[sdiy] Interesting article on top octave generators.

[brianw]

I don't understand how these Chebyshev polynomials would work, or how they would help.

If you have a sine wave at f, presumably as a table of samples in DSP memory like many of them have, and you want another sine wave at 3f, then you can simply skip 2 samples and use every 3rd sample. What would the T3(x) function even do? It requires four multiply operations and subtraction. A DSP would be able to read every third address in a table without even requiring an extra opcode.

A quick look at NCO operation shows a Phase Accumulator and a Phase-to-Amplitude Converter, which usually contains a Look-Up Table. It seems like it would be easier to just have a second NCO borrow most of the PA of the first NCO, but triple the rate of the phase accumulation.

I suppose there could be a savings if the PAC uses complicated interpolation that's more expensive than the four multiply-accumulate operations that the T3(x) function has.

Brian

On Dec 28, 2024, at 3:50 AM, Richie Burnett wrote:
> In "DSP land" you can use Chebyshev polynomials of the first kind to multiply the frequency of a sine-wave by an integer...
> 
> https://en.wikipedia.org/wiki/Chebyshev_polynomials#First_kind
> 
> So, if you have a sine-wave at f and want another sine-wave at 3f you can "wave-shape" it using the T3(x) function in that table.  This can be quicker than running another NCO at 3f and another (interpolated) LUT to convert it's phase-angle to a sine-wave, provided your processor is quick at maths. The f and 3f waveforms are guaranteed to be "phase locked", if that is what you're after.  If you don't want them phase locked then running a bunch of "Magic circle" algorithms is probably the quickest way to get a load of sine-waves at arbitrary frequencies.
> 
> Real tone-wheels don't generate pure sine-waves though (>.<)
> 
> -Richie,
> 
> 
> -----Original Message----- From: Tom Wiltshire
> Sent: Friday, December 27, 2024 11:48 PM
> To: Roman
> Cc: synth-diy at synth-diy org
> Subject: Re: [sdiy] Interesting article on top octave generators.
> 
> I think the tonewheel "n/m" thing is actually quite a bit harder to do. The obvious way would be an NCO, since that gives you the fractional divider structure, the n/m, but then you have a problem with jitter, which is not something the gearwheels in the original generator have to deal with. With rotating gears, you can multiply a frequency *up* or *down* by whatever tooth-ratio you like, within reason. Since in that situation you might be after sinewaves instead of squares, perhaps some kind of pulse-density-modulated sine output would be ok. Yes, there'd be some small amount of jitter, but after the PDM is filtered, it might well be inaudible.
> It's actually a situation where DSP with a fairly low sample rate is more than acceptable, since we're only producing sinewaves and the highest sine frequency in the Hammond organ isn't that high - just under 6KHz. Hence a fairly modest DSP running at even a 20KHz sample rate could generate the sines with no aliasing.
> 
> 
> The classic divide-down TOG can be done without jitter since it's always a whole number of master clock pulses. With the original chips running at about 2MHz, there's enough time to calculate the outputs on a basic processor going somewhat faster (20MHz, 32MHz, whatever).
> 
> I completely agree that RPi2040 is massive overkill even for a 12-note TOG, let alone six. It's been done on an AVR, so there's really no need for such heavy firepower!
> 
> On 27 Dec 2024, at 14:06, Roman via Synth-diy <synth-diy at synth-diy.org> wrote:
>> Following that route - why not make TOG replacement which produses exact frequencies of tonewheels, or better yet, switchable frequencies of every historical or another weird scales?
>> Frankly, using RP2040 for 6 tone TOG is not so impressive to make video about it. Microcontroller based TOG replacements are available for decades now.
>> 
>> Roman
>> 
>> ---- Użytkownik Tom Wiltshire napisał ----
>> .
>>> 
>>> I agree that it does seem a bit crazy to replace some basic logic with a whole uP, but the arguments for doing it are pretty convincing. Doing twelve 9-bit dividers in hardware is going to be a lot of chips, whereas the uP is only one. That makes it much easier to fit inside whatever it is you're fixing. And it's cheaper - a boardful of simple logic costs *more* than a processor these days because of the economies of scale. It *is* crazy to use a million transistors when 10,000 would do, but when they all come on one chip and costs pennies, it starts to make a lot of sense.
>>> 
>>> 
>>> It terms of the technological limitations, all of the uP-based divider solutions I've seen are pretty much exact clones and just as limited as the original chips! They certainly keep the original division ratios and consequent frequency error.
>>> 
>>> 
>>> There are a few original divider chips out there, but the people that have them want $30 a chip or worse for them. $1 processor wins over that every time.
>