Is TI-86 calculator capable of producing sounds?
In what ways the sound is produced (like timers, beeper bits or alike) if yes?
If no, are there any mods that add this capability?
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No, there is no speaker or buzzer integrated into the ti-86.
It does however have a jack port (2,5mm if I remember correctly) which can be used to transfer data between calculators. It is possible to control this output with a program written in assembly or Axeparser (i'm not sure about ti-basic). You can then connect a speaker to this port and play sounds with it. However this output is digital, not analog, so you will not be able to make anything pretty.
See https://axe.eeems.ca/Commands.html for a reference of Axe Parser (my favorite language for programming on a ti) In the "Link Port" section, it describes the usage of commande
Freq(WAVE, TIME) to generate sound.
See https://www.ticalc.org/pub/text/calcinfo/86ports.txt for a reference of how to control output to the port in assembly.
This is a really cool example of what can be achieved Thanks @tofro for this one.
No, TI-86 does not have a purpose-built sound output. However, it has a "link port" (2.5mm Mono Jack connector). People seem to usually connect this jack to a piezo-buzzer or to a mono headphones, see e.g. here. At the same time, it is actually possible to control the stereo channels separately, so you can, if you want to, connect it to stereo headphones and interpret it as a stereo-beeper.
The sound is produced via port #7 using direct CPU control. The wiring and relevant values to be outputted to the port are explained here. The available interrupts are not useful for sound as they only happen at 200Hz rate.
I personally consider ZX Spectrum the most mature platform for beeper sound technologies. Beeper music has been very substantially explored both during the commercial era for ZX Spectrum, as well as during a sort of rebirth of the interest to 1-bit music during the last 10 years. The example of beeper track discussed in the other answer is cool, but is merely a single example. You can find many more ZX Spectrum beeper tracks, both old and new, here.
Last but not least, higher CPU frequency in the TI-8x calculators, as well as the availability of the stereo outputs (with just a tiny bit of modding) actually makes TI-8x calculators a very nice platform for beeper music. So it may come as no big surprise to you that probably the most active beeper music coder and composer utz, who also goes under the label Irrlicht Project, created a purpose-built tracker for TI-82, TI-83/82STATS, and TI-83+/84+/SE called HoustonTracker 2, see also here. Of course, TI-86 belongs to a different family and there is a certain amount of porting that would have to happen to replicate same results on TI-86. However, there is nothing to prevent your TI-86 to output, with a bit of perseverance, something akin to