I recently learned that the wavelength of a note is half that of the same note one octave lower. Do the wavelengths of the notes in a chord have some sort of similarities? Is there another reason that the notes sound good together?
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Music notes on the 12 step scale (A to G plus accidentals) are all mathematical ratios to each other. But because it’s 12 steps with comparatively equal ratios, it’s not a simple rational ratio. The ratio between two notes a half step apart is 1:2^(1/12), or about 1:1.059. Replace the 1 in 1/12 with the amount of steps apart the notes are to get any specific note’s ratio; 3/12 (1:1.189) for a minor third, 4/12 (1:1.260) for a major third, 7/12 (1:1.498) for a fifth, etc. This turns into 1:2^(12/12) for an entire octave, which simplifies to 1:2 as you know.
These are the precise increments for technical definitions, but there are alternate tunings that take into account individual preferences and ears that might round it to more convenient numbers (such as 1:1.5 for the fifth). There’s also multiple scale definitions that use a different amount of notes that might not be equidistant (such as a major scale only has 7 notes).
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