> for an electric guitar, a triangle wave

That is only true when the string is plucked exactly at the middle - not a regular occurrence. Usually it's more of a sawtooth wave, just without the upper harmonics.

My experience with distortion in guitars is that a huge component here is how different it can sound depending on articulation. Some pieces for example require the player to not alternate their picking, but pluck only in one direction, as the difference is audible.

This is not always the case of course as some amplifiers like those made by Mesa Boogie get their signature tone by exploiting the limited gain-bandwidth product(GBP) of amplifiers, creating an even sound that at the high end of the gain setting is largely without dynamics.

In some cases(like the BOSS DS-1) the manufacturer killed the sound by introducing a technically better amplifier chip - the original had somewhat poor GBP and poor settling time, which in combination produced a nice lowpass filter with a resonance peak at the cutoff frequency, which in turn emphasized articulation.

It's all a surprisingly huge topic.

Distortion of monophonic tones (single notes/voices) became accepted in the mainstream quite rapidly because the circuits used to produce it quickly improved, informed by feedback (no pun intended) from musicians.

Nobody hates ugly sounding distortion more than he or she who practices 5 hours a day with it.

Single note distortion, at its best, is a harmonically rich sound which shares something with bowed instruments and reed woodwinds.

Nasty sounding single distortion has not gained complete mainstream acceptance. Musicians who do that on purpose will remain niche, even today. From time to time, such nasty sounds make appearances in mainstream pop, but only as a kind of "cameo". The statement is, "we are inserting this ugly thing here specifically for its idiosyncratic effect, haha! But only a few seconds, we promise".

Distortion (other than perhaps mild distortion) has never been fully accepted in roles where the multiple voices of a complex harmony part would be distorted together.

Nowhere was that better seen than in jazz/rock fusion, which accepted ragingly distorted guitars for solo work, but not so much for the rest of the music: except, of course, in passages where the guitars provide the "sound of rock": distorted fourths and fifths and whatnot, or double stop bluesy cliches and whatnot.

The music best known for distortion and that couldn't exist without distortion (and a lot of it) is of course heavy metal, which is a big landscape of styles and sounds.

In metal, the harmonies from an individual guitar part tend to consist of only a few notes. The clean chord is transformed into something else, which perhaps cannot be described in music notation. Complexity comes from the distortion. Distortion includes the sum and difference products, which relate to the tonality and scale of the music in unsual ways. Those notes are not identified. If notation is used at all, the underlying clean notes are notated: e.g. C-F# tritone on the A and D strings, over open E bass. Heavy metal uses syncopated and alternating rhythms to separate bass notes from upper notes in three and four note chords. This is not only to create rhythmical excitement, but to better separate the notes.

The notes of a distorted chord are also easier to for the ear to identify if they are introduced separately as a lasciare suonare arpeggio; that's a thing in metal.

Harmonic textures are also created by combining distorted guitar parts. Using two lead guitars originated in rock, with groups like Wishbone Ash. Multitrack recording allows an unlimited number of parts to be layered.

Years ago, I hatched a theory, that one of the reasons my generation gravitated to rock and roll as teenagers, is that the sound of a distorted guitar closely resembles the sound of an engine, and is thus a symbol of freedom, just like getting one's driver's license.

I shared this theory with my then-little-in-both-senses sister, who asked "what, exactly, is distortion, in a musical sense?"

I trotted to my bedroom and retrieved Pink Floyd's Pulse. I popped disc 1 into the living-room CD player, cued up track 9 (Sorrow), and turned it up. We potched out onto the floor.

https://www.youtube.com/watch?v=NO9Kp_Wn_2A

"Oh. Okay. Yeah, alright. I think I get it."

“Whatever you now find weird, ugly, uncomfortable and nasty about a new medium will surely become its signature. CD distortion, the jitteriness of digital video, the crap sound of 8-bit - all of these will be cherished and emulated as soon as they can be avoided. It’s the sound of failure: so much modern art is the sound of things going out of control, of a medium pushing to its limits and breaking apart. The distorted guitar sound is the sound of something too loud for the medium supposed to carry it. The blues singer with the cracked voice is the sound of an emotional cry too powerful for the throat that releases it. The excitement of grainy film, of bleached-out black and white, is the excitement of witnessing events too momentous for the medium assigned to record them.” - Brian Eno, 1996