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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Yes. Harmony are integer ratios of base frequencies, like 2/1 for an octave, 3/2 for a quint, 4/3 for a quart, etc.
Those ratios sound pleasant to the ear, for possibly three reasons:
(1) the acoustic waveform remains strictly periodic only for integer ratios of frequencies.
(2) naturally created sounds are rich in overtones, the ratios between overtones are integer ratios.
(3) base frequencies in integer ratios have their corresponding overtones aligned to each other.
These are all mathematical features. Why they sound appealing is not explained by math, but it’s likely that our ear evolved around naturally produced sounds which do exhibit those overtone features. Chords are then artificial sounds that emphasize and exaggerate those overtone features.
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