Is there a ‘mathematical’ reason that the music notes in chords ‘work’ together?

Question

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?

In: 21

Answer 1

> Do the wavelengths of the notes in a chord have some sort of similarities?

Yes, they do! Well… kinda.

A major triad includes a root, major 3rd, and a perfect 5th. A “pure” perfect fifth is exactly 3:2 the wavelength of the root note, and “pure” major 3rd is exactly 5:4 the wavelength of the root note.

These relationships aren’t just theoretical – the root note itself, like any other notes, includes more than just the one fundamental frequency. The note called “A” at 440Hz vibrates not just at 440Hz but also 2x that (880Hz) and 3x and 4x etc, in theory into infinity but they get progressively less loud as you go higher and you can only hear the first 8 or so depending on the instrument.

The other non-fundamental frequencies are called “harmonics” or “overtones” and the relationship between them called the “harmonic” or “overtone” series.

But, we don’t actually use pure intervals. Since the 1600s or something like that western music has been using “equal temperment” instead. This is what allows an instrument to play in any of the 12 keys. Prior to equal temperment there were different harpsichords for different keys and/or more complicated instruments with separate sharps and flats, because the Eb in the key of C minor was actually a different note than the D# in the key of E major, for example.

> Is there another reason that the notes sound good together?

Yes, there are many other reasons, and important amongst them is cultural conditioning. If this is the first you are hearing about equal temperment, and that the intervals you have heard you entire life are not “pure” – did they sound bad to you the whole time? Did you ever get the sense that the notes were out of tune?

The overwhelming response from honest people to those questions is “no.” To those of us who grew up listening to music from the classical era to today, nothing ever sounded out of tune. It’s only to people who grew up hearing the other tuning systems, to which our music would sound out of tune.

So, there must be more to it than the ‘mathematical’ reasons, or you might even say that these ‘mathematical’ reasons don’t matter at all to why they sound ‘good’ together, but there certainly are mathematical ways to describe why the notes of the chord are the way they are.