As you already figured out, MIDI has nothing to do with timbres or waveforms. MIDI is an interface and a command language for musical instruments to talk to each other, saying things like "Note On, C-5, velocity 100". How the recipient device reacts to that Note-On message could be literally anything. It could cause a stage light to move to a different position, or it could cause a MIDI-controlled tape deck to start playing, or it could cause a synthesizer to start making sound, or it could cause a MIDI-controlled garage door to open. I don't know if there are MIDI controlled garage doors, but if there is, the garage door will probably have an electric motor, which will produce an electrical waveform of some sort. If the waveform has a frequency between around 20Hz and 20kHz, and if it causes air vibrations at those frequencies, then the garage door motor makes a sound and the sound can be heard by a human. Whether a MIDI sequence playd on the garage door counts as electronic music, is a matter of opinion.
Notation applications can produce MIDI messages, working as sequencers. There are free notation applications such as MuseScore, which you could use for sending out the MIDI commands just fine, no "digital audio workstation" needed at all.
The device receiving the MIDI messages and producing the audio waveforms can be a piece of software or it can be an outboard MIDI synth, i.e. external hardware box or keyboard-type instrument that you connect to your computer via a MIDI interface and a MIDI cable. If it's a virtual synth or sampler or other sound-producing program, you connect it to the sequencer (for example MuseScore) via a virtual MIDI cable, which is a piece of software that routes MIDI messages from application A to application B. Or you could save out ("export") a MIDI file from the notation application and load it into a combined sequencer/synth application for playback, bypassing the need for a virtual MIDI cable.
I think this much you could have found out yourself by reading an introductory text or watching a tutorial about what MIDI is.
But then the more interesting question. How to create a waveform that you could legitimately call "fractal shaped"? Fractals are supposed to have "infinite detail", detail inside detail inside detail, reproducing variations of the same pattern or formula at many resolutions and scales, right? How do you do that, because audio only has finite resolution - how do you demonstrate fractal properties in an audio waveform? Is it enough if it just sounds weird and is accompanied by an explanation "something something fractal something", creating a science-mystic layer? Or do you have to be able to justify and defend the fractal claims?
This is just mumbling out my ad-hoc ideas, but to believe something is a fractal, I'd want to see repetitions of characteristics of a pattern occurring at many different scales of magnification. Our display screens have finite resolution just like audio, and fractal properties in images are demonstrated by zooming, i.e. changing the horizontal and/or vertical magnification scale of the display. So the demonstration cannot be just one static image, you have to provide several images or an animation with different scales. If we extend this thinking to audio, you'd have to show some kind of a display, something happening either in the time dimension (time domain) or frequency spectrum (frequency domain). Maybe if the visual shape of the waveform in time domain at some scale has a shape, and then, when looked at closer, similar shapes or characteristics could be found? Or what if the waveform itself morphed, creating a zooming in/out image, while keeping the oscilloscope zoom level the same? This would also cause changes in the sound itself, creating a morphing timbre! Or what if the spectrogram in frequency domain looked like a fractal?
One way to do this properly, in my opinion, would be by programming the audio-generating (virtual) device yourself. See topo's answer for ideas.
You had a requirement of the fractal-like properties being found in waveforms, and that suits the idea of playing an existing composition, but fractal-like properties have been claimed to be found in the rhythmic patterns/timings of Jeff Porcaro's drumming. https://phys.org/news/2015-08-fractals-patterns-drummer-music.html Fascinating? Nonsense? You decide.