anything modulation of the volume/amplitude should be possible too. Besides that it offers dedicated RM in the mixer. Last but not least FM is possible too.

Another synth for such stuff could be Synthmaster 2.8 that was already mentioend.

Overall theer do not seem to be too many synths that offer doing "proper" AM besides ring modulation.

As in DUNE 2.5 you could route the oscillators to anything modulation of the volume/amplitude should be possible too. Besides that it offers dedicated RM in the mixer. Last but not least FM is possible too.

Another synth for such stuff could be Synthmaster 2.8 that was already mentioend.

Overall theer do not seem to be too many synths that offer doing "proper" AM besides ring modulation.

Ingo Weidner

HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

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HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

i5-3350P / Win 10 x64 / Live 9 / Cubase Pro 9.5 / Bitwig 2 / Studio One 3

- KVRAF
- 6410 posts since 16 Mar, 2003, from Porto - Portugal

I did some research, and there are two synths where we can easily compare AM and RM (which are both in the same place, as indeed RM is a special kind of AM, as has been said).

One is Synthmaster, from which a demos can be easily downloaded. AM/RM are immediately accessible, if we use the Init patch and use one modulator to multiply the oscillator (the upper slot of modulation above the oscillator). To change between AM and RM we just have top change the DC offset of the modulator (the only control available for it, BTW), which at 0 is RM and at 1 is AM (with all changes in between, which means we can "morph" between RM and AM).

The other is Linplug Alpha. Osc 2 also has an AM/RM control, which, again lets us morph between pure Rm and pure AM. Linplu implementatios seems somehow more simplistic, since, acording to the manual, to Peter its just a matter of adding the carrier spectrum or not (as we saw, it's not just that since the amplitude of the generated sidebands generated also change).

Synthmaster implementation seems better. As I said, the demo is easily accessible. I can provide a screenshot of the patch, if wanted, but its really quite simple.

As I said, AM tends to be "mellower" than RM, and the end results doesn't differ too much from the carrier (it's like having the carrier slightly distorted through some kind of convolution). Honestly, I never found it interesting enough to justify a synth solely based on it, as the OP enquired. As a complement to other synthesis techniques, it's OK to have it, but as a main resource, it will fall short, IMO.

One is Synthmaster, from which a demos can be easily downloaded. AM/RM are immediately accessible, if we use the Init patch and use one modulator to multiply the oscillator (the upper slot of modulation above the oscillator). To change between AM and RM we just have top change the DC offset of the modulator (the only control available for it, BTW), which at 0 is RM and at 1 is AM (with all changes in between, which means we can "morph" between RM and AM).

The other is Linplug Alpha. Osc 2 also has an AM/RM control, which, again lets us morph between pure Rm and pure AM. Linplu implementatios seems somehow more simplistic, since, acording to the manual, to Peter its just a matter of adding the carrier spectrum or not (as we saw, it's not just that since the amplitude of the generated sidebands generated also change).

Synthmaster implementation seems better. As I said, the demo is easily accessible. I can provide a screenshot of the patch, if wanted, but its really quite simple.

Ingonator wrote: Overall there does not seem to be too many synths that offer doing "proper" AM besides ring modulation.

As I said, AM tends to be "mellower" than RM, and the end results doesn't differ too much from the carrier (it's like having the carrier slightly distorted through some kind of convolution). Honestly, I never found it interesting enough to justify a synth solely based on it, as the OP enquired. As a complement to other synthesis techniques, it's OK to have it, but as a main resource, it will fall short, IMO.

Fernando (FMR)

- KVRAF
- 11552 posts since 21 Mar, 2008, from Hannover, Germany

While i just opened Serum where i wanted to compare AM, RM and FM (from the Warp modes menu) i just had the idea that if AM and/or RM is a multiplication of the amplitudes i could create a corresponding formula in the formula editor of Serum.

As i used two Sines in my examples at the previous page and one was 2 octaves higher this should correspond to a formula like: sin(4*x*pi)*sin(x*pi)

This is how the coresponding waveform looks in Serum using the formula parser:

https://dl.dropboxusercontent.com/u/532 ... on%201.png

This is indeed seems to closely nail my example from the Bass Station 2 ring modulation that i posted at the previous page.

Here is the screenshot of only the Bass Station 2 ring modulation (without the other screenshots of the single Oscs etc. for comparison):

https://dl.dropboxusercontent.com/u/532 ... 2001_2.png

If you add a second variable z to that formula in Serum you could create a full wavetable from one formula where z is related to the wave index in the wavetable.

As i used two Sines in my examples at the previous page and one was 2 octaves higher this should correspond to a formula like: sin(4*x*pi)*sin(x*pi)

This is how the coresponding waveform looks in Serum using the formula parser:

https://dl.dropboxusercontent.com/u/532 ... on%201.png

This is indeed seems to closely nail my example from the Bass Station 2 ring modulation that i posted at the previous page.

Here is the screenshot of only the Bass Station 2 ring modulation (without the other screenshots of the single Oscs etc. for comparison):

https://dl.dropboxusercontent.com/u/532 ... 2001_2.png

If you add a second variable z to that formula in Serum you could create a full wavetable from one formula where z is related to the wave index in the wavetable.

Last edited by Ingonator on Fri Aug 11, 2017 10:54 am, edited 1 time in total.

Ingo Weidner

HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

i5-3350P / Win 10 x64 / Live 9 / Cubase Pro 9.5 / Bitwig 2 / Studio One 3

HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

i5-3350P / Win 10 x64 / Live 9 / Cubase Pro 9.5 / Bitwig 2 / Studio One 3

- KVRAF
- 6410 posts since 16 Mar, 2003, from Porto - Portugal

Ingonator wrote:While i just opened Serum where i wanted to compare AM, RM and FM (from the Warp modes menu) i just had the idea that if AM and/or RM is a multiplication of the amplitudes i could create a corresponding formula in the formula editor of Serum.

As i used two Sines in my examples at the previous page and one was 2 octaves higher this should correspond to a formula like: sin(4*x*pi)*sin(x*pi)

According to the book I quoted earlier, the formula for RM is as follows: cos(C)*cos(M)=0.5*[cos[C-M)+cos(C+M)]

And the formula for AM is as follows: Ac*cos(C)+(I*Ac)/2*cos(C+M)+(I*Ac)/2*cos(C-M)

Ac is the amplitude of the carrier, I is the modulation index, C is the carrier frequency and M is the modulator frequency.

Fernando (FMR)

- KVRAF
- 11552 posts since 21 Mar, 2008, from Hannover, Germany

fmr wrote:Ingonator wrote:While i just opened Serum where i wanted to compare AM, RM and FM (from the Warp modes menu) i just had the idea that if AM and/or RM is a multiplication of the amplitudes i could create a corresponding formula in the formula editor of Serum.

As i used two Sines in my examples at the previous page and one was 2 octaves higher this should correspond to a formula like: sin(4*x*pi)*sin(x*pi)

According to the book I quoted earlier, the formula for RM is as follows: cos(C)*cos(M)=0.5*[cos[C-M)+cos(C+M)]

And the formula for AM is as follows: Ac*cos(C)+(I*Ac)/2*cos(C+M)+(I*Ac)/2*cos(C-M)

Ac is the amplitude of the carrier, I is the modulation index, C is the carrier frequency and M is the modulator frequency.

As i used two Sine waveforms in the synths i used Sines in the formula and multiplicated them and obviously the result in the synth seems to fit to the result from the formula.

The thought behind my formula was that i used two Sines with different frequency (actuiually one Sine at 2 octaves higher = 4 times the frequency) and wanted to multiplicate them.

If i dissect that formula sin(4*x*pi)*sin(x*pi) into the two oscillators it means:

- Oscillator 1 with a Sine at Octave 8': sin(x*pi)

- Oscillator 2 with a Sine at Octave 2': sin(4*x*pi)

Last edited by Ingonator on Fri Aug 11, 2017 11:07 am, edited 1 time in total.

Ingo Weidner

HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

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HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

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- KVRAF
- 6410 posts since 16 Mar, 2003, from Porto - Portugal

Ingonator wrote:fmr wrote: According to the book I quoted earlier, the formula for RM is as follows: cos(C)*cos(M)=0.5*[cos[C-M)+cos(C+M)]

And the formula for AM is as follows: Ac*cos(C)+(I*Ac)/2*cos(C+M)+(I*Ac)/2*cos(C-M)

Ac is the amplitude of the carrier, I is the modulation index, C is the carrier frequency and M is the modulator frequency.

As i used two Sine waveforms in the synths i used Sines in the formula and multiplicated them and obviously the result in the synth seems to fit to the result from the formula.

The thought behind my formula was that i used two Sines with different frequency (actuiually one Sine at 2 octaves higher = 4 times the frequency) and wanted to multiplicate them.

All is well, except that your formula gives you RM but not AM. To get AM you have to take into consideration the modulation index for the depth of the sidebands (divided by 2) and add the carrier frequency, which will be absent in RM. The use of sine waves doesn't change anything for this matter. Actually, sines and cosines are sonically the same thing (both produce sine waves, it's just a matter of phase).

Fernando (FMR)

- KVRAF
- 11568 posts since 7 Dec, 2004

Facepalm...

- KVRAF
- 11552 posts since 21 Mar, 2008, from Hannover, Germany

nvm

Last edited by Ingonator on Fri Aug 11, 2017 1:05 pm, edited 1 time in total.

HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

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- KVRAF
- 11552 posts since 21 Mar, 2008, from Hannover, Germany

In Serum i now used two Sines like in the other examples, again one at 2 octaves higher where Osc B is used as a "modulator". I then used the AM, RM and FM Warp modes in Osc A and used the oscillospcope plugin to see the resulting shapes.

The AM and RM were used at 100% amount and FM at 50% (100% gives quite strange results...).

Here is a comparison of the results with the additional calculated RM using the formula sin(4*pi*x)*sin(pi*x) (waveshape created with the formula parser in Serum) in the lower right:

https://dl.dropboxusercontent.com/u/532 ... on%201.png

UPDATE:

A simple formula for the AM is: y(t) = [1 + m(t)]*c(t) (with m = modulator and c = carrier)

In this specific example done in Serum (posted above) it would lead to a AM formula like: (1+sin(4*pi*x))*sin(pi*x)

This is how it looks using the formula parser in Serum:

https://dl.dropboxusercontent.com/u/532 ... la%201.png

The result fits to the screenshot of the AM Warp mode in Serum posted above.

The factor 0.5 is due to normalization purposes. With a factor 1 the resulting waveform is partly cut off.

The calculated RM waveshape based on the formula sin(4*x*pi)*sin(x*pi) for direct comparison:

https://dl.dropboxusercontent.com/u/532 ... on%201.png

The simplified formulas for AM and RM in comparison:

AM: y(t) = [1 + m(t)]*c(t) (with m = modulator and c = carrier)

RM: y(t) = m(t)*c(t) (with m = modulator and c = carrier)

The terms m(t) and c(t) include both the mathematical formula of the actual waveshape in the synth and the frequency ratios.

In this specifc example posted above m(t) = sin(4*pi*x) and c(t) = sin(pi*x) where the waveshape is represented by the Sine function and the frequency ratio is 4 (having teh modulator 2 octaves above the other carrier means 4 times the frequency and 3 octaves above 8 times the frequency).

The AM and RM were used at 100% amount and FM at 50% (100% gives quite strange results...).

Here is a comparison of the results with the additional calculated RM using the formula sin(4*pi*x)*sin(pi*x) (waveshape created with the formula parser in Serum) in the lower right:

https://dl.dropboxusercontent.com/u/532 ... on%201.png

UPDATE:

A simple formula for the AM is: y(t) = [1 + m(t)]*c(t) (with m = modulator and c = carrier)

In this specific example done in Serum (posted above) it would lead to a AM formula like: (1+sin(4*pi*x))*sin(pi*x)

This is how it looks using the formula parser in Serum:

https://dl.dropboxusercontent.com/u/532 ... la%201.png

The result fits to the screenshot of the AM Warp mode in Serum posted above.

The factor 0.5 is due to normalization purposes. With a factor 1 the resulting waveform is partly cut off.

The calculated RM waveshape based on the formula sin(4*x*pi)*sin(x*pi) for direct comparison:

https://dl.dropboxusercontent.com/u/532 ... on%201.png

The simplified formulas for AM and RM in comparison:

AM: y(t) = [1 + m(t)]*c(t) (with m = modulator and c = carrier)

RM: y(t) = m(t)*c(t) (with m = modulator and c = carrier)

The terms m(t) and c(t) include both the mathematical formula of the actual waveshape in the synth and the frequency ratios.

In this specifc example posted above m(t) = sin(4*pi*x) and c(t) = sin(pi*x) where the waveshape is represented by the Sine function and the frequency ratio is 4 (having teh modulator 2 octaves above the other carrier means 4 times the frequency and 3 octaves above 8 times the frequency).

Last edited by Ingonator on Fri Aug 11, 2017 1:32 pm, edited 1 time in total.

HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

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- KVRAF
- 11568 posts since 7 Dec, 2004

When you don't understand something like "two quadrant vs. four quadrant modulation" it won't help you to just change the subject to something entirely unrelated.

I'll explain what you two are very clearly missing out on:

In an operation there are two variables: x, y

Each of these variables has two "quadrants": negative and positive.

This reduces to a simple multiplication:

1 quadrant * 2 quadrants = 2 quadrants (two quadrant multiplication / modulation)

2 quadrant * 2 quadrants = 4 quadrants (four quadrant multiplication / modulation)

https://www.youtube.com/watch?v=fJIjoE27F-Q

Now you understand the very basic stuff you need to understand in order to re-read my post and understand it. Nothing I said in my post is incorrect and the arguments against it have been unbelievably dumb. Please, re-read my post with this newfound knowledge.

I get that my post is condescending like a slap in the face. It is very difficult to hold back, but I know it isn't your fault.

I'll explain what you two are very clearly missing out on:

In an operation there are two variables: x, y

Each of these variables has two "quadrants": negative and positive.

This reduces to a simple multiplication:

1 quadrant * 2 quadrants = 2 quadrants (two quadrant multiplication / modulation)

2 quadrant * 2 quadrants = 4 quadrants (four quadrant multiplication / modulation)

https://www.youtube.com/watch?v=fJIjoE27F-Q

Now you understand the very basic stuff you need to understand in order to re-read my post and understand it. Nothing I said in my post is incorrect and the arguments against it have been unbelievably dumb. Please, re-read my post with this newfound knowledge.

I get that my post is condescending like a slap in the face. It is very difficult to hold back, but I know it isn't your fault.

- KVRAF
- 11552 posts since 21 Mar, 2008, from Hannover, Germany

@aciddose:

First i am not sure if you did actually read my last post above which i updated with some additions a few times.

Well, i had used AM, RM and FM in an actual synth (Xfer Serum) and in the same synth i was able to nail the resulting AM and RM waveshapes with quite simple formulas using the formula parser in the waveform editor of that synth. Not sure what should be "dumb" or wrong about that.

If you have a better example including screenshots, audio demos etc. nobody holds you back to do that.

Personally i stopped just talking and today did some "real world" examples based on actual synths while i admit that those are mostly oscillaoscope screenshots. If i see it makes sense i would also post audio examples but posts like yours do not really encourage me to do that.

First i am not sure if you did actually read my last post above which i updated with some additions a few times.

Well, i had used AM, RM and FM in an actual synth (Xfer Serum) and in the same synth i was able to nail the resulting AM and RM waveshapes with quite simple formulas using the formula parser in the waveform editor of that synth. Not sure what should be "dumb" or wrong about that.

If you have a better example including screenshots, audio demos etc. nobody holds you back to do that.

Personally i stopped just talking and today did some "real world" examples based on actual synths while i admit that those are mostly oscillaoscope screenshots. If i see it makes sense i would also post audio examples but posts like yours do not really encourage me to do that.

HW: Blofeld, Pulse 2, UltraNova, Bass Station 2

SW: PPG 3.V, Largo, Nave, Icarus, Avenger, Serum, Legend, Saurus 2, Diva, Repro-5, VC 6, Komplete 10, many more

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- KVRAF
- 6410 posts since 16 Mar, 2003, from Porto - Portugal

aciddose wrote:When you don't understand something like "two quadrant vs. four quadrant modulation" it won't help you to just change the subject to something entirely unrelated.

I'll explain what you two are very clearly missing out on:

In an operation there are two variables: x, y

Each of these variables has two "quadrants": negative and positive.

This reduces to a simple multiplication:

1 quadrant * 2 quadrants = 2 quadrants (two quadrant multiplication / modulation)

2 quadrant * 2 quadrants = 4 quadrants (four quadrant multiplication / modulation)

Idiotic post as usual. First, you confused tremolo with vibrato (a noob mistake), and said that RM is similar to FM. Now, you come with this, maybe attempting to launch a cloud over your previous dumb assertion. I only pointed the mistakes, politely (without video links).

here is what you wrote:

aciddose wrote: In fact ring mod is more like FM than AM, which is a partial (reduced depth) version of ring mod. In that sense ring mod is like FM; AM is like vibrato.

Your last answer is the usual, and as usual, nothing of what you said above adds anything useful to the discussion.

And, of course, the usual pict and video links, that "enlighten" a lot. Typical

Fernando (FMR)

- KVRAF
- 10125 posts since 13 Oct, 2009

fmr wrote:ghettosynth wrote:fmr wrote:aciddose wrote: In that sense ring mod is like FM; AM is like vibrato.

AM is NOT like vibrato. AM is tremolo.

Vibrato is modulation applied to frequency (by a low frequency oscillator). Tremolo is modulation applied to amplitude (by also a low frequency oscillator).

RM, when thought of as is it used in synthesizers, is a form of AM. While what results from any modulation process is some combination of the input frequencies, there is a subtle technical difference. In FM modulation, the frequency of the carrier is being modulated. In AM and RM, the amplitude of the carrier is being modulated.

I didn't deny that RM is "a form of AM". Actually, the mistake I pointed, and that caused my post, was calling vibrato to AM, a very common mistake.

It sounded like you were saying that RM is something very different from AM and FM and that his post was incorrect when, other than the misunderstanding of the vibrato analogy, it was spot on. He started out by saying that RM is a form of AM and my interest was in clarifying that from a practical understanding.

Radio actually helps with this because modulation is generally related to practical information and the modulator is often substantially different from the carrier. Phone or voice modulation makes this really clear, the information is the voice message, the carrier is just the vehicle by which we get that voice message to the other station.

So, the point that I was trying to make clear, and still want to make clear, if not for you, then for other readers, is that modulation is applied to some property of an oscillator/signal. AM, FM, and even PM, are clearly named appropriately because signals have amplitude, frequency, and phase. However, RM, and any other similarly misnamed modulation method must be some other form of modulation because signals don't have "rings."

Going back to radio, you will never hear someone from that field referring to the use of a diode ring mixer as "ring modulation." Balanced modulator, yes, as that describes what it does. Ring modulation is something of a misnomer which is why I was disagreeing with your statement that is something very different from AM or FM, it isn't. As generally used in synthesis, it's a type of AM. Granted, it's use in telephony predates it's use in radio largely for technical reasons, however, the construction of the phrase is still different in a sense that we aren't talking about a property of a signal and I think that this can cause some confusion.

About the difference between RM and AM in synthesis terms - Let's grab some help from Curtis Roads, and the book "The Computer Music Tutorial":

"Like RM, AM generates a pair of sidebands for every sinusoidal component in the carrier and the modulator. .../... The sonic difference between RM and AM is that the AM spectrum contains the carrier frequency as well. The amplitude of the sidebands increases in proportion to the amount of modulation, but never exceeds half the level of the carrier"

This is the main difference. In RM, the carrier frequency disappears, according to Curtis Roads, when the frequency of the modulator is in the audible range (which is when we actually have "our" RM synthesis). Even if it doesn't disappears, as some here said, certainly its frequency will me much lower than the one of the sidebands, resulting in quite different spectra.

Yes, acidose made this absolutely clear, moreover, as I've been dabbling in radio for years it's certainly not news to me. I mentioned this in my post above. Understanding this is foundational in radio. SSB (single sideband) came into widespread use between the 50s and 70s replacing AM as the primary HF (high frequency, i.e., shortwave) phone method for non-broadcast phone transmission.

Most SSB radios start by generating a DSB or double sideband signal, i.e., they use a ring mixer, and the unwanted sideband is then removed with a narrow bandpass filter.

AM will always be much "mellower", which is indeed what we can easily check in real life, and maybe the reason why it doesn't get so much interest as RM and FM.

Yes, this is a function of the carrier being dominant in the output signal.

Furthermore, if C and M are an integer ratio of one another, the sidebands generated are harmonic. Otherwise, they are inharmonic. That's why RM is often used to produce sounds that are difined as "clangorous" or "metallic".

Another curious effect mentioned is that, in analog implementations, depending on the type of diodes, the circuits introduce extraneous frequencies. One example quoted is that, in circuits based is silicon diodes, the diodes clip the carrier, truning it into a quasi-square wave, when it reaches the momentary level of the modulator, creating the effect of several sums and diferences on odd harmonics of the carrier (maybe this is where the "clangorous" effect comes from).

AFAIK, nothing of this happens in "pure" AM.

Well, yes and no, which is why relying on derived ideas from practical synthesis will lead you astray. An AM signal contains sidebands as well, they are just not dominant owing to the carrier signal being present. So it does, in fact, "happen", further, any type of modulator may introduce unwanted frequencies.

From wiki:

Therefore, the modulated signal has three components: the carrier wave c(t) which is unchanged, and two pure sine waves (known as sidebands) with frequencies slightly above and below the carrier frequency fc.

...

Again the carrier c(t) is present unchanged, but for each frequency component of m at fi there are two sidebands at frequencies fc + fi and fc - fi. The collection of the former frequencies above the carrier frequency is known as the upper sideband, and those below constitute the lower sideband.

RM, as used in synthesis, is AM with a suppressed carrier. It is the carrier suppression that is responsible for what you hear.

So yes, practically speaking in our context of interest, RM sounds different from AM, but as soon as we try to understand it more technically, we see that RM is really a bit of a misnomer. Also, yes, different methods can sound very different. Modular guys know this and it's why you can always use more than one type of ring mod module.

Last edited by ghettosynth on Fri Aug 11, 2017 3:47 pm, edited 1 time in total.

- KVRAF
- 13238 posts since 20 Oct, 2007, from No Location

fmr wrote:aciddose wrote:When you don't understand something like...

First, you confused tremolo with vibrato (a noob mistake), and said that RM is similar to FM.

Yeah that cuts right through a whole pile of bullshit. To confound tremolo (amplitude variance) with vibrato (frequency or pitch variance) indicates he will misconstruct the very essence of FM, or be simply mistaken about it.

- KVRAF
- 10125 posts since 13 Oct, 2009

Ingonator wrote:

The simplified formulas for AM and RM in comparison:

AM: y(t) = [1 + m(t)]*c(t) (with m = modulator and c = carrier)

RM: y(t) = m(t)*c(t) (with m = modulator and c = carrier)

We can simplify this further by expanding the formula and eliminating notation unnecessary for discussion, that is, we know that they're all a function of 't'.

AM: y = c +m*c

RM: y = m*c

There you have it. The difference is whether or not the carrier is suppressed. Said differently:

RM: y = AM -c

BTW: As I mentioned back on page one. FM6 from the Reaktor library implements these in a way that you can experiment with AM (and RM) practically as with FM. In fact, you can combine them having chains of AM, RM, and FM in the same patch and at any place in the algorithm. It implements other operator types as well, but that is beyond this conversation.

As with FM, it becomes more interesting harmonically when you have multiple modulators. While you can do this with wavetable editors that allow waveforms to be defined via formula, it strikes me that it's more practical and intuitive to use the Yamaha approach, insofar as that can be called intuitive.

The primary reason for this, IMHO, is that you need envelopes for each operator, just as you do with FM, to get any interesting time dependant variation in the sound. A modulator that has a slow attack time, for example, turns a sine wave slowly into a clangorous ring mod sound. You can repeat this with the very same chains that are used for FM. That is, one carrier with multiple modulators, or chains of modulators, to build up complex and interesting sounds.

I will say that I think that there are some issues with the envelope implementation in FM6 which is one of the reasons I'm not yet ready to post my own variant.

The essence of my point here though is that, like FM and additive synthesis, AM and RM both are more interesting in the context of being able to modulate the effect of the modulation in complex ways and that, AFAIK, none of the synths mentioned so far have that kind of flexibility.