That is true when the chords are sung or played in an untempered scaled. That is, when the intervals between notes in the chord have simple ratios to each other. For example, in the untempered scale the notes of a major third have a ratio of 5/4 (386 cents difference) and the two notes share the most harmonic frequencies. In the tempered scale a major third has a ratio of 1.26 (400 cents difference). Instead of getting a lot of harmonics, you get beats due to slight dissonances. That is the secret of barbershop harmonies. Instruments that are tuned to a tempered scale can achieve the same effect if the musicians bend their pitches to a tempered scale on sustained chords. See HERE

Cheers, S. in Seattle