Which mode is described as having half steps between notes 2-3 and 6-7, similar to playing all white keys from D to D?

The mode being described is the Dorian mode. This mode is characterized by the following interval pattern: whole, half, whole, whole, whole, half, whole. When applied to the D note (playing all white keys from D to D), the 2-3 and 6-7 degrees of the scale indeed have half steps, which aligns with the Dorian mode structure.

In the case of the Dorian mode starting on D, the notes would be D, E, F, G, A, B, and C, resulting in half steps between E and F (2-3) and B and C (6-7). This unique feature gives the Dorian mode its distinctive sound, often described as a minor scale with a raised sixth degree.

While the other modes have different interval structures, it is the specific half steps mentioned in the question that pinpoint the Dorian mode as correct. For example, the Mixolydian mode has a different half step placement (between 3-4 and 6-7), the Aeolian mode features half steps between 2-3 and 5-6, and the Phrygian mode has a half step between 1-2 and 5-6