Yes, it does look neater if you have a striped background and the lines of text align with that. However, you won't use a striped background. There is nothing else those lines of text align with unless you stick the text into multiple columns, which you probably won't do since it's annoying (zigzag scrolling) and pointless (there is no height limit on websites).
I think it's more important to pick distinct headline sizes which still look good when they wrap around. If they don't perfectly align with some imaginary stripe pattern which no one is imaging, then so be it.
Yes, it does look neater if you have a striped background and the lines of text align with that. However, you won't use a striped background. There is nothing else those lines of text align with unless you stick the text into multiple columns, which you probably won't do since it's annoying (zigzag scrolling) and pointless (there is no height limit on websites).
I think it's more important to pick distinct headline sizes which still look good when they wrap around. If they don't perfectly align with some imaginary stripe pattern which no one is imaging, then so be it.