Organ Tone Wheel development.

i thought the parasitic FM thing was a good idea too but i didn't see any indication in that example. maybe the recording just had a high noise floor?

I read that the mechanical wobble and the motor modulation are a thing, worked out they would be AM and FM respectively (obvious), then read another analysis that came to a similar conclusion. I also did some listening tests that suggested that a small still audible amount does give some character that makes it sound a little more real… although that could just be cognitive bias.

what i'd actually do is test phase offset idea with the setup you're using now to see if there's anything to it. heck that's assuming it even relatively easy to derive cosine from a frequency doubled sine? maybe frequency doubled triangle would be more straightforward instead

The maths for the phase offset is a thing, you can't just interpolate, because the magnitude of the resulting sinusoid changes.

a = sin(x)

b = cos(x)

d [0..1] is the crossfade amount between sine and cosine.

a linear crossfade would be this: c = db + (1 - d)a

but we need to scale the final magnitude….

lets call it: g = sqrt( d^2 + (1-d)^2 )

then we scale c by 1/g and we get a sinusoid where d is the phase between sine and cosine… so 90 degrees.

For wider phase offsets, I suppose we're looking at separating into quadrants or something ([a, b], [-a, b], [-a,-b], [a,-b]). That would be a pain because the routing logic can be a cpu hog in some circumstances.

There is probably a better way to do this. if you know one, post please :)

https://www.desmos.com/calculator/fgxeuhjsbp

In terms of accessing partials turning them on and off etc. That's maybe just a scaling thing too, we're already applying a coefficient, so just multiply that with an 'amount' from 0 to 1...

As long as that phase is not accessible at audio rate, the math is very cheap per partial, maybe cheaper to just turn off by multiplying with a zero gain than by checking if its 'on' and using routing logic. Maybe MUCH cheaper depending on the implementations.

Managing the partials has always been the difficult and expensive part (hence partials framework ;-)).

beside the whole phase thing i would put money on the next most significant addition to be some kind of waveshaping/harmonic distortion. from that audio file i uploaded its pretty easy to see the extra pointiness compare to if it was a pure sine:

Parasitic harmonics coming from nearby tonewheels can modify the resulting waveform in various ways, including 'extra pointiness'!

Here's a sine:

here's the same sine with an octave up harmonic mixed in at around -16dB

A few other lower level related parasitic harmonics could make it even more like your pic…

Here's a 2 octaves up -180 degrees -21dB is mixed in

Some of the parasitics will be in or out of phase due to the physical position in relation to the pickup.

The FM thing isn't something we would see as an obvious waveform change. The FM is low level, and at a frequency unrelated and mostly some distance away from the fundamental of whatever tonewheel (except form the very low ones that are weird and their own way - very far from sinusoidal)… but it is audible IMO as a grunge/distortion that doesn't sound like coloured noise, or like power supply noise.

I think its likely visible on the spectragam you posted, but we would need to know what to look for. It would be at frequencies relating to the motor frequency and the driving gears (there are 12 of them), so 12 basic frequencies. Then for some partial, one of those frequencies depending on which driving gear is being used would FM it to get some sideband action, which would depend also on the driven gear, and the number of teeth on the tonewheel in question.

Pretty sure I read that on a development version of the organ it was quite objectionable, and the generator was redesigned so that this FM effect might seem more like some intentional (he was fooling nobody :-)) vibrato effect… by better gearing choices etc.

there's always some trick, like zgabr just showed in another thread how to create subharmonics of a sine by taking the square root and inverting polarity every other half wave

My circuit is still different from the one described in the document, because I gave up the rectifier and square root and built the algorithm a little differently, although my circuit also has a counter, but in my circuit the counter is triggering the integrator. And the counter itself is triggered by the upper half-wave obtained by separating....

dunno whether you saw in that example osc i posted but,

when a phase shifted partial is required, just generate a separate shifted fundamental partial, and multiply up… (might as well generate the partial individual unless there are others with the same phase shift)

Start with a range of fundamentals that can be chosen between… so not just cos and sin, but many more. Still limited in terms of how high the partial can be before we get the same problem.

Do phase shift times 2… so for a one octave higher shifted partial, create a 45 degrees shifted fundamental using the fundamental sin and cos, then use that with one of the fundamentals to generate the shifted partial… maybe this is doable in a way that's still cheaper than brute forcing every partial… maybe

none of this actually ends up being necessary. thought maybe there wouldn't be an easy way to generate a cosine from a frequency doubled sine but you can just repeatedly frequency double a triangle wave and you get a sine and cosine for just a couple more multiply adds a piece

cr@p all these cheap sine cosine waves could even be a good way to do constant Q spectrum analysis on the cheap