What is the designation for a diminished chord combined with a minor seventh?

A diminished chord combined with a minor seventh is referred to as a half-diminished seventh chord, often notated as a "m7♭5" or simply "ø7". This chord includes a diminished triad (root, minor third, and diminished fifth) along with a minor seventh interval above the root. The presence of the minor seventh (rather than a diminished seventh) is key to distinguishing the half-diminished seventh from a fully diminished seventh chord, which features a diminished seventh interval above the root.

To understand why this designation is appropriate, consider how the notes are constructed: in a half-diminished seventh chord, you have a diminished triad plus that minor seventh, which gives it a distinctly different harmonic function and sound compared to other chord types such as a fully diminished seventh, minor seventh, or dominant seventh chords. Each of those chords has a specific structure that does not include the combination of a diminished triad with a minor seventh, making the half-diminished seventh unique.