Let's say you're notating patterns up a scale and it becomes obvious how the pattern continues, but you don't want to write out the full thing. You'd like to use the equivalent of "..." or "etc." to show that it continues, then notate the final instances of the pattern to show how it ends. What is the sheet music equivalent of ellipses or "etc."?
There really isn't something like that. The closest thing is a simile where you could write a rhythm pattern and then put the chord changes over the similes as shown in this answer.
Music notation in general is very exact to make sure the musician playing the song know exactly how to play it.
There are several existing forms of shorthand that have been accepted as part of standard notation, such as full-/half-/two-measure repeat symbols,
sim. (for articulations and dynamics) and turns (which define note sequences by relative pitch and are probably the closest thing to what you are looking for [I'll add a pic shortly]), but those assume you've already written out what you want at least once.
Example usages of the one- and two-bar repeat signs:
It depends on your audience. A jazz guitarist might be able to pick up on "Oh, a two-measure repeat (where the noteheads were slashes but were placed on like the actual lines/spaces) plus a chord change, several times over? I guess I'll just move the scale pattern to fit the chord", but a French horn player or clarinetist is infinitely less likely to interpret your markings that way.
Also, as stated above, trills are shorthand that are universally understood because they've been established. Who knows, if you use your own invented shorthand enough, people might just catch on.
You could, of course, just write like in the upper or lower margin,
Play an A scale, starting on the " and" of beat 2 and beginning on the fifth scale degree, in sixteenth notes, with accents on the "e" and "a" of 4. Upon reaching scale degree 1 for the second time, descend back to starting point.
and then afterwards say
same, but relative to E3 and on the E scale, every other 2nd scale degree naturaled.
but that's just way harder to read and also probably to write out.
The kind of shorthand you're looking for that depends on the reader's ability to "pick up on" or "get" what you mean is good for exercise books,as pointed out in an earlier answer, but not for actual sheet music, where composers are known to "change things up" to make things interesting and music becomes much less predictable and standardized.
Hope this helps; the other answers are great too!
I have seen "etc." being used in e.g. books of exercises. First there's say two bars establishing the pattern and how it progresses, then one bar with just "etc." written in it followed by the end point of the exercise. But then the type of pattern has been explained earlier e.g. by writing out one exercise fully. So it can be used in certain contexts, but should not be assumed to be a known practice. For scale practice, it should be acceptable I think.
Miškinis’s Pater Noster, which begins with various repeating short motifs by each voice. The motifs do not start together either.
My edition, to express this, writes the motif once, then the first few notes of the next repetition of the motif (to ensure the next motif enters where expected), then use a “percent” sign followed by a black line around in the middle of the staff which continues across pages until the point where everyone stops.
"Etc." is fine, if it's clear. The more musical term is "sim.". Just make sure it IS clear :-) If in doubt, write it out.