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by andete. Original documents available at: https://github.com/andete/ym2413/tree/master/results
The YM2413 contains 18 operators. 2 operators combined form 1 channel, so the YM2413 has 9 melodic channels. Alternatively 6 of the operators can be used to produce 5 rhythm sounds, resulting in 6 melodic channels plus 5 rhythm channels.
So far all the reverse engineering was focused on a single operator. In this post I'll finally combine 2 operators into 1 melodic channel.
The YM2413 datasheet contains the following diagram:
++ ++  ++   ++   > SIN >(+)> SIN > F(t)   ++   ^ ++    ^    ^     I     A   ++ ++   ++ ++    PG   EG     PG   EG    ++ ++   ++ ++  ++ ++
F(t) = A sin(wc * t + I * sin(wm * t))
with
and
Each of the 2 big squares is 1 operator. By chaining them together in this configuration (the only possible configuration for YM2413; Y8950 and YMF262 also have other configurations) the output of the 1st operator (called the 'modulator', or 'mod') can modulate the phase of the 2nd operator (called the 'carrier' or 'car').
So this configuration implements phase modulation. Mathematically phase modulation and frequency modulation are closely related. So often this configuration is also called frequency modulation. Note that this is a different form of frequency modulation as what was discussed in the previous post (that was FM within 1 operator, here it's 2 operators combined that produce FM).
From all the information in the previous posts we already have a fairly good idea how a single operator works. The new part here is chaining them together. In the diagram this chain in drawn as a simple line. In reality it's a bit more complex. E.g. is the output of the modulator still multiplied with some factor before being added to the phase of the carrier? How many bits does this signal have? Is the addition done before or after the 10.9 bits fixedpoint phase counter is truncated to an integer?
I captured waveforms using the following settings:
Operator  AM  PM  EG  KR  ML  KL  TL  WF  FB  AR  DR  SL  RR 

modulator  0  0  1  0  00  0  nn  0  0  15  00  00  15 
carrier  0  0  1  0  00  0  0  15  00  00  15 
reg#0x20 = 0x00  keyoff 
reg#0x10 = 0x40  fnumlow=0x40 
reg#0x30 = 0x00  max volume / custom instrument 
reg#0x20 = 0x10  keyon / block=0 
with TL varied between 0..63
Note: for TL=63 this gives the (almost) sine wave I've been using in most of my previous posts. For lower values of TL the modulator operator gets less attenuated and thus there's more modulation.
I'm using a very low frequency: 16384 samples for 1 period (~3Hz). So each of the 1024 entries in the sine table is repeated 16 times.
The results are presented in a table (I suggest to look at these images from back to front, 63 to 0):
+0  +1  +2  +3  

0  
4  
8  
12  
16  
20  
24  
28  
32  
36  
40  
44  
48  
52  
56  
60 
Some observations:
(These graphs are for TL=8, but similar features can be seen for all other TL values). Instead of seeing the same value for 16 consecutive samples we only see 15 samples with the same value, followed by 1 other value again followed by 15 constant samples. This could be explained by assuming the modulator and carrier are 1 step out of phase.
In all these measurements both the modulator and the carrier have the same frequency, we also believe they are both (almost) in phase. So that means every halfperiod both operators internal sinewave has a zero crossing. So the output of the total channel should also be zero. We can indeed see this in the combined image: every 8192 samples all graphs cross zero. (Remember values 255 and 256 on the yaxis represent YM2413 values +0 and 0.)
If we zoom in near the green and red box we respectively get:
This shows that every half period, the waveforms for all TLvalues indeed exactly overlap (the image only shows 3 graphs, but this property holds for all TL values).
Inside the red circle you see a spike. I started all the graphs in the big table at the sample immediately right from this spike.
At the left of this graph, the waveform of the previous measurement is visible. Near the start of the green line the YM2413 registers are reprogrammed, we very briefly set keyoff followed by keyon again. This triggers the 'damp' phase, that is instead of immediately going to the attack phase, the envelope generator first (at a fairly fast pace) drops to maximum attenuation, only when that point is reached we enter the attack phase (I think this is different from the other OPLx chips. I need to investigate this is in more detail later). The whole 'damp' phase is indicated by the green line.
On the right is the red circle. There are 2 interesting features in this circle:
As you can see the actual signal is not flat at all, it oscillates between 2 values with spikes to a 3rd value. More on this below.
The next step is to try to reproduce the measured values with a model of the system. But let me first repeat the model we had so far for a single operator:
To extend this model to a 2operator channel, the main loop needs to have the following form:
Note that the implementation of lookupExp() drops the lowest 4 bits from the value found in the exponentlookuptable. Immediately after we multiply that value by 16. So maybe these bits didn't need to be dropped in the first place? Remember, in an earlier post I showed the die shots of the YM2413 and YMF262 ROM tables, and they were quasi identical, suggesting the YM2413 and YMF262 use the same exponent ROM table. In that post I also said I found it strange that in our YM2413 model we drop the lowest 4 bits of exponent table (why include the bits in the table if they are not used). So maybe here, in the modcar calculation, all the bits of this table are being used?
Further experiments show that for nonzero values of TL, the model matches (the overall shape of) the measurements for 'unknown1' equal to 32. I've also changed the model to produce 16 consecutive samples with a constant value (same as in the measurements). With all these changes the model becomes:
Earlier I already guessed the mod and car operators are 1 sample out of phase. Let's test this hypothesis by tweaking the model:
So this can reproduce the spikes, but the other oscillations are still missing. It took some time, but then I got the idea to reduce the resolution of the expTable again (drop 1 bit instead of 4). This can be modeled like this:
With this last change the graph matches exactly with the measured data, not only for TL=37, but for all values!
Actually, it's not a 100% match for the following two reasons:
In the experiments in the previous posts I often got the problem that the prediction didn't match exactly with the measured data because I ignored the modulator operator in the predictions. Now we know exactly how this works, so we could revisit those experiments.
Another thing we didn't look at yet is the 'feedback' mechanism: the output of the modulator operator can be feedback to the input of the same operator (to modulate itself).