I've been finding that most (all?) of the things Mr. Springstead tells us in band have a reason grounded in physics. For example: when playing a major chord consisting of the root, third, and fifth, he often tells us the third note should be slightly lower.
Notes that sound "good" together, such as the notes of a basic major chord, sound that way because their wavelengths "meet up" at regular intervals. Mathematically, we can look at the ratio of each note's frequency. The ratio of C4 (middle C) to G, the fifth note of a C major scale, is (approximately) 3/2, so every third G wave meets up with every second C wave. The ratio of C to E is approximately 5/4. But in reality, that ratio is slightly off. 329.63 Hz, the frequency of E4, divided by 261.63, the frequency of middle C, is actually 1.25991. (Likewise, the frequency of G4 divided by middle C is 1.498.) Clearly, none of these ratios are perfect, but the third of the notes is slightly more off than the fifth with a percentage difference of 0.79% vs. 0.133%.
And so, to correct this (barely perceptible) problem, we try to play the third note slightly lower than normal.