Let's say we have a coffee cup as shown below.
The cup is, in contrast to what's shown on the image, empty (I had some trouble finding free good-quality top-view coffee cup images). When we tap on $\frac14,\frac12,\frac34$-th of the coffee cup (measured from the "ear", not sure what that's called in English), showed with the red dots, the cup makes a lower sound then when we tap on $\frac 18,\frac38,\frac58,\frac78$th of the cup (shown by the blue dots). I have tested this on multiple types of cups and is thus definitely not an opinion-based question.
Why is this? Please make your answer as simple as possible, as I'm just a simple mathematician, and know nothing in particular about physics (although I have a basic understanding, of course).