Reflection, Diffusion, Absorption
Sound behaves in much the same way as light. Shine a flashlight at a mirror in a dark room, and a hard glare will be reflected right back into your eyes. Shine the same flashlight onto a frosted piece of glass, and you'll notice that the light is evenly distributed in a pleasing circle on the surface of the glass, which will also reflect more light around the dark room than the mirror did. Apply this to music in an enclosed space, and you can understand why diffusion — the random scattering of sound — is far better than hard reflection. The latter makes the music itself sound hard and brittle, while diffusion provides clarity, warmth, and an evenness of sound throughout the room. And because diffusion more evenly distributes high- and mid-frequency sound throughout a room, it adds greatly to musical clarity.
Absorption is useful in reducing the amount of sonic energy in a room. Many people make the mistake of cutting down reflections by deadening their music rooms with heavy draperies, thick carpets, and overstuffed furniture. However, this will not absorb all frequencies evenly, and can make a room sound dull in the upper octaves and too heavy in the bass — or the other way around. While in "live" rooms some absorption is desirable, even necessary, I suggest a combination of absorption and diffusion. This can be done by placing books, bookcases, artworks, chairs, and other randomly shaped objects along the walls to break up reflections, as well as scattering around the room some soft surfaces, such as upholstered furniture. Some of the best music rooms have mostly hard surfaces with little absorption, but they all have many diffusive surfaces that break up the reflections, which keeps the sound live, warm, and resonant. Partially closed wooden blinds or other irregularly shaped treatments for windows and glass doors will help diffuse reflections coming off of those glass surfaces. Note that flat artworks, even when not covered with glass, can cause degrading reflections unless they have a very irregular diffusive surface. Fabric wall hangings, especially quilts and other thick, soft, irregular surfaces, can absorb a lot of high-frequency reflections, when used in moderation — but not heavy drapes, unless the room is especially "live" and reverberant.
What you put under your grand piano can make a huge difference in its sound. In designing a music room, whether or not it will contain a piano, I normally specify hard floor surfaces, whether of hardwood, ceramic tile, or marble. The center of the floor should be covered with an acoustically absorbent surface, such as a carpet or rug. The idea here is to have sound absorption in the central part of the floor to cut down on reflections, while keeping the edges of the room more "live" for resonance. If the best-sounding location for your piano is not far enough out into the room for the instrument to be placed on the carpet or rug, place under the piano a separate area rug large enough to cover the piano's entire footprint. The bottom of a grand piano's soundboard produces a great deal of sound that a hard floor will reflect, thus making the sound harsh and brittle—unless something is there to help absorb that energy. If you don't mind how it looks, you can store piles or boxes of music or recordings on the floor directly under the piano, which will provide absorption and diffusion. In very "live" rooms, a thick fabric cover (similar to a full piano cover) can be suspended under the instrument's soundboard. This is especially useful in practice rooms, where clarity is more important than generating a big sound.
Vertical pianos, normally placed against or near walls, don't interact with hard floor surfaces as intimately as do grands. However, if your vertical is in the middle of a very "live" space, such as a dance studio or theater rehearsal room, it can benefit from some sort of floor covering under it that extends a few feet out from the piano on all sides. If a vertical's sound is still too resonant or bright, whether the piano is up against a wall or out in the middle of the room, you can eliminate some of this by hanging a heavy fabric cover or blanket over the back of the instrument. Not very stylish, but it works.
Some high-end piano dealers will give you time to audition an instrument in your home or studio before you make a final commitment to purchase. I strongly recommend taking advantage of any such offer—the room in which you place your piano is as important as the instrument itself in determining the ultimate sound.