Why does the music coming from a band marching in spread out formation on football field sometimes sound discordant?

The main reason for this is the speed of sound and the distance between the instruments. Speed of sound is 1125 feet per second, which means that a sound travels about 1 foot in 1 millisecond. Thus a source of sound 50 feet away will be heard 50 milliseconds after it is produced. Since musicians in a marching band are spread out over a greater distance, some sounds will be heard earlier or later than others. These time delays can create a phasing effect which makes the music sound discordant.

Here is an illustration for further clarification. Consider a band in spread-out formation with three instruments: a trumpet, a clarinet, and a saxophone. The trumpet is 50 feet from the audience while the clarinet and saxophone are 100 and 150 feet away, respectively. When all three play a note at the same time, the audience hears the trumpet's note first, then the clarinet's note 50 milliseconds later, and then the saxophone's note 100 milliseconds after the trumpet. This time difference can create a phasing effect which makes the music sound discordant.