What you should do really depends on what you want to do. The only thing that is somewhat universal is that a line of text should not be too long. This ensures that readers can jump from the end of a line to the start of the next with ease. Markus Kohm (the author of KOMA-Script) tells us that a line should contain at most around 65 characters and a dozen words [1, 2]. (He also mentions that this limit can sometimes be stretched to up to around 75 characters, but since your lecture notes are probably not the easiest read, I wouldn't test those limits.)
So, what do you want?
I want a beautiful printed document that is easily read
Large margins help with that. You should probably just use the KOMA-Script defaults. Since your header is wide enough to appear as part of the type area,
headinclude is in order. Also:
- If the document will be printed one-sided, it would be better to make header and footer be the same on odd and even pages, i.e. use
oneside. I would opt for having both either centered or right-aligned.
- If the document will be printed two-sided, it would be better to ditch
twoside=semi in order for the margin between the left and right column to be the same width as the outer margins.
I want a readable document with as few pages as possible
It is certainly justifiable to reduce the margins somewhat in order to reduce costs and conserve resources, accepting a somewhat uglier document. Other possibilities than just reducing the margins may be helpful here. First and foremost, reducing the font size (
12pt is really rather large, I would go with at most
10pt in this case). You may switch to a more slender font. In order to accommodate both smaller margins and a smaller (and maybe narrower) font, you will need to use more columns to avoid breaking the universal rule mentioned above. Maybe a three-columned landscape format is an option? (Using several columns also allows considerably smaller horizontal margins without offending the eye too much.)
This approach will of course require a bit more effort when typesetting large figures, tables or mathematical expressions that don't easily fit in one column.
I want a document that is easy to work with on a screen
First, note that this is a very different goal from the two above. The number of pages is irrelevant here and margins don't mean the same as on a sheet of paper, because readers often don't see the whole page at once and the margin of the screen itself will surround the document. Since you don't know if your document will be the only thing on the screen or if it will be surrounded by other windows, how big the screen will be and what portion of the document will be seen at once, aesthetic considerations are pretty much impossible.
However, screen space is often valuable real estate and small screens may not have any space for margins when the text is zoomed to a readable size. Because of this, it is probably best to reduce each margin to the bare minimum and have no footer (put the page number in the header together with the running title). The document should only have one column and the margins should be the same on every page (
oneside). A good example of this is the online KOMA-Script documentation , which would be hideous printed as is.
In order to make the document readable on small devices like phone screens, the column should probably be a bit narrower than usual (maybe around 50 characters, though I have not tested this). This will also allow readers to have the document open next to another program (e.g. a LaTeX editor, answering exercise questions). If you suspect that your students may read the document on low resolution screens, consider selecting a sans serif font (I wouldn't worry about this nowadays, though).
I want all of the above
Well, tough luck. You could of course maintain three different versions of your document (or four, if you want to optimize for both one-sided and two-sided printing), but this will mean quite some overhead, as you would have to adjust equations, figures and tables for each of them. Personally, I wouldn't want to do that.
A good compromise could be to take what you have, manually reduce the bottom margin a bit (but not too much) and recommend people print it as A5. Starting with a 12 pt font, this will result in an 8.5 pt font, which in my own experience as a student printing lecture notes, is still fine.
If you are willing to accept a little bit of extra work and want to at least have equal horizontal margins (center and outer) on the resulting page, you could prepare a second version without
twoside=semi (this will leave the size of the type area unchanged, so you won't have to worry about the content) and "print" two pages per page into a PDF. (Rather than just providing the
twoside version in order to prevent the common mistake of printing odd pages on the left and even pages on the right.)
A real-world example
Since originally writing this answer I have kind of done this for a document summarizing some LaTeX basics . I made four versions (
screen_dark for reading on a screen,
print for printing and
generic as a compromise for the undecided) that only differ in their margins, headers, footers and (in the case of
screen_dark) colors. The content of the type area is the same on every page between the four versions.
I don't really expect this document to be printed a lot, so reducing the number of pages wasn't a consideration for me. This approach is not a cure-all for the points raised in the question, but automatically typesetting these versions turned out not to be that hard and may be worth the trouble to target both the printing audience and the screen readers.
Why should margins be like that anyway? Is that even widely accepted?
It is widely accepted, yes. Go to your bookshelf, pick out five random novels and have a look at their type areas. My guess is that at least four of them follow the rule that center and outer margins be equal and that the bottom margin be twice as high as the top margin1. You may also notice that those with more generous margins seem more harmonious and feel a bit easier to read. Even most textbooks follow these rules, though you will find more exceptions there.
Why is that so? Well, that's a bit difficult to answer. In my opinion it comes down to "That's just what's pleasing to the eye.". How do we know this? That in turn comes down to "Centuries of expert have found it is so, trust them." for me.
Reading about this, you will often find attempts to explain this that go something like this: When the margins are chosen as described, all of the grey rectangles in the drawing below have the same aspect ratio as either a page or the double page and the top and outer corners of the type area lie on the indicated diagonals. Also, we like the golden ratio and we can find loads of ratios in this drawing that approximate it (though not very well2).
In the end, this still comes down to the same two "answers" mentioned before when we ask "Why, though?" once more, but it seems to make the whole thing a bit more intuitive for some people.
1These margins are always far from the classical ideal of having about half of the page occupied by the type area (or about a third of the page's width/height being margin), which serves to conserve both costs and room on your bookshelf.
2There are other type area constructions, of course, that choose an exact golden ratio over equal aspect ratios of the indicated rectangles.
: The KOMA-Script documentation (English version, German version).
: A German article on type areas by Markus Kohm.
: A LaTeX Primer showing one way of providing various versions of a document.