Even when I properly match the system and web fonts in Muse the font don't display properly and the wrong weights of the font will be displayed.
Does this happen in Muse or after it is published or during preview? If you are using web fonts, you will be able to see them only after you publish the file. For system fonts, ensure that you have uploaded the files to the correct location.
Can you provide us with a screenshot?
This is happening during the preview. I have access to the desktop and web font files. When I assign the web font files to the same system files then type in the web font the wrong font appears. I have been experimenting with the font files and now Muse doesn't even recognizing all the system fonts even though the rest of CC can use all the system fonts.
I have sent you a link to the font files in you DM Günter. I feel like every time I uninstall and reinstall the font files Muse will recognize a different set of weights.