There is some relationship when you consider the nature of frequency response. The higher the frequencies in the overtone series, the brighter the tone. Frequencies are a form of measurable pitch.
Overtones are measurable as well. But as Dr. Carl Seashore pointed out, tonal memory is different. It's the ability to recall melodic phrases and may have something to do with the ability to memorize chords. Tone quality relies on pitch discrimination to some degree.

Frank Hamilton