There are a few reasons why D minor might be seen as a sad key. First, it contains a number of dissonant intervals, such as the tritone (between the notes F and B) and the diminished fifth (between the notes C and F♯). These intervals can create a sense of tension and unease. Additionally, the key of D minor is relatively dark and low-pitched compared to other keys, which can contribute to its somber mood.
However, it's important to note that the key of D minor is not always used to express sadness. It can also be used to express a wide range of other emotions, such as longing, regret, and even anger. For example, Mozart's opera Don Giovanni features several scenes in D minor that express the character's guilt and remorse.
Ultimately, the meaning of a particular piece of music is determined by the composer's intent, and not by the key it is written in. However, the key of D minor can often provide a strong emotional context for a piece of music, and it is a key that has been used by many great composers to create some of their most memorable works.