Mix and oscillator amplitudes, more research

I just can't let this one go. The code appears to add a linear amount of the side oscillators to the sum, but Adam Szabo says differently - the center oscillator is attenuated and the outer ones follow a curved gain.

I just had a happy accident. I am trying to find the contribution of each oscillator to the total if one normalize the sum - saying the total should always be 1.

However, I only added one single outer oscillator - but the output was very interesting:

Here it is compared to the graph in the article:

 

Here, the value of oscillator 1 is 1 / (1 + mix), whereas the plot for the others is mix / (1 + mix). The plots are quite similar! It really makes me want to understand this even more!

But - if I assume that ALL 6 outer oscillators should be included in the normalization, everything breaks down, so clearly that's not correct.

Going back to the article, we have this graph (Figure 9):

Note that the amplitudes of 5, 6 and 7 are higher than 1, 2, 3.

In the text, Szabo says that 2, 3, 5, 6 and 7 are removed, leaving 1 and 4, as illustrated in the first graph.

I don't know if he DID measure those too, but there is a chance that they don't follow the exact same curve. We'll see if we can figure that one out.

Also, let's renumber the spikes in the plot to match the order in the detune_table:

[0, 318, -318, 1020, -1029, 1760, -1800], let's call them A-G to keep them separated

That gives us:

A = 4, B = 5, C = 3, D = 6, E = 2, F = 7, G = 1

In case it matters, what is called 1 here is actually the last element added in the summing in the ESP code. 

 

Now, I'm not entirely sure how to interpret Figure 9 in terms of "max wave amplitude". The spectrum has peaks of a certain width, not just a single frequency, and there are no units on the Y axis. If the scale is linear and one assumes that the amplitude of each wave in the result is actually propotional to the max value of the center oscillator, we get the numbers in the table (each pure frequency is a sine wave, so the highest peak would correspond to the root frequency of the saw waves, wouldn't it)?

It looks like that's what Szabo means that they are, so let's accept that.

If so, the sum of all amplitudes is definitely not 1. Could the perceived total "loudness" be equal if one uses dB instead of a linear scale? The total energy or something?  

And in any case, how does one go from the sum += saw[i] * mix to this thing? 

My own measurements

I wanted to confirm my understanding of Szabo's graph, so I fired up JE-8086 and coded a spectrum analyzer in web audio. I fed the audio from JE-8086 back to the input of my mac using the virtual microphone/input "VB-cable".

I used a linear Y-axis and a logarithmic X-axis. Then, with detuning at max, for every line on the mix pot (11 in total) I screenshot'ed the spectrum. Finally, I went through every graph, measuring the height in pixels.

Here is the first and last spectrum plots:

The measurements were of course wildly inaccurate, and the spacing between the mix values not quite even as I couldn't see the slider value in the display (and also, the slider resolution wasn't high enough). 

Here are the values:

And, more importantly, the plot of the values relative to max value of the center oscillator:

 

This is indeed very cool. It does confirm most of what Szabo described - the center oscillator is fairly linear and the others are definitely curved, and the center oscillator ends at an amplitude lower than the outer ones. There are some small differences though:

- I don't get the feeling that the outer oscillators actually go DOWN in amplitude at the end. The 7th oscillator appears to go slightly  down again, but I think it's more likely a measuring error. 

- The higher pitch / right hand oscillators have a higher amplitude than the lower ones. This matches what can be seen in Figure 9 in Szabo. The top oscillator is even higher though, and that does not match. I am still not sure if this is an artifact of the spectrum analyzer or if it is real. It could be an artifact of approximate multiplication or running average or something. 

- Something else to note: Both in mine and Szabo's spectrum analyzers, the minimum amplitudes for the outer oscillators is not zero. If the summing is actually saw[i] * spread, there should be no trace of the oscillator if spread is 0. Very strange. Also - why do they call it spread and not mix?