How/why did certain similar typefaces (like these, which are apparently called ‘wood fonts’) become associated with the American Wild West?
The Clarendon typeface was fairly popular int he 1800s. Towards the end of the 1800s (in other words, right around the Old West times) this was modified to create the French Clarendon typeface, which is basically what we tend to think of as your basic Old West font.
The French Clarendon typeface is basically a reverse contrast version of the Clarendon typeface. Reverse contrast typefaces emphasize the horizontal lines and use very thin lines for the verticals, which is generally the opposite of how most normal fonts were at the time. Reverse contrast typefaces started being used in the early 1800s, and by the late 1800s the French Clarendon reverse typeface became popular for posters and signs.
One advantage of reverse contrast typefaces like the French Clarendon is that the thinner letters allow you to cram more letters together in the same amount of space. Another advantage is that it’s kinda backwards from what your eyes expect, as we tend to expect the vertical liens to be either the same thickness or maybe even be a bit thicker than the horizontal lines. This makes the typeface more attention-getting, which is the whole point behind signs and posters. You want to get their attention.
Because of its historical use and its attention grabbing nature, Western style movies in the 1950s and 1960s used this type of typeface in their advertising posters, title screens, etc. That pretty much cemented this style of font for Old West style stuff in popular culture.
Wikipedia’s article on reverse contrast typefaces (French Clarendon is mentioned towards the bottom of the article):
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