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I am trying to learn whether or not there is a sort of "Turing completeness" type of situation for synthesizers. Here is a definition from Wikipedia of Turing completeness:

In colloquial usage, the terms "Turing-complete" and "Turing-equivalent" are used to mean that any real-world general-purpose computer or computer language can approximately simulate the computational aspects of any other real-world general-purpose computer or computer language.

So, what I am asking is, can any FM synthesizer with N number of operators enabled likewise make the exact same sounds as any other FM synthesizer with N number of operators enabled (e.g. Ableton's Operator vs. FM8), can any wavetable synthesizer with N number of oscillators make the exact same sounds as any other wavetable synthesizer with N number of oscillators, and so on...? Why or why not?

Please do not consider non-sonic differences such as interface differences, or auxiliary features such as builtin effects.

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    Synthesizers are not comparable to computers related to the definition you cite. View synths more like a small subset, instead.
    – MS-SPO
    Mar 27 at 6:09
  • One interesting and maybe strange thing about this question is that an analog synthesizer is basically a specific kind of analog computer. So the Turing completeness definition could already be applied to synthesizers as computers. We just listen to the output instead of looking at an oscilloscope or reading the numerical values from a meter.
    – Todd Wilcox
    Apr 11 at 3:58

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The basic fundamentals are equal on all FM synthesizers. (Not exactly: most so-called FM synthesizers are Phase Modulation synths, which is strictly different from FM even if related). But different synthesizers can have:

  • different modulation index tables,
  • different keyboard scaling curves,
  • different sets of algorithms,
  • different wavetable sets,
  • perhaps many other subtleties.

Note that FM can be very picky with the modulation indexes: a small change can have an huge impact especially with high modulator output levels since the real modulation index is usually an exponential of this number. Then if the modulation index table is different from one synthesizer to another, it can be hard to emulate one with the other.

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    This was true even back in the days of the original DX7 & DX9. Even if you trimmed the 7 back to emulate a voice on the 9, it always sounded 'richer'.
    – Tetsujin
    Mar 27 at 10:58
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can any FM synthesizer with N number of operators enabled likewise make the exact same sounds as any other FM synthesizer with N number of operators enabled (e.g. Ableton's Operator vs. FM8), can any wavetable synthesizer with N number of oscillators make the exact same sounds as any other wavetable synthesizer with N number of oscillators, and so on...? Why or why not?

My reading of this is that "and so on" means you have the same question about other types of synthesizers beyond FM and wavetable. If that's true, then I can speak to analog synthesizers. It is absolutely not a given that all analog synthesizers can make the exact same sounds. It's not even true that all analog subtractive synthesizers with three oscillators and a four-pole low pass filter can make the same sounds.

This is because in analog synthesizers, circuit design matters. Even from the same company, different oscillator and filter circuits will sound different. I have four analog synthesizers made by Moog. Two of them seem to have the same or similar oscillator circuit designs and they sound quite similar. The other two have different designs and sound close to each other but quite different from the first two. Once we consider other products, such as Sequential Circuits, Korg, or Arturia products, the designs differ by even greater degrees and the sonic differences are much more obvious.

I don't quite understand this part:

Please do not consider non-sonic differences such as interface differences

Any design difference can have a sonic effect, so I can't figure out what a "non-sonic" difference would be.

I also want to mention that the quoted definition of Turing Completeness uses the word "approximately", but your question uses the word "exactly". Subtractive analog synths with the same number of oscillators and four-pole low pass filters can make approximately the same sounds as each other. There are other analog synth designs, such as subharmonic synthesizers, and the variation in components and internal patching usually means each synth has its own capabilities, which is why people will buy a variety of otherwise similar analog synths.

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