Final answer:
The relationship between two tones an octave apart is that the higher tone's frequency is double that of the lower tone. This ratio of 2:1 creates the same note at different pitches. The unique sound of instruments like trumpets and clarinets is due to their distinct overtone series.
Step-by-step explanation:
The relationship between two tones where one is an octave above the other is that they have a frequency ratio of 2:1. This means that the higher tone's frequency is exactly double that of the lower tone. In musical terms, the two tones are perceived as being the same note, but at different pitches. The concept of overtones comes into play when discussing the timbre of an instrument. For instance, when middle C is played on different instruments like a trumpet or a clarinet, each instrument produces a unique sound quality based on its overtone series. While the fundamental frequency remains the same, the overtones vary in number and intensity, which contributes to the distinct sound of each instrument. A tube open at both ends will have both even and odd multiples of the fundamental frequency as overtones, while a tube closed at one end will only have odd multiples.